International macroe.. - Free

To Shirley, Laurie, and Leslii

viCONTENTS5 International Real Business Cycles 1375.1 Calibrating the One-Sector Growth Model ........1385.2 Calibrating a Two-Country Model .............1496 Foreign Exchange Market Efficiency 1616.1 Deviations From UIP ....................1626.2 Rational Risk Premia ....................1726.3 Testing Euler Equations ..................1776.4 Apparent Violations of Rationality ............1836.5 The ‘Peso Problem’ .....................1866.6 Noise-Traders ........................1937 The Real Exchange Rate 2077.1 Some Preliminary Issues ..................2087.2 Deviations from the Law-Of-One Price ..........2097.3 Long-Run Determinants of the Real Exchange Rate ...2137.4 Long-Run Analyses of Real Exchange Rates .......2178 The Mundell-Fleming Model 2298.1 A Static Mundell-Fleming Model .............2298.2 Dornbusch’s Dynamic Mundell—Fleming Model ......2378.3 A Stochastic Mundell—Fleming Model ...........2418.4 VAR analysis of Mundell—Fleming .............2499 The New International Macroeconomics 2639.1 The Redux Model ......................2649.2 Pricing to Market ......................28610 Target-Zone Models 30710.1 Fundamentals of Stochastic Calculus ...........30810.2 The Continuous—Time Monetary Model ..........31010.3 InÞnitesimal Marginal Intervention ............31310.4 Discrete Intervention ....................31910.5 Eventual Collapse ......................32010.6 Imperfect Target-Zone Credibility .............322

CONTENTSvii11 Balance of Payments Crises 32711.1 A First-Generation Model .................32811.2 A Second Generation Model ................335

Chapter 1Some InstitutionalBackgroundThis chapter covers some institutional background and develops somebasic relations that we rely on in international macroeconomics andÞnance. First, you will get a basic description some widely held internationalÞnancial instruments and the markets in which they trade.This discussion allows us to quickly derive the fundamental parity relationsimplied by the absence of riskless arbitrage proÞts that relate assetprices in international Þnancial markets. These parity conditions areemployed regularly in international macroeconomic theory and serveas jumping off points for more in-depth analyses of asset pricing in theinternational environment. Second, you’ll get a brief overview of thenational income accounts and their relation to the balance of payments.This discussion identiÞes some of the macroeconomic data that we wanttheory to explain and that are employed in empirical work. Third, youwill see a discussion of the central bank’s balance sheet—an understandingof which is necessary to appreciate the role of international (foreignexchange) reserves in the central bank’s foreign exchange market interventionand the impact of intervention on the domestic money supply.1

2 CHAPTER 1. SOME INSTITUTIONAL BACKGROUND1.1 International Financial MarketsWe begin with a description of some basic international Þnancial instrumentsand the markets in which they trade. As a point of reference,we view the US as the home country.Foreign ExchangeForeign exchange is traded over the counter through a spatially decentralizeddealer network. Foreign currencies are mainly bought andsold by dealers housed in large money center banks located around theworld. Dealers hold foreign exchange inventories and aim to earn tradingproÞts by buying low and selling high. The foreign exchange marketis highly liquid and trading volume is quite large. The Federal ReserveBank of New York [51] estimates during April 1998, daily volume of foreignexchange transactions involving the US dollar and executed withinin the U.S was 405 billion dollars. Assuming a 260 business day calendar,this implies an annual volume of 105.3 trillion dollars. The totalvolume of foreign exchange trading is much larger than this Þgure becauseforeign exchange is also traded outside the US—in London, Tokyo,and Singapore, for example. Since 1998 US GDP was approximately 9trillion dollars and the US is approximately 1/7 of the world economy,the volume of foreign exchange trading evidently exceeds, by a greatamount, the quantity necessary to conduct international trade.During most of the post WWII period, trading of convertible currenciestook place with respect to the US dollar. This meant thatconverting yen to deutschemarks required two trades: Þrst from yen todollars then from dollars to deutschemarks. The dollar is said to be thevehicle currency for international transactions. In recent years crosscurrencytrading, that allows yen and deutschemarks to be exchangeddirectly, has become increasingly common.The foreign currency price of a US dollar is the exchange rate quotedin European terms. The US dollar price of one unit of the foreigncurrencyistheexchangerateisquotedinAmerican terms. InAmericanterms, an increase in the exchange rate means the dollar currency hasdepreciated in value relative to the foreign currency. In this book, wewill always refer to the exchange rate in American terms.

1.1. INTERNATIONAL FINANCIAL MARKETS 3The equilibrium condition in cross-rate markets is given by the absenceof unexploited triangular arbitrage proÞts. To illustrate, assumethat there are no transactions costs and consider 3 currencies–the dollar,the euro, and the pound. Let S 1 be the dollar price of the pound, S 2be the dollar price of the euro, and S3 x be the euro price of the pound.Thecross-ratemarketisinequilibriumiftheexchangeratequotationsobeyS 1 = S3 x S 2. (1.1)The opportunity to earn riskless arbitrage proÞts are available if (1.1)is violated. For example, suppose that you get price quotations of S 1 =1.60 dollars per pound, S 2 =1.10 dollars per euro, and S3 x =1.55eurosper pound. An arbitrage strategy is to put up 1.60 dollars to buyone pound, sell that pound for 1.55 euros and then sell the euros for1.1 dollars each. You begin with 1.6 dollars and end up with 1.705dollars, which is quite a deal. But when you take money out of theforeign exchange market it comes at the expense of someone else. Veryshort-lived violations of the triangular arbitrage condition (1.1) mayoccasionally occur during episodes of high market volatility, but we donot think that foreign exchange dealers will allow this to happen on aregular basis.Transaction TypesForeign exchange transactions are divided into three categories. TheÞrst are spot transactions for immediate (actually in two working days)delivery. Spot exchange rates are the prices at which foreign currenciestrade in this spot market.Second, swap transactions are agreements in which a currency sold(bought) today is to be repurchased (sold) at a future date. The priceof both the current and future transaction is set today. For example,you might agree to buy 1 million euros at 0.98 million dollars today andsell the 1 million euros back in six months time for 0.95 million dollars.The swap rate is the difference between the repurchase (resale) priceand the original sale (purchase) price. The swap rate and the spot ratetogether implicitly determine the forward exchange rate.The third category of foreign exchange transactions are outrightforward transactions. These are current agreements on the price, quan-

4 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDtity, and maturity or future delivery date for a foreign currency. Theagreed upon price is the forward exchange rate. Standard maturitiesfor forward contracts are 1 and 2 weeks, 1,3,6, and 12 months. We saythat the forward foreign currency trades at a premium when the forwardrate exceeds the spot rate in American terms. Conversely if thespot rate is exceeds the forward rate, we say that the forward foreigncurrency trades at discount.Spot transactions form the majority of foreign exchange tradingand most of that is interdealer trading. About one—third of the volumeof foreign exchange trading are swap transactions. Outright forwardtransactions account for a relatively small portion of total volume.Forward and swap transactions are arranged on an informal basis bymoney center banks for their corporate and institutional customers.Short-Term DebtA Eurocurrency is a foreign currency denominated deposit at a banklocated outside the country where the currency is used as legal tender.Such an institution is called an offshore bank. Although they are calledEurocurrencies, the deposit does not have to be in Europe. A US dollardeposit at a London bank is a Eurodollar deposit and a yen depositat a San Francisco bank is a Euro-yen deposit. Most Eurocurrencydeposits are Þxed-interest time-deposits with maturities that matchthose available for forward foreign exchange contracts. A small part ofthe Eurocurrency market is comprised of certiÞcates of deposit, ßoatingrate notes, and call money.London Interbank Offer Rate (LIBOR) is the rate at which banks arewilling to lend to the most creditworthy banks participating in theLondon Interbank market. Loans to less creditworthy banks and/orcompanies outside the London Interbank market are often quoted as apremium to LIBOR.Covered Interest ParitySpot, forward, and Eurocurrency rates are mutually dependent throughthe covered interest parity condition. Let i t be the date t interest rate

1.1. INTERNATIONAL FINANCIAL MARKETS 5on a 1-period Eurodollar deposit, i ∗ t betheinterestrateonanEuroeurodeposit rate at the same bank, S t , the spot exchange rate (dollars pereuro), and F t , the 1-period forward exchange rate. Because both Eurodollarand Euroeuro deposits are issued by the same bank, the twodeposits have identical default and political risk. They differ only by thecurrency of their denomination. 1 Covered interest parity is the conditionthat the nominally risk-free dollar return from the Eurodollar andthe Euroeuro deposits are equal. That is1+i t =(1+i ∗ t )F t. (1.2)S tWhen (1.2) is violated a riskless arbitrage proÞt opportunity is availableand the market is not in equilibrium. For example, suppose there areno transactions costs, and you get the following 12-month eurocurrency,forward exchange rate and spot exchange rate quotationsi t =0.0678, i ∗ t =0.0422, F t =0.9961, S t =1.0200.You can easily verify that these quotes do not satisfy (1.2). Thesequotes allow you to borrow 0.9804 euros today, convert them to 1/S t =1 dollar, invest in the eurodollar deposit with future payoff 1.0678 butyou will need only (1 + i ∗ t )F t /S t =1.0178 dollars to repay the euroloan. Note that this arbitrage is a zero-net investment strategy since itis Þnanced with borrowed funds. Arbitrage proÞts that arise from suchquotations come at the expense of other agents dealing in the internationalÞnancial markets, such as the bank that quotes the rates. Sincebanks typically don’t like losing money, swap or forward rates quoted bybank traders are routinely set according to quoted eurocurrency ratesand (1.2).Using the logarithmic approximation, (1.2) can be expressed aswhere f t ≡ ln(F t ), and s t ≡ ln(S t ).i t ' i ∗ t + f t − s t (1.3)1 Political risk refers to the possibility that a government may impose restrictionsthat make it difficult for foreign investors to repatriate their investments. Coveredinterest arbitrage will not in general hold for other interest rates such as T-bills orcommercial bank prime lending rates.

6 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDTesting Covered Interest ParityCovered interest parity won’t hold for assets that differ greatly in termsof default or political risk. If you look at prices for spot and forwardforeign exchange and interest rates on assets that differ mainly in currencydenomination, the question of whether covered interest parityholds depends on whether there there exist unexploited arbitrage proÞtopportunities after taking into account the relevant transactions costs,how large are the proÞts, and the length of the window during whichthe proÞts are available.Foreign exchange dealers and bond dealers quote two prices. Thelow price is called the bid. If you want to sell an asset, you get thebid (low) price. The high price is called the ask or offer price. If youwant to buy the asset from the dealer, you pay the ask (high) price. Inaddition, there will be a brokerage fee associated with the transaction.Frenkel and Levich [63] applied the neutral-band analysis to testcovered interest parity. The idea is that transactions costs create aneutral band within which prices of spot and forward foreign exchangeand interest rates on domestic and foreign currency denominated assetscan ßuctuate where there are no proÞt opportunities. The question ishow often are there observations that lie outside the bands.Let the (proportional) transaction cost incurred from buying or sellinga dollar debt instrument be τ, the transaction cost from buying orselling a foreign currency debt instrument be τ ∗ , the transaction costfrom buying or selling foreign exchange in the spot market be τ s andthe transaction cost from buying or selling foreign exchange in the forwardmarket be τ f . A round-trip arbitrage conceptually involves fourseparate transactions. A strategy that shorts the dollar requires you toÞrst sell a dollar-denominated asset (borrow a dollar at the gross rate1+i).Afterpayingthetransactioncostyournetis1− τ dollars. Youthen sell the dollars at 1/S which nets (1 − τ)(1 − τ s ) foreign currencyunits. You invest the foreign money at the gross rate 1 + i ∗ , incurringa transaction cost of τ ∗ . Finally you cover the proceeds at the forwardrate F , where you incur another cost of τ f .Let¯C ≡ (1 − τ)(1 − τ s )(1 − τ ∗ )(1 − τ f ),and f p ≡ (F − S)/S. The net dollar proceeds after paying the transac-

1.1. INTERNATIONAL FINANCIAL MARKETS 7tions costs are ¯C(1 + i ∗ )(F/S). The arbitrage is unproÞtable if¯C(1 + i ∗ )(F/S) ≤ (1 + i), or equivalently iff p ≤ ¯f p ≡ (1 + i) − ¯C(1 + i ∗ ). (1.4)¯C(1 + i ∗ )By the analogous argument, it follows that an arbitrage that is long inthe dollar remains unproÞtable iff p ≥ f p≡ ¯C(1 + i) − (1 + i ∗ ). (1.5)(1 + i ∗ )[f p, ¯f p ]deÞne a neutral band of activity within which f p can ßuctuatebut still present no proÞtable covered interest arbitrage opportunities.The neutral-band analysis proceeds by estimating the transactions costs¯C. These are then used to compute the bands [f p, ¯f p ] at various pointsin time. Once the bands have been computed, an examination of theproportion of actual f p that lie within the bands can be conducted.Frenkel and Levich estimate τ s and τ f to be the upper 95 percentileof the absolute deviation from spot and 90-day forward triangular arbitrage.τ is set to 1.25 times the ask-bid spread on 90-day treasurybills and they set τ ∗ = τ. They examine covered interest parity for thedollar, Canadian dollar, pound, and the deutschemark. The sampleis broken into three periods. The Þrst period is the tranquil peg precedingBritish devaluation from January 1962—November 1967. Theirestimates of τ s range from 0.051% to 0.058%, and their estimates of τ frange from 0.068% to 0.076%. For securities, they estimate τ = τ ∗ tobe approximately 0.019%. The total cost of transactions fall in a rangefrom 0.145% to 0.15%. Approximately 87% of the f p observations liewithin the neutral band.Thesecondperiodistheturbulent peg from January 1968 to December1969, during which their estimate of ¯C rises to approximately0.24%. Now, violations of covered interest parity are more pervasivewith the proportion of f p that lie within the neutral band ranging from0.33 to 0.67.The third period considered is the managed ßoat from July 1973 toMay 1975. Their estimates for ¯C rises to about 1%, and the proportion

8 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDof f p within the neutral band also rises back to about 0.90. The conclusionis that covered interest parity holds during the managed ßoat andthe tranquil peg but there is something anomalous about the turbulentpeg period. 2Taylor [130] examines data recorded by dealers at the Bank of England,and calculates the proÞt from covered interest arbitrage betweendollar and pound assets predicted by quoted bid and ask prices thatwould be available to an individual. Let an “a” subscript denote anask price (or ask yield), and a “b” subscript denote the bid price. Ifyou buy pounds, you get the ask price S a . Buying pounds is the sameas selling dollars so from the latter perspective, you can sell the dollarsatthebidprice1/S a . Accordingly, we adopt the following notation.S a : Spot pound ask price. F a : Forward pound ask price.1/S a : Spot dollar bid price. 1/F a : Forward dollar bid price.S b : Spot pound bid price. F b : Forward pound bid price.1/S b : Spot dollar ask price. 1/F b : Forward dollar ask price.i a : Eurodollar ask interest rate. i ∗ a : Euro-pound ask interest rate.i b : Eurodollar bid interest rate. i ∗ b : Euro-pound bid interest rate.Itwillbethecasethati a >i b , i ∗ a >i ∗ b, S a >S b ,andF a >F b . Anarbitrage that shorts the dollar begins by borrowing a dollar at thegross rate 1 + i a , selling the dollar for 1/S a pounds which are investedat the gross rate 1 + i ∗ b and covered forward at the price F b . The perdollar proÞt is(1 + i ∗ b) F b− (1 + i a ).S aUsing the analogous reasoning, it follows that the per pound proÞt thatshorts the pound is(1 + i b ) S b− (1 + i ∗ aF ).aTaylor Þnds virtually no evidence of unexploited covered interest arbitrageproÞts during normal or calm market conditions but he is ableto identify some periods of high market volatility when economicallysigniÞcant violations may have occurred. The Þrst of these is the 19672 Possibly, the period is characterized by a ‘peso problem,’ which is covered inchapter 6.

1.1. INTERNATIONAL FINANCIAL MARKETS 9British devaluation. Looking at an eleven-day window spanning theevent an arbitrage that shorted 1 million pounds at a 1-month maturitycould potentially have earned a 4521-pound proÞt on WednesdayNovember 24 at 7:30 a.m. but by 4:30 p.m. Thursday November 24, theproÞt opportunity had vanished. A second event that he looks at is the1987 UK general election. Examining a window that spans from June1 to June 19, proÞt opportunities were generally unavailable. Amongthe few opportunities to emerge was a quote at 7:30 a.m. WednesdayJune 17 where a 1 million pound short position predicted 712 poundsof proÞt at a 1 month maturity. But by noon of the same day, thepredicted proÞt fell to 133 pounds and by 4:00 p.m. the opportunitieshad vanished.To summarize, the empirical evidence suggests that covered interestparity works pretty well. Occasional violations occur after accountingfor transactions costs but they are short-lived and present themselvesonly during rare periods of high market volatility.Uncovered Interest ParityLet E t (X t+1 )=E(X t+1 |I t ) denote the mathematical expectation of therandom variable X t+1 conditioned on the date-t publicly available informationset I t . If foreign exchange participants are risk neutral, theycare only about the mean value of asset returns and do not care at allabout the variance of returns. Risk-neutral individuals are also willingto take unboundedly large positions on bets that have a positiveexpected value. Since F t − S t+1 is the proÞt from taking a position inforward foreign exchange, under risk-neutrality expected forward speculationproÞts are driven to zero and the forward exchange rate must,in equilibrium, be market participant’s expected future spot exchangerateF t =E t (S t+1 ). (1.6)Substituting (1.6) into (1.2) gives the uncovered interest parity condition1+i t =(1+i ∗ t )E t[S t+1 ]. (1.7)S tIf (1.7) is violated, a zero-net investment strategy of borrowing in onecurrency and simultaneously lending uncovered in the other currency

10 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDhas a positive payoff in expectation. We use the uncovered interestparity condition as a Þrst-approximation to characterize internationalasset market equilibrium, especially in conjunction with the monetarymodel (chapters 3, 10, and 11). However, as you will see in chapter 6,violations of uncovered interest parity are common and they present animportant empirical puzzle for international economists.(2)⇒Risk Premia. What reason can be given if uncovered interest paritydoes not hold? One possible explanation is that market participantsare risk averse and require compensation to bear the currency risk involvedin an uncovered foreign currency investment. To see the relationbetween risk aversion and uncovered interest parity, consider the followingtwo-period partial equilibrium portfolio problem. Agents takeinterest rate and exchange rate dynamics as given and can invest a fractionα of their current wealth W t in a nominally safe domestic bondwith next period payoff (1+i t )αW t . The remaining 1−α of wealth canbe invested uncovered in the foreign bond with future home-currencypayoff (1 + i ∗ t ) S t+1S t(1 − α)W t . We assume that covered interest parityis holds so that a covered investment in the foreign bond is equivalentto the investment in the domestic bond. Next period nominal wealthis the payoff from the bond portfolioW t+1 =·α(1 + i t )+(1− α)(1 + i ∗ t ) S t+1S t¸W t . (1.8)Domestic market participants have constant absolute risk aversion utilitydeÞned over wealth, U(W )=−e −γW where γ ≥ 0isthecoefficientof absolute risk aversion. The domestic agent’s problem is to choosethe investment share α to maximize expected utilityE t [U(W t+1 )] = −E t³e−γW t+1´. (1.9)Notice that the right side of (1.9) is the moment generating function ofnext period wealth. 33 The moment generating function for the normally distributed random variableX ∼ N(µ, σ 2 )isψ X (z) =E ¡ e zX¢ ¢ ¡µz+= eσ2 z 22. Substituting W for X, −γ for z,E t W t+1 for µ, andVar(W t+1 )forσ 2 and taking logs results in (1.12).

1.1. INTERNATIONAL FINANCIAL MARKETS 11If people believe that W t+1 is normally distributed conditional oncurrently available information, with conditional mean and conditionalvarianceE t W t+1 =·α(1 + i t )+(1− α)(1 + i ∗ t ) E tS t+1S t¸W t , (1.10)Var t (W t+1 )= (1 − α)2 (1 + i ∗ t )2 Var t (S t+1 )Wt2St2 . (1.11)It follows that maximizing (1.9) is equivalent to maximizing the simplerexpressionE t W t+1 − γ 2 Var(W t+1). (1.12)We say that traders are mean-variance optimizers. These individualslike high mean values of wealth, and dislike variance in wealth.Differentiating (1.12) with respect to α and re-arranging the Þrstorderconditions for optimality yields(1 + i t ) − (1 + i ∗ t ) E t[S t+1 ]= −γW t(1 − α)(1 + i ∗ t ) 2 Var t (S t+1 ), (1.13)S twhich implicitly determines the optimal investment share α. Even ifthere is an expected uncovered proÞt available, risk aversion limits thesize of the position that investors will take. If all market participantsare risk neutral, then γ = 0 and it follows that uncovered interest paritywill hold. If γ > 0, violations of uncovered interest parity can occur andthe forward rate becomes a biased predictor of the future spot rate, thereason being that individuals need to be paid a premium to bear foreigncurrency risk. Uncovered interest parity will hold if α = 1, regardlessof whether γ > 0. However, the determination of α requires us to bespeciÞc aboutthedynamicsthatgovernS t and that is information thatwe have not speciÞedhere. Thepointthatwewanttomakehereisthat the forward foreign exchange market can be in equilibrium andthere are no unexploited risk-adjusted arbitrage proÞts even thoughthe forward exchange rate is a biased predictor of the future spot rate.We will study deviations from uncovered interest parity in more detailin chapter 6.S 2 t

12 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDFutures ContractsParticipation in the forward foreign exchange market is largely limitedto institutions and large corporate customers owing to the size of thecontracts involved. The futures market is available to individuals andis a close substitute to the forward market. The futures market isan institutionalized form of forward contracting. Four main featuresdistinguish futures contracts from forward contracts.First, foreign exchange futures contracts are traded on organizedexchanges. In the US, futures contracts are traded on the InternationalMoney Market (IMM) at the Chicago Mercantile Exchange. In Britain,futures are traded at the London International Financial Futures Exchange(LIFFE). Some of the currencies traded are, the Australian dollar,Brazilian real, Canadian dollar, euro, Mexican peso, New Zealanddollar, pound, South African rand, Swiss franc, Russian ruble and theyen.Second, contracts mature at standardized dates throughout theyear. The maturity date is called the last trading day. Delivery occurson the third Wednesday of March, June, Sept, and December,provided that it is a business day. Otherwise delivery takes place onthe next business day. The last trading day is 2 business days priorto the delivery date. Contracts are written for Þxed face values. Forexample, for the face value of an euro contract is 125,000 euros.Third, the exchange serves to match buyers to sellers and maintainsazeronetposition. 4 Settlement between sellers (who take short positions)and buyers (who take long positions) takes place daily. Youpurchase a futures contract by putting up an initial margin with yourbroker. If your contract decreases in value, the loss is debited from yourmargin account. This debit is then used to credit the account of theindividual who sold you the futures contract. If your contract increasesin value, the increment is credited to your margin account. This settlementtakes place at the end of each trading day and is called “markingto market.” Economically, the main difference between futures andforward contracts is the interest opportunity cost associated with the4 If you need foreign exchange before the maturity date, you are said to haveshort exposure in foreign exchange which can be hedged by taking a long positionin the futures market.

1.1. INTERNATIONAL FINANCIAL MARKETS 13funds in the margin account. In the US, some part of the initial margincan be put up in the form of Treasury bills, which mitigates the loss ofinterest income.Fourth, the futures exchange operates a clearinghouse whose functionis to guarantee marking to market and delivery of the currenciesupon maturity. Technically, the clearing house takes the other side ofany transaction so your legal obligations are to the exchange. But asmentioned above, the clearinghouse maintains a zero net position.Most futures contracts are reversed prior to maturity and are notheld to the last trading day. In these situations, futures contracts aresimply bets between two parties regarding the direction of future exchangerate movements. If you are long a foreign currency futurescontract and I am short, you are betting that the price of the foreigncurrency will rise while I expect the price to decline. Bets in the futuresmarket are a zero sum game because your winnings are my losses.How a Futures Contract WorksFor a futures contract with k days to maturity, denote the date T − kfutures price by F T −k , and the face value of the contract by V T . Thecontract value at T − k is F T −k V T .Table 1.1 displays the closing spot rate and the price of an actual12,500,000 yen contract that matured in June 1999 (multiplied by 100)and the evolution of the margin account. When the futures price increases,the long position gains value as reßectedbyanincrementinthe margin account. This increment comes at the expense of the shortposition.Suppose you buy the yen futures contract on June 16, 1998 at0.007346 dollars per yen. Initial margin is 2,835 dollars and the spotexchange rate is 0.006942 dollars per yen. The contract value is 91,825dollars. If you held the contract to maturity, you would take deliveryof the 12,500,000 yen on 6/23/99 at a unit price of 0.007346 dollars.Suppose that you actually want the yen on December 17, 1998. Youclose out your futures contract and buy the yen in the spot market.The appreciation of the yen means that buying 12,500,000 yen costs20675 dollars more on 12/17/98 than it did on 6/16/98, but most ofthehighercostisoffset by the gain of 21197.5-2835=18,362.5 dollars

14 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDTable 1.1: Yen futures for June 1999 deliveryLong yen positionDate F T −k S T −k ∆F T −k ∆(F T −k V T ) Margin φ T −k6/16/98 0.7346 0.6942 0.0000 0.0 2835.0 1.05816/17/98 0.772 0.7263 0.0374 4675.0 7510.0 1.06287/17/98 0.7507 0.7163 -0.0213 -2662.5 4847.5 1.04798/17/98 0.7147 0.6859 -0.0360 -4500.0 347.5 1.04189/17/98 0.7860 0.7582 0.0713 8912.5 9260.0 1.036510/16/98 0.8948 0.8661 0.1088 13600.0 22860.0 1.033011/17/98 0.8498 0.8244 -0.0450 -5625.0 17235.0 1.030812/17/98 0.8815 0.8596 0.0317 3962.5 21197.5 1.025401/19/99 0.8976 0.8790 0.0161 2012.5 23210.0 1.021102/17/99 0.8524 0.8401 -0.0452 -5650.0 17560.0 1.014603/17/99 0.8575 0.8463 0.0051 637.5 18197.5 1.0131on the futures contract.The hedge comes about because there is a covered interest paritylikerelation that links the futures price to the spot exchange rate witheurocurrency rates as a reference point. Let i T −k be the Eurodollar rateat T − k which matures at T , i ∗ T −k be the analogous one-year Euroeurorate, assume a 360 day year, and letφ T −k = 1+ ki T −k360,1+ ki∗ T −k360be the ratio of the domestic to foreign gross returns on an eurocurrencydeposit that matures in k days. The parity relation for futures pricesisF T −k = φ T −k S T −k . (1.14)Here, the futures price varies in proportion to the spot price with φ T −kbeing the factor of proportionality. As contract approaches last tradingday, k → 0. It follows that φ T −k → 1, and F T = S T .Thismeansthatyou can obtain the foreign exchange in two equivalent ways. You canbuy a futures contract on the last trading day and take delivery, or you

1.2. NATIONAL ACCOUNTING RELATIONS 15can buy the foreign currency in the interbank market because arbitragewill equate the two prices near the maturity date.(1.14) also tells you the extent to which the futures contract hedgesrisk. If you have long exposure, an increase in S T −k (a weakening of thehome currency) makes you worse off while an increase in the futuresprice makes you better off. The futures contract provides a perfecthedge if changes in F T −k exactly offset changes in S T −k but this onlyhappens if φ T −k = 1. To obtain a perfect hedge when φ T −k 6=1,youneed to take out a contract of size 1/φ and because φ changes overtime, the hedge will need to be rebalanced periodically.1.2 National Accounting RelationsThis section gives an overview of the National Income Accounts andtheir relation to the Balance of Payments. These accounts form some ofthe international time—series that we want our theories to explain. TheNational Income Accounts are a record of expenditures and receiptsat various phases in the circular ßow of income, while the Balance ofPayments is a record of the economic transactions between domesticresidents and residents in the rest of the world.National Income AccountingIn real (constant dollar) terms, we will use the following notation.Y Gross domestic product,Q National income,C Consumption,I Investment,G Government Þnal goods purchases,A aggregate expenditures (absorption), A = C + I + G,IM Imports,EX Exports,R Net foreign income receipts,T Tax revenues,

16 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDS Private saving,NFA Net foreign asset holdings.Closed economy national income accounting. We’ll begin with a quickreview of the national income accounts for a closed economy. Abstractingfrom capital depreciation, which is that part of total Þnal goodsoutput devoted to replacing worn out capital stock. The value of outputis gross domestic product Y . When the goods and services aresold the sales become income Q. If we ignore capital depreciation, thenGDP is equal to national incomeY = Q. (1.15)In the closed economy, there are only three classes of agents–households,businesses, and the government. Aggregate expenditures on goods andservices is the sum of the component spending by these agentsA = C + I + G. (1.16)The nation’s output Y has to be purchased by someone A. If thereis any excess supply, Þrms are assumed to buy the extra output inthe form of inventory accumulation. We therefore have the accountingidentityY = A = Q. (1.17)The Open Economy. To handle an economy that engages in foreigntrade, we must account for net factor receipts from abroad R, whichincludes items such as fees and royalties from direct investment, dividendsand interest from portfolio investment, and income for laborservices provided abroad by domestic residents. In the open economynational income is called gross national product (GNP) Q =GNP.This is income paid to factors of production owned by domestic residentsregardless of where the factors are employed. GNP can differ fromGDP since some of this income may be earned from abroad. GDP canbe sold either to domestic agents (A − IM) or to the foreign sector

1.2. NATIONAL ACCOUNTING RELATIONS 17EX. This can be stated equivalently as the sum of domestic aggregateexpenditures or absorption and net exportsY = A +(EX − IM). (1.18)National income (GNP) is the sum of gross domestic product and netfactor receipts from abroadSubstituting (1.18) into (1.19) yieldsQ = Y + R. (1.19)Q = A + (EX− IM) + R| {z }Current Account(1.20)A country uses the excess of national income over absorption to Þnancean accumulation of claims against the rest of the world. This is nationalsaving and called the balance on current account. A country with acurrent account surplus is accumulating claims on the rest of the world.Thus rearranging (1.20) giveswhich we summarize byQ − A = ∆(NFA)= (EX − IM)+R= Q − (C + I + G)= [(Q − T ) − C] − I +(T − G)= (S − I)+(T − G),∆(NFA) = EX − IM + R =[S − I]+[T − G] =Q − A. (1.21)The change in the country’s net foreign asset position ∆NFA in (1.21)is the nation’s accumulation of claims against the foreign sector andincludes official (central bank) as well as private capital transactions.The distinction between private and official changes in net foreign assetsis developed further below.Although (1.21) is an accounting identity and not atheory,itcanbe used for ‘back of the envelope’ analyses of current account problems.For example, if the home country experiences a current account

18 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDsurplus (EX − IM + R>0) and the government’s budget is in balance(T = G), you see from (1.21) that the current account surplusarises because there are insufficient investment opportunities at home.To satisfy domestic resident’s desired saving, they accumulate foreignassets so that ∆NFA > 0. If the inequality is reversed, domestic savingswould seem to be insufficient to Þnance the desired amount ofdomestic investment. 5 On the other hand, the current account mightalso depend on net government saving. If net private saving is in balance(S = I), then the current account imbalance is determined bythe imbalance in the government’s budget. Some people believed thatUS current account deÞcits of the 1980s were the result of governmentbudget deÞcits.Because current account imbalances reßect a nation’s saving decision,the current account is largely a macroeconomic phenomenon aswell as an intertemporal problem. The current account will dependon ßuctuations in relative prices of goods such as the real exchangerate or the terms of trade, only to the extent that these prices affectintertemporal saving decisions.The Balance of PaymentsThe balance of payments is a summary record of the transactions betweenthe residents of a country with the rest of the world. Theseinclude the exchange of goods and services, capital, unilateral transferpayments, official (central bank) and private transactions. A credittransaction arises whenever payment is received from abroad. Creditscontribute toward a surplus or improvement of the balance of payments.Examples of credit transactions include the export of goods, Þnancialassets, and foreign direct investment in the home country. The lattertwo examples are sometimes referred to as inßows of capital. Creditsare also generated by income received for factor services renderedabroad, such as interest on foreign bonds, dividends on foreign equities,and receipts for US labor services rendered to foreigners, receipts of foreignaid, and cash remittances from abroad are credit transactions in5 This was a popular argument used to explain Japan’s current account surpluseswith the US

1.2. NATIONAL ACCOUNTING RELATIONS 19the balance of payments. Debit transactions arise whenever payment ismade to agents that reside abroad. Debits contribute toward a deÞcitor worsening of the balance of payments. 6SubaccountsThe precise format of balance of payments subaccount reporting differsacross countries. For the US, the main subaccounts of the balanceof payments that you need to know are the current account, whichrecords transactions involving goods, services, and unilateral transfers,the capital account, which records transactions involving real or Þnancialassets, and the official settlements balance, which records foreignexchange transactions undertaken by the central bank.Credit transactions generate a supply of foreign currency and alsoa demand for US dollars because US residents involved in credit transactionsrequire foreign currency payments to be converted into dollars.Similarly, debit transactions create a demand for foreign exchange anda supply of dollars. As a result, the combined deÞcits on the currentaccount and the capital account can be thought of as the excess demandfor foreign exchange by the private (non central bank) sector.This combined current and capital account balance is commonly calledthe balance of payments.Under a system of pure ßoating exchange rates, the exchange rateis determined by equilibrium in the foreign exchange market. Excessdemand for foreign exchange in this case is necessarily zero. It followsthat it is not possible for a country to have a balance of payments problemunder a regime of pure ßoating exchange rates because the balanceof payments is always zero and the current account deÞcit always isequal to the capital account surplus.When central banks intervene in the foreign exchange market eitherby buying or selling foreign currency, their actions, which are designedto prevent exchange rate adjustment, allow the balance of payments tobe non zero. To prevent a depreciation of the home currency, a privatelydetermined excess demand for foreign exchange can be satisÞedby sales of the central bank’s foreign exchange reserves. Alternatively,6 Note the unfortunate terminology: Capital inßows reduce net foreign assetholdings, while capital outßows increase net foreign asset holdings.

20 CHAPTER 1. SOME INSTITUTIONAL BACKGROUNDif the home country spends less abroad than it receives there will bea privately determined excess supply of foreign exchange. The centralbank can absorb the excess supply by accumulating foreign exchangereserves. Changes in the central bank’s foreign exchange reserves arerecorded in the official settlements balance, which we argued above isthe balance of payments. Central bank foreign exchange reserve lossesare credits and their reserve gains are debits to the official settlementsaccount.1.3 The Central Bank’s Balance SheetThe monetary liabilities of the central bank is called the monetary base,B. It is comprised of currency and commercial bank reserves or depositsat the central bank. The central bank’s assets can be classiÞed into twomain categories. The Þrst is domestic credit, D. In the US, domesticcredit is extended to the treasury when the central bank engages inopen market operations and purchases US Treasury debt and to thecommercial banking system through discount lending. The second assetcategory is the central bank’s net holdings of foreign assets, NFA cb .These are mainly foreign exchange reserves held by the central bankminus its domestic currency liabilities held by foreign central banks.Foreign exchange reserves include foreign currency, foreign governmentTreasury bills, and gold. We state the central bank’s balance sheetidentity asB =D+NFA cb . (1.22)Since the money supply varies in proportion to changes in the monetarybase, you see from (1.22) that in the open economy there aretwo determinants of the money supply. The central bank can alter themoney supply either through a change in discount lending, open marketoperations, or via foreign exchange intervention. Under a regimeof perfectly ßexible exchange rates, ∆NFA cb = 0, which implies that,the central bank controls the money supply just as it does in the closedeconomy case.

1.3. THE CENTRAL BANK’S BALANCE SHEET 21Mechanics of InterventionSuppose that the central bank wants to the dollar to fall in value againstthe yen. To achieve this result, it must buy yen which increases NFA cb ,B, and hence the money supply M. If the Fed buys the yen fromCitibank (say), in New York, the Fed pays for the yen by creditingCitibank’s reserve account. Citibank then transfers ownership of a yendeposit at a Japanese bank to the Fed.If the intervention ends here the US money supply increases but theJapanese money supply is unaffected. In Japan, all that happens is aswap of deposit liabilities in the Japanese commercial bank. The Fedcould go a step further and convert the deposit into Japanese T-bills.It might do so by buying T-bills from a Japanese resident which it paysfor by writing a check drawn on the Japanese bank. The Japaneseresident deposits that check in a bank, and still, there is no net effecton the Japanese monetary base.If, on the other hand, the Fed converts the deposit into currency,the Japanese monetary base does decline. The reason for this is thatthe Japanese monetary base is reduced when the Fed withdraws currencyfrom circulation. The Fed would never do this, however, becausecurrency pays no interest. The intervention described above is referredto as an unsterilized intervention because the central bank’s foreign exchangetransactions have been allowed to affect the domestic moneysupply. A sterilized intervention, on the other hand occurs when thecentral bank offsets its foreign exchange operations with transactionsin domestic credit so that no net change in the money supply occurs.To sterilize the yen purchase described above, the Fed would simultaneouslyundertake an open market sale, so that D would decrease byexactly the amount that NFA cb increases from the foreign exchange intervention.It is an open question whether sterilized interventions canhave a permanent effect on the exchange rate.

22 CHAPTER 1. SOME INSTITUTIONAL BACKGROUND

Chapter 2Some Useful Time-SeriesMethodsInternational macroeconomic and Þnance theory is typically aimed atexplaining the evolution of the open economy over time. The naturalway to empirically evaluate these theories are with time-series methods.This chapter summarizes some of the time-series tools that areused in later chapters to estimate and to test predictions by the theory.The material is written assuming that you have had a Þrst course ineconometrics covering linear regression theory and is presented withoutproofs of the underlying statistical theory. There are now severalaccessible textbooks that contain careful treatments of the associatedeconometric theory. 1 If you like, you may skip this chapter for now anduse it as reference when the relevant material is encountered.You will encounter the following notation and terminology. Underlinedvariables will denote vectors and bold faced variables will denotematrices. a = plim(X T ) indicates that the sequence of random variables{X T } converges in probability to the number a as T →∞.Thismeans that for sufficiently large T , X T can be treated as a constant.N(µ, σ 2 ) stands for the normal distribution with mean µ and varianceσ 2 , U[a, b] stands for the uniform distribution over the interval [a, b],iidX t ∼ N(µ, σ 2 ) means that the random variable X t is independentlyand identically distributed as N(µ, σ 2 iid), X t ∼ (µ, σ 2 )meansthatX t is1 See Hamilton [66], Hatanaka [74], and Johansen [81].23

24 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(3)⇒independently and identically distributed according to some unspeci-Þed distribution with mean µ and variance σ 2 , Y D T → N(µ, σ 2 ) indicatesthat as T →∞, the sequence of random variables Y T converges in distributionto the normal with mean µ and variance σ 2 andiscalledtheasymptotic distribution of Y T .Thismeansthatforsufficiently large T ,the random variable {Y T } has the normal distribution with mean µ andvariance σ 2 . We will say that a time-series {x t } is covariance stationaryif its Þrst and second moments are Þnite and are time invariant—forexample, if E(x t ) = µ, and E(x t x t−j ) = γ j . AR(p) stands for autoregressionof order p, MA(n) stands for moving average of ordern, ARIMA stands for autoregressive-integrated-moving-average, VARstands for vector autoregression, and VECM stands for vector errorcorrection model.2.1 Unrestricted Vector AutoregressionsConsider a zero-mean covariance stationary bivariate vector time-series,=(q 1t ,q 2t ) 0 and assume that it has the p-th order autoregressiveq t(4)⇒representation 2 q t=pXj=1A j q t−j+ ² t , (2.1)Ã !a11,j awhere A j =12,jand the error vector has mean, E(²a 21,j a t )=022,jand covariance matrix E(² t ² 0 t)=Σ. The unrestricted vector autoregressionVAR is a statistical model for the vector time-series q t. Thesame variables appear in each equation as the independent variables sothe VAR can be efficiently estimated by running least squares (OLS)individually on each equation.To estimate a p−thorderVARforthis2−equation system, letz 0 t =(q 1t−1 ,...,q 1t−p ,q 2t−1 ,...,q 2t−p ) and write (2.1) out asq 1t = z 0 tβ 1+ ² 1t ,q 2t = z 0 t β + ² 2 2t.Let the grand coefficient vector be β = (β 0 1 , β0 2 )0 , and let2 q twill be covariance stationary if E(q t)=µ, E(q t− µ)(q t−j− µ) 0 = Σ j .

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 25Q = plim ³ 1 P Tt=1qT tq´,beapositivedeÞnite 0 tmatrix of constants which ⇐(5)exists by the law of large numbers and the covariance stationarity assumption.Then, as T →∞√T ( ˆβ − β) → D N(0, Ω), (2.2)where Ω = Σ ⊗ Q −1 . The asymptotic distribution can be used to test ⇐(6)hypotheses about the β vector.Lag-Length DeterminationUnless you have a good reason to do otherwise, you should let the datadetermine the lag length p. If the q tare drawn from a normal distribution,the log likelihood function for (2.1) is −2ln|Σ| + c where c is aconstant. 3 If you choose the lag-length to maximize the normal likelihoodyou just choose p to minimize ln | ˆΣ p |,where ˆΣ p = 1 P Tt=p+1 ˆ²T −p tˆ² 0 tis the estimated error covariance matrix of the VAR(p). In applicationswith sample sizes typically available to international macroeconomists—100 or so quarterly observations—using the likelihood criterion typicallyresults in choosing ps that are too large. To correct for the upwardsmall-sample bias, two popular information criteria are frequently usedfor data-based lag-length determination. They are AIC suggested byAkaike [1], and BIC suggested by Schwarz [125]. Both AIC and BICmodify the likelihood by attaching a penalty for adding additional lags.Let k be the total number of regression coefficients (the a ij,r coef-Þcients in (2.1)) in the system. In our bivariate case k =4p. 4 Thelog-likelihood cannot decrease when additional regressors are included.Akaike [1] proposed attaching a penalty to the likelihood for addinglags and to choose p to minimizeAIC = 2 ln | ˆ Σ p | + 2kT .3 |Σ| denotes the determinant of the matrix Σ.4 This is without constants in the regressions. If constants are included in theVAR then k =4p +2.

26 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSEven with the penalty, AIC often suggests p to be too large. An alternativecriterion, suggested by Schwarz [125] imposes an even greaterpenalty for additional parameters isBIC = 2 ln | ˆ Σ p | + k ln TT . (2.3)Granger Causality, Econometric Exogeniety and CausalPriorityIn VAR analysis, we say q 1t does not Granger cause q 2t if lagged q 1t donot appear in the equation for q 2t . That is, conditional upon currentand lagged q 2t , current and lagged q 1t do not help to predict future q 2t .You can test the null hypothesis that q 1t does not Granger cause q 2t byregressing q 2t on lagged q 1t and lagged q 2t and doing an F-test for thejoint signiÞcance of the coefficients on lagged q 1t .If q 1t does not Granger cause q 2t ,wesayq 2t is econometrically exogenouswith respect to q 1t . Ifitisalsotruethatq 2t does Granger causeq 1t ,wesaythatq 2t is causally prior to q 1t .TheVectorMoving-AverageRepresentationGiven the lag length p, you can estimate the A j coefficients by OLS andinvert the VAR(p)togettheWold vector moving-average representationq t==⎛⎝I −∞Xj=0pXj=1⎞−1A j L j ⎠ ² tC j L j ² t , (2.4)where L is the lag operator such that L j x t = x t−j for any variable x t .Tosolve for the C j matrices, you equating coefficients on powers of the lagoperator. From (2.4) you know that ( P ∞j=0 C j L j )(I − P pj=1 A j L j )=I.

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 27Write it out as (7) (see line 2)I = C 0 +(C 1 − C 0 A 1 )L +(C 2 − C 1 A 1 − C 0 A 2 )L 2=+(C 3 − C 2 A 1 − C 1 A 2 − C 0 A 3 )L 3+(C 4 − C 3 A 1 − C 2 A 2 − C 1 A 3 − C 0 A 4 )L 4 + ···⎛⎞∞XjX⎝C j − C j−k A k⎠ L j .j=0k=1Now to equate coefficients on powers of L, Þrst note that C 0 = I andthe rest of the C j follow recursivelyC 1 = A 1 ,C 2 = C 1 A 1 + A 2 ,C 3 = C 2 A 1 + C 1 A 2 + A 3 ,C 4 = C 3 A 1 + C 2 A 2 + C 1 A 3 + A 4 ,.C k =kXj=1C k−j A j .For example if p =2,setA j = 0 for j ≥ 3. Then C 1 = A 1 , C 2 =C 1 A 1 + A 2 , C 3 = C 2 A 1 + C 1 A 2 , C 4 = C 3 A 1 + C 2 A 2 ,andsoon.(8)(formulaeto end of section)⇐(9)Impulse Response AnalysisOnce you get the moving-average representation you will want employimpulse response analysis to evaluate the dynamic effect of innovationsin each of the variables on (q 1t ,q 2t ). When you go to simulate the dynamicresponse of q 1t and q 2t toashockto² 1t ,youareimmediatelyconfronted with two problems. The Þrst one is how big should the ⇐(10)shock be? This becomes an issue because you will want to compare theresponse of q 1t across different shocks. You’ll have to make a normalizationfor the size of the shocks and a popular choice is to considershocks one standard deviation in size. The second problem is to getshocks that can be unambiguously attributed to q 1t and to q 2t .If² 1t and² 2t are contemporaneously correlated, however, you can’t just shock ² 1tand hold ² 2t constant.

28 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(11) (eq. 2.6)To deal with these problems, Þrst standardize the innovations. Sincethe correlation matrix is given bywhere Λ =⎛⎝1√σ11001√σ22R = ΛΣΛ =⎞Ã1ρρ 1!,⎠ is a matrix with the inverse of the standarddeviations on the diagonal and zeros elsewhere. The error covariancematrix can be decomposed as Σ = Λ −1 RΛ −1 . This means the Woldvector moving-average representation (2.4) can be re-written asq t==⎛∞X⎝j=0⎛∞X⎝j=0C j L j ⎞⎠ Λ −1 (Λ² t )D j L j ⎞⎠ v t . (2.5)where D j ≡ C j Λ −1 ,v t ≡ Λ² t and E(v t v 0 t)=R. The newly deÞnedinnovations v 1t and v 2t both have variance of 1.Now to unambiguously attribute an innovation to q 1t , you mustorthogonalize the innovations by taking the unique upper triangularCholeski matrix à decomposition ! of the correlation matrix R = S 0 S,s11 swhere S =12. Now insert SS −1 into the normalized moving0 s 22average (2.5) to get⎛ ⎞∞Xq t= ⎝ D j L j ⎠ S ³ ∞XS −1 v t´= B j L j η t , (2.6)j=0where B j ≡ D j S = C j Λ −1 S and η t ≡ S −1 v t , is the 2×1 vectorofzeromeanorthogonalized innovations with covariance matrix E(η t η 0 = I). tNote that S −1 is also upper triangular.Now write out the individual equations in (2.6) to getq 1t =q 2t =∞Xj=0∞Xj=0b 11,j η 1,t−j +b 21,j η 1,t−j +∞Xj=0∞Xj=0j=0b 12,j η 2,t−j , (2.7)b 22,j η 2,t−j . (2.8)

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 29The effect on q 1t at time k of a one standard deviation orthogonalizedinnovation in η 1 at time 0, is b 11,k . Similarly, the effect on q 2k is b 21,k .Graphing the transformed moving-average coefficients is an efficientmethod to examine the impulse responses.You may also want to calculate standard error bands for the impulseresponses. You can do this using the following parametric bootstrapprocedure. 5 Let T be the number of time-series observations you haveand let a ‘tilde’ denote pseudo values generated by the computer, then ⇐(12) ‘tilde’1. Take T + M independent draws from the N(0, ˆΣ) to form thevector series {˜² t }.2. Set startup values of q tat their mean values of 0 then recursivelygenerate the sequence {˜q t } of length T + M according to (2.1)using the estimated A j matrices.3. Drop the Þrst M observations to eliminate dependence on startingvalues. Estimate the simulated VAR. Call the estimated coefficientsà j .4. Form the matrices ˜B j = ˜C j ˜Λ −1˜S. You now have one realization ⇐(13)of the parametric bootstrap distribution of the impulse responsefunction.5. Repeat the process say 5000 times. The collection of observationson the ˜B j forms the bootstrap distribution. Take the standarddeviation of the bootstrap distribution as an estimate of the standarderror.Forecast-Error Variance DecompositionIn (2.7), you have decomposed q 1t into orthogonal components. Theinnovation η 1t is attributed to q 1t and the innovation η 2t is attributed5 The bootstrap is a resampling scheme done by computer to estimate the underlyingprobability distribution of a random variable. In a parametric bootstrapthe observations are drawn from a particular probability distribution such as thenormal. In the nonparametric bootstrap, the observations are resampled from thedata.

30 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSto q 2t . You may be interested in estimating how much of the underlyingvariability in q 1t is due to q 1t innovations and how much is due toq 2t innovations. For example, if q 1t is a real variable like the log realexchange rate and q 2t is a nominal quantity such as money and youmight want to know what fraction of log real exchange rate variabilityis attributable to innovations in money. In the VAR framework, youcan ask this question by decomposing the variance of the k-step aheadforecast error into contributions from the separate orthogonal components.At t + k, the orthogonalized and standardized moving-averagerepresentation isq t+k= B 0 η t+k+ ···+ B k η t+ ··· (2.9)Take expectations of both sides of (2.9) conditional on informationavailable at time t to getE t q t+k= B k η t+ B k+1 η t−1+ ··· (2.10)Now subtract (2.10) from (2.9) to get the k-period ahead forecast errorvectorq t+k− E t q t+k= B 0 η t+k+ ···+ B k−1 η t+1. (2.11)Because the η tare serially uncorrelated and have covariance matrix I,the covariance matrix of these forecast errors isE[q t+k− E t q t+k][q t+k− E t q t+k] 0 = B 0 B 0 0 + B 1 B 0 1 + ···+ B k−1 B 0 k−1kXkX= B j B 0 j = ³ Ãbb1,j ,b 2,j´ 0 !1,jb 0 2,j=j=0kXb 1,j b 0 1,j +j=0| {z }(a)j=0kXb 2,j b 0 2,j, (2.12)j=0| {z }(b)where b 1,j is the Þrst column of B j and b 2,j is the second column of B j .As k →∞,thek-period ahead forecast error covariance matrix tendstowards the unconditional covariance matrix of q t.The forecast error variance of q 1t attributable to the orthogonalizedinnovations in q 1t is Þrst diagonal element in the Þrst summation which

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 31is labeled a in (2.12). The forecast error variance in q 1t attributable toinnovations in q 2t is given by the Þrst diagonal element in the secondsummation (labeled b). Similarly, the second diagonal element of a isthe forecast error variance in q 2t attributable to innovations in q 1t andthe second diagonal element in b is the forecast error variance in q 2tattributable to innovations in itself.A problem you may encountered in practice is that the forecast errordecomposition and impulse responses may be sensitive to the orderingof the variables in the orthogonalizing process, so it may be a goodidea to experiment with which variable is q 1t and which one is q 2t . Asecond problem is that the procedures outlined above are purely of astatistical nature and have little or no economic content. In chapter(8.4) we will cover a popular method for using economic theory toidentify the shocks.Potential Pitfalls of Unrestricted VARsCooley and LeRoy [32] criticize unrestricted VAR accounting becausethe statistical concepts of Granger causality and econometric exogeneityare very different from standard notions of economic exogeneity.Their point is that the unrestricted VAR is the reduced form of somestructural model from which it is not possible to discover the true relationsof cause and effect. Impulse response analyses from unrestrictedVARs do not necessarily tell us anything about the effect of policy interventionson the economy. In order to deduce cause and effect, youneed to make explicit assumptions about the underlying economic environment.We present the Cooley—LeRoy critique in terms of the two-equationmodel consisting of the money supply and the nominal exchange ratem = ² 1 , (2.13)s = γm + ² 2 , (2.14)iidwhere the error terms are related by ² 2 = λ² 1 + ² 3 with ² 1 ∼ N(0, σ1),2iid² 3 ∼ N(0, σ3)andE(² 2 1 ² 3 ) = 0. Then you can rewrite (2.13) and (2.14)asm = ² 1 , (2.15)

32 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSs = γm + λ² 1 + ² 3 . (2.16)m is exogenous in the economic sense and m = ² 1 determines part of ² 2 .The effect of a change of money on the exchange rate ds =(λ + γ)dmis well deÞned.A reversal of the causal link gets you into trouble because you willnot be able to unambiguously determine the effect of an m shock ons. Suppose that instead of (2.13), the money supply is governed byiidtwo components, ² 1 = δ² 2 + ² 4 with ² 2 ∼ N(0, σ2), 2 iid² 4 ∼ N(0, σ4)and2E(² 4 ² 2 )=0. Thenm = δ² 2 + ² 4 , (2.17)s = γm + ² 2 . (2.18)If the shock to m originates with ² 4 ,theeffect on the exchange rateis ds = γd² 4 . If the m shock originates with ² 2 , then the effect isds =(1+γδ)d² 2 .Things get really confusing if the monetary authorities follow a feedbackrule that depends on the exchange rate,where E(² 1 ² 2 ) = 0. The reduced form ism = θs + ² 1 , (2.19)s = γm + ² 2 , (2.20)m = ² 1 + θ² 21 − γθ , (2.21)s = γ² 1 + ² 21 − γθ . (2.22)Again, you cannot use the reduced form to unambiguously determinethe effect of m on s because the m shock may have originated with ² 1 ,² 2 , or some combination of the two. The best you can do in this caseis to run the regression s = βm + η, andgetβ =Cov(s, m)/Var(m)which is a function of the population moments of the joint probabilitydistribution for m and s. If the observations are normally distributed,then E(s|m) =βm, so you learn something about the conditional expectationof s given m. But you have not learned anything about theeffects of policy intervention.

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 33To relate these ideas to unrestricted VARs, consider the dynamicmodelm t = θs t + β 11 m t−1 + β 12 s t−1 + ² 1t , (2.23)s t = γm t + β 21 m t−1 + β 22 s t−1 + ² 2t , (2.24)iidwhere ² 1t ∼ N(0, σ1), 2 iid² 2t ∼ N(0, σ2), 2 and E(² 1t ² 2s ) = 0 for all t, s.Without additional restrictions, ² 1t and ² 2t are exogenous but both m tand s t are endogenous. Notice also that m t−1 and s t−1 are exogenouswith respect to the current values m t and s t .If θ = 0, then m t is said to be econometrically exogenous withrespect to s t . m t ,m t−1 ,s t−1 would be predetermined in the sense thatan intervention due to a shock to m t can unambiguously be attributedto ² 1t and the effect on the current exchange rate is ds t = γdm t . Ifβ 12 = θ =0,thenm t is strictly exogenous to s t .Eliminate the current value observations from the right side of (2.23)and (2.24) to get the reduced formwherem t = π 11 m t−1 + π 12 s t−1 + u mt , (2.25)s t = π 21 m t−1 + π 22 s t−1 + u st , (2.26)π 11 = (β 11 + θβ 21 )(1 − γθ), π 12 = (β 12 + θβ 22 ),(1 − γθ)π 21 = (β 21 + γβ 11 ), π 22 = (β 22 + γβ 12 )(1 − γθ)(1 − γθ)u mt = (² 1t + θ² 2t )(1 − γθ) , u st = (² 2t + γ² 1t )(1 − γθ) ,Var(u mt )= (σ2 1 + θ 2 σ 2 2)(1 − γθ) 2 , Var(u st) = (γ2 σ 2 1 + σ 2 2)(1 − γθ) 2 ,Cov(u mt ,u st )= (γσ2 1 + θσ2)2(1 − γθ) 2 . ⇐(14) (last 3If you were to apply the VAR methodology to this system, youwould estimate the π coefficients. If you determined that π 12 =0,expressions)

34 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSyou would say that s does not Granger cause m (and therefore m iseconometrically exogenous to s). But when you look at (2.23) and(2.24), m is exogenous in the structural or economic sense when θ =0but this is not implied by π 12 = 0. The failure of s to Granger cause mneed not tell us anything about structural exogeneity.Suppose you orthogonalize the error terms in the VAR. Letδ = Cov(u mt ,u st )/Var(u mt )betheslopecoefficient from the linearprojection of u st onto u mt . Then u st − δu mt is orthogonal to u mt byconstruction. An orthogonalized system is obtained by multiplying(2.25) by δ and subtracting this result from (2.26)m t = π 11 m t−1 + π 12 s t−1 + u mt , (2.27)s t = δm t +(π 21 − δπ 11 )m t−1 +(π 22 − δπ 12 )s t−1 + u st − δu mt . (2.28)The orthogonalized system includes a current value of m t in the s tequation but it does not recover the structure of (2.23) and (2.24). Theorthogonalized innovations areu mt = ² 1t + θ² 2t1 − γθ , (2.29)u st − δu mt = (γ² 1t + ² 2t ) − ³ ´γσ1 2+θσ2 2σ1 2+θ2 σ22 (²1t + θ² 2t ), (2.30)1 − γθwhich allows you to look at shocks that are unambiguously attributableto u mt in an impulse response analysis but the shock is not unambiguouslyattributable to the structural innovation, ² 1t .To summarize, impulse response analysis of unrestricted VARs providesummaries of dynamic correlations between variables but correlationsdo not imply causality. In order to make structural interpretations,you need to make assumptions of the economic environment andbuild them into the econometric model. 66 You’ve no doubt heard the phrase made famous by Milton Friedman, “There’sno such thing as a free lunch.” Michael Mussa’s paraphrasing of that principle indoing economics is “If you don’t make assumptions, you don’t get conclusions.”

2.2. GENERALIZED METHOD OF MOMENTS 352.2 Generalized Method of MomentsOLS can be viewed as a special case of the generalized method of moments(GMM) estimator studied by Hansen [70]. Since you are presumablyfamiliar with OLS, you can build your intuition about GMMby Þrst thinking about using it to estimate a linear regression. Aftergetting that under your belt, thinking about GMM estimation in morecomplicated and possibly nonlinear environments is straightforward.OLS and GMM. Suppose you want to estimate the coefficients inthe regressionq t = z 0 t β + ² t, (2.31)where β is the k-dimensional vector of coefficients, z t is a k-dimensionaliidvector of regressors and ² t ∼ (0, σ 2 )and(q t ,z t ) are jointly covariancestationary. The OLS estimator of β is chosen to minimize1TTXt=1² 2 t = 1 T= 1 TTXt=1TX(q t − β 0 z t )(q t − z 0 tβ)t=1q 2 t − 2β 1 TTXt=1z t q t + β 0 1TTXt=1(z t z 0 t )β. (2.32)When you differentiate (2.32) with respect to β and set the result tozero, you get the Þrst-order conditions,− 2 TTX|t=1{z }(a)TXz t ² t = −2 1 T t q t )+2βt=1(z 1 TTX| {zt=1}(b)(z t z 0 t ) =0. (2.33)If the regression is correctly speciÞed, the Þrst-order conditions form aset of k orthogonality or ‘zero’ conditions that you used to estimate β.These orthogonality conditions are labeled (a) in (2.33). OLS estimationis straightforward because the Þrst-order conditions are the set ofk linear equations in k unknowns labeled (b) in (2.33) which are solvedby matrix inversion. 7 Solving (2.33) for the minimizer ˆβ, you get, ⇐(16) (last7 In matrix notation, we usually write the regression as q = Zβ + ² where qis the T-dimensional vector of observations on q t , Z is the T × kdimensionalmatrix of observations on the independent variables whose t-th row is z 0 t, β is thek-dimensional vector of parameters that we want to estimate, ² is the T-dimensionalvector of regression errors, and ˆβ =(Z 0 Z) −1 Z 0 q.line of footnote)

36 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSÃ1ˆβ =TTXt=1z t z 0 t! −1 Ã1TTXt=1(z t q t )!. (2.34)Let Q = plim 1 P zt z 0 T t and let W = σ 2 Q. Because {² t } is an iidsequence, {z t ² t } is also iid. It follows from the Lindeberg-Levy centrallimit theorem that √ Tt=1 1 PTz t ² D t → N(0, W). Let the residuals beˆ² t = q t − z 0 tˆβ, the estimated error variance be ˆσ 2 = 1 P Tt=1ˆ² 2 T t , and letPŴ = ˆσ2 Tt=1zT t z 0 t. While it may seem like a silly thing to do, you canset up a quadratic form using the orthogonality conditions and get theOLS estimator by minimizingà ! 0 à !1TX1TX(zT t ² t ) Ŵ −1 (zT t ² t ) , (2.35)t=1with respect to β. This is the GMM estimator for the linear regression(2.31). The Þrst-order conditions to this problem aret=1Ŵ −1 1 TXzt ² t = 1 TXzt ² t =0,(15)⇒which are identical to the OLS Þrst-order conditions (2.33). You alsoknow that the asymptotic distribution of the OLS estimator of β is√T (ˆβ − β) D → N(0, V), (2.36)where V = σ 2 Q −1 . If you let D =E(∂(z t ² t )/∂β 0 )=Q, theGMMcovariance matrix V can be expressed as V = σ 2 Q −1 =[D 0 W −1 D] −1 .The Þrst equality is the standard OLS calculation for the covariancematrix and the second equality follows from the properties of (2.35).You would never do OLS by minimizing (2.35) since to get theweighting matrix Ŵ−1 , you need an estimate of β which is what youwant in the Þrst place. But this is what you do in the generalizedenvironment.Generalized environment. Suppose you have an economic theory thatrelates q t toavectorx t . The theory predicts the set of orthogonalityconditionsE[z t ² t (q t ,x t , β)] = 0,

2.2. GENERALIZED METHOD OF MOMENTS 37where z t is a vector of instrumental variables which may be differentfrom x t and ² t (q t ,x t , β) may be a nonlinear function of the underlyingk-dimensional parameter vector β and observations on q t and x t . 8 Toestimate β by GMM, let w t ≡ z t ² t (q t ,x t , β) where we now write the ⇐(17)vector of orthogonality conditions as E(w t )=0. Mimickingthestepsabove for GMM estimation of the linear regression coefficients, you’llwant to choose the parameter vector β to minimizeÃ1TTXt=1w t! 0Ŵ −1 Ã1T!TXw tt=1, (2.37)where Ŵ is a consistent estimator of the asymptotic covariance matrix1 Pof √ wt T. It is sometimes called the long-run covariance matrix. Youcannot guarantee that w t is iid in the generalized environment. It maybe serially correlated and conditionally heteroskedastic. To allow forthese possibilities, the formula for the weighting matrix isW = Ω 0 +∞Xj=1(Ω j + Ω 0 j ), (2.38)where Ω 0 =E(w t w 0 t)andΩ j =E(w t w 0 t−j). A popular choice for estimatingŴ is the method of Newey and West [114]Ŵ = ˆΩ 0 + 1 TmXµj=11 − j +1T ³ˆΩ j + ˆΩ 0 j´, (2.39)where ˆΩ 0 = 1 P Tt=1wT t w 0 t,andˆΩ j = 1 P Tt=j+1wTt w 0 t−j. The weightingfunction 1 − (j+1) is called the Bartlett window. WhenŴ Tconstructedby Newey and West, it is guaranteed to be positive deÞnite which isa good thing since you need to invert it to do GMM. To guaranteeconsistency, the Newey-West lag length (m) needsgotoinÞnity, but ataslowerratethanT . 9 You might try values such as m = T 1/4 .Totest8 Alternatively, you may be interested in a multiple equation system in which thetheory imposes parameter restrictions across equations so not only may the modelbe nonlinear, ² t could be a vector of error terms.9 Andrews [2] and Newey and West [115] offer recommendations for letting thedata determine m.⇐(18)(eq. 2.37)

38 CHAPTER 2. SOME USEFUL TIME-SERIES METHODShypotheses, use the fact that√T (ˆβ − β) D → N(0, V), (2.40)(19)(eq. 2.41)⇒µwhere V =(D 0 W −1 D) −1 ∂w, and D =E t. To estimate D, youcanµP Tt=1 ∂ŵ t.∂β 0Let R be a k×q restriction matrix and r is a q dimensional vector ofuse ˆD = 1 T∂β 0 constants. Consider the q linear restrictions Rβ = r on the coefficientvector. The Wald statistic has an asymptotic chi-square distributionunder the null hypothesis that the restrictions are trueW T = T (Rˆβ − r) 0 [RVR 0 ] −1 (Rˆβ − r) D → χ 2 q. (2.41)It follows that the linear restrictions can be tested by comparing theWald statistic against the chi-square distribution with q degrees of freedom.GMM also allows you to conduct a generic test of a set of overidentifyingrestrictions. The theory predicts that there are as manyorthogonality conditions, n, as is the dimensionality of w t . The parametervector β is of dimension k

2.3. SIMULATED METHOD OF MOMENTS 39β is the vector of parameters to be estimated.{q t } T t=1 is the actual time-series data of length T .Letq0 =(q 1 ,q 2 ,...,q T )denote the collection of the observations.{˜q i (β)} M i=1 is a computer simulated time-series of length M which isgenerated according to the underlying economic theory. Let˜q 0 (β) =(˜q 1 (β), ˜q 2 (β),...,˜q M (β)) denote the collection of theseM observations.h(q t ) is some vector function of the data from which to simulate themoments. For example, setting h(q t )=(q t ,q 2 t ,q 3 t ) 0 will pick offthe Þrst three moments of q t .H T (q) = 1 TP Tt=1h(q t ) is the vector of sample moments of q t .H M (˜q(β)) = 1 MP Mi=1h(˜q i (β)) is the corresponding vector of simulatedmoments where the length of the simulated series is M.u t = h(q t ) − H T (q) ish in deviation from the mean form.ˆΩ 0 = 1 TP Tt=1u t u 0 t is the sample short-run variance of u t .ˆΩ j = 1 TP Tt=1u t u 0 t−j is the sample cross-covariance matrix of u t .Ŵ T = ˆΩ 0 + 1 P mj=1(1 − j+1TT)( ˆΩ j + ˆΩ 0 j) is the Newey-West estimate ofthe long-run covariance matrix of u t .g T,M(β) =H T (q) − H M (˜q(β)) is the deviation of the sample momentsfrom the simulated moments.The SMM estimator is that value of β that minimizes the quadraticdistance between the simulated moments and the sample momentsg T,M (β) 0 h W −1T,MigT,M (β), (2.42)where W T,M = h³ 1+ T M´WTi.LetˆβS be SMM estimator. It is asymptoticallynormally distributed with√T (ˆβS − β) D → N(0, V S ),

40 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSas T and M → ∞ where V S = h B h³ ´ i −1 0 1+ T M Wi B and ⇐(20)B = E∂h[˜q j(β)]. You can estimate the theoretical value of B using its∂βsample counterparts.When you do SMM there are three points to keep in mind. First,you should choose M to be much larger than T . SMM is less efficientthan GMM because the simulated moments are only estimates of thetrue moments. This part of the sampling variability is decreasing inM and will be lessened by choosing M sufficiently large. 10 Second,the SMM estimator is the minimizer of the objective function for aÞxed sequence of random errors. The random errors must be held Þxedin the simulations so each time that the underlying random sequenceis generated, it must have the same seed. This is important becausethe minimization algorithm may never converge if the error sequenceis re-drawn at each iteration. Third, when working with covariancestationary observations, it is a good idea to purge the effects of initialconditions. This can be done by initially generating a sequence of length2M, discarding the Þrst M observations and computing the momentsfrom the remaining M observations.2.4 Unit Roots(21)⇒Unit root analysis Þgures prominently in exchange rate studies. A unitroot process is not covariance stationary. To Þx ideas, consider theAR(1) process(1 − ρL)q t = α(1 − ρ)+² t , (2.43)iidwhere ² t ∼ N(0, σ² 2 )andL is the lag operator. 11 Most economic timeseriesdisplay persistence so for concreteness we assume that 0 ≤ ρ ≤1. 12 {q t } is covariance stationary if the autoregressive polynomial (1 −ρz) is invertible. In order for that to be true, we need ρ < 1, whichisthesameassayingthattherootz in the autoregressive polynomial10 Lee and Ingram suggest M = 10T , but with computing costs now so low it mightbe a good idea to experiment with different values to ensure that your estimatesare robust to M.11 For any variable X t , L k X t = X t−k .12 If we admit negative values of ρ, werequire−1 ≤ ρ ≤ 1.

2.4. UNIT ROOTS 41(1 − ρz) = 0 lies outside the unit circle, which in turn is equivalent tosaying that the root is greater than 1. 13The stationary case. To appreciate some of the features of a unit roottime-series, we Þrst review some properties of stationary observations.If 0 ≤ ρ < 1 in (2.43), then {q t } is covariance stationary. It is straightforwardto show that E(q t )=α and Var(q t )=σ² 2 /(1 − ρ 2 ), which areÞnite and time-invariant. By repeated substitution of lagged values ofq t into (2.43), you get the moving-average representation with initialcondition q 0⎛ ⎞Xt−1q t = α(1 − ρ) ⎝ ρ j ⎠ + ρ t Xt−1q 0 + ρ j ² t−j . (2.44)j=0The effect of an ² t−j shock on q t is ρ j . More recent ² t shocks have a ⇐(22)larger effect on q t than those from the more distant past. The effects (eq.2.44)of an ² t shock are transitory because they eventually die out.To estimate ρ, we can simplify the algebra by setting α =0sothat{q t } from (2.43) evolves according toq t+1 = ρq t + ² t+1 ,where 0 ≤ ρ < 1. The OLS estimator is ˆρ = ρ+[( P T −1t=1 q t ² t+1 )/( P T −1t=1 q 2 t )].Multiplying both sides by √ T and rearranging gives√T (ˆρ − ρ) =1 √TP T −11Tj=0t=1 q t ² t+1. (2.45)P T −1t=1 q 2 tThe reason that you multiply by √ T is because that is the correctnormalizing factor to get both the numerator and the denominator onthe right side of (2.45) to remain well behaved as T →∞. By the lawof large numbers, plim 1 P T −1T t=1 qt 2 =Var(q t )=σ² 2 /(1 − ρ 2 ), so for thatsufficiently large T , the denominator can be treated like σ² 2 /(1 − ρ 2 )iidwhich is constant. Since ² t ∼ N(0, σ² 2 )andq t ∼ N(0, σ² 2 /(1 − ρ 2 )),13 Most economic time-series are better characterized with positive values of ρ,but the requirement for stationarity is actually |ρ| < 1. We assume 0 ≤ ρ ≤ 1 tokeep the presentation concrete.

42 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSthe product sequence {q t ² t+1 } is iid normal with mean E(q t ² t+1 ) =0 and variance Var(q t ² t+1 )=E(² 2 t+1)E(qt 2 )=σ² 4 /(1 − ρ 2 ) < ∞. Bythe Lindeberg-Levy central limit theorem, you have √ 1 P T −1T t=1 q t² D t+1 →N (0, σ 4 /(1 − ρ 2 )) as T →∞. For sufficiently large T , the numeratoris a normally distributed random variable and the denominator is aconstant so it follows that√T (ˆρ − ρ) → D N(0, 1 − ρ 2 ). (2.46)You can test hypotheses about ρ by doing the usual t-test.Estimating the Half-Life to ConvergenceIf the sequence {q t } follows the stationary AR(1) process, q t = ρq t−1 +² t ,its unconditional mean is zero, and the expected time, t ∗ , for it toadjust halfway back to 0 following a one-time shock (its half life) canbe calculated as follows. Initialize by setting q 0 =0. Thenq 1 = ² 1and E 1 (q t )=ρ t q 1 = ρ t ² 1 . The half life is that t such that the expectedvalue of q t has reverted to half its initial post-shock size–the t thatsets E 1 (q t )= ² 12.Sowelookforthet ∗ that sets ρ t∗ ² 1 = ² 12t ∗ = − ln(2)ln(ρ) . (2.47)If the process follows higher-order serial correlation, the formulain (2.47) only gives the approximate half life although empirical researcherscontinue to use it anyways. To see how to get the exact halflife, consider the AR(2) process, q t = ρ 1 q t−1 + ρ 2 q t−2 + ² t ,andlet" # " # " #qtρ1 ρy t= ; A =2²t, uq t−1 1 0 t = .0Now rewrite the process in the companion form,y t= Ay t−1 + u t , (2.48)and let e 1 =(1, 0) be a 2 × 1rowselection vector. Now q t =e 1 y t,E 1 (q t )=e 1 A t y 1,whereA 2 = AA, A 3 = AAA, and so forth. The halflife is the value t ∗ such thate 1 A t∗ y 1 = 1 2 e 1y 1= 1 2 ² 1.

2.4. UNIT ROOTS 43The extension to higher-ordered processes is straightforward.The nonstationary case.If ρ =1,q t has the driftless random walkprocess 14 q t = q t−1 + ² t .Setting ρ = 1 in (2.44) gives the analogous moving-average representationXt−1q t = q 0 + ² t−j .The effect on q t from an ² t−j shock is 1 regardless of how far in the pastit occurred. The ² t shocks therefore exert a permanent effect on q t .The statistical theory developed for estimating ρ for stationary timeseriesdoesn’t work for unit root processes because we have terms like1 − ρ in denominators and the variance of q t won’t exist. To seewhy that is the case, initialize the process by setting q 0 =0. Thenq t =(² t + ² t−1 + ··· + ² 1 ) ∼ N(0,tσ² 2 ). You can see that the varianceof q t grows linearly with t. Now a typical term in the numeratorof (2.45) is {q t ² t+1 } which is an independent sequence with meanE(q t ² t+1 ) = E(q t )E(² t+1 ) = 0 but the variance isVar(q t ² t+1 ) = E(qt 2 )E(² 2 t+1) = tσ²4 which goes to inÞnity over time.Since an inÞnite variance violates the regularity conditions of the usualcentral limit theorem, a different asymptotic distribution theory is requiredto deal with non-stationary data. Likewise, the denominator in(2.45) does not have a Þxed mean. In fact, E( 1 PT q2t )=σ 2 P t = T 2doesn’t converge to a Þnite number either.The essential point is that the asymptotic distribution of the OLSestimator of ρ is different when {q t } has a unit root than when theobservations are stationary and the source of this difference is that thevariance of the observations grows ‘too fast.’ It turns out that a differentscaling factor is needed on the left side of (2.45). In the stationary case,we scaled by √ T , but in the unit root case, we scale by T .T (ˆρ − ρ) =j=01 P T −1T1 P T −1T 2 t=1 qt2t=1 q t ² t+1, (2.49)14 When ρ = 1, we need to set α = 0 to prevent q t from trending. This willbecome clear when we see the Bhargava [12] formulation below.

44 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSconverges asymptotically to a random variable with a well-behaved distributionand we say that ˆρ converges at rate T whereas in the stationarycase we say that convergence takes place at rate √ T . The distributionfor T (ˆρ − ρ) is not normal, however, nor does it have a closedform so that its computation must be done by computer simulation.Similarly, the studentized coefficient or the ‘t-statistic’ for ˆρ reportedby regression packages τ = T ˆρ( P Tt=1 q 2 t )/( P Tt=1 ² 2 t ), also behaves has awell-behaved but non-normal asymptotic distribution. 15Test ProceduresThe discussion above did not include a constant, but in practice one isalmost always required and sometimes it is a good idea also to includea time trend. Bhargava’s [12] framework is useful for thinking aboutincluding constants and trends in the analysis. Let ξ t be the deviationof q t from a linear trendq t = γ 0 + γ 1 t + ξ t . (2.50)If γ 1 6= 0, the question is whether the deviation from the trend is stationaryorifitisadriftlessunitrootprocess.Ifγ 1 =0andγ 0 6=0,the question is whether the deviation of q t from a constant is stationary.Let’s ask the Þrst question–whether the deviation from trend isstationary. Letξ t = ρξ t−1 + ² t , (2.51)iidwhere 0 < ρ ≤ 1and² t ∼ N(0, σ² 2 ). You want to test the null hypothesisH o : ρ = 1 against the alternative H a : ρ < 1. Under the null hypothesis∆q t = γ 1 + ² t ,and q t is a random walk with drift γ 1 . Add the increments to getq t =tXj=1∆q j = γ 1 t +(² 0 + ² 1 + ···+ ² t )=γ 0 + γ 1 t + ξ t , (2.52)(23)⇒where γ 0 = ² 0 and ξ t =(² 1 +² 2 +···+² t ). You can initialize by assuming15 In fact, these distributions look like chi-square distributions so the least squaresestimator is biased downward under the null that ρ = 1.

2.4. UNIT ROOTS 45² 0 = 0, which is the unconditional mean of ² t . Now substitute (2.51)into (2.50). Use the fact that ξ t−1 = q t−1 − γ 0 − γ 1 (t − 1) and subtractq t−1 from both sides to get∆q t =[(1− ρ)γ 0 + ργ 1 ]+(1− ρ)γ 1 t +(ρ − 1)q t−1 + ² t . (2.53)(2.53) says you should run the regression∆q t = α 0 + α 1 t + βq t−1 + ² t , (2.54)where α 0 =(1− ρ)γ 0 + ργ 1 , α 1 =(1− ρ)γ 1 ,andβ = ρ − 1. The nullhypothesis, ρ = 1, can be tested by doing the joint test of the restrictionβ = α 1 = 0. To test if the deviation from a constant is stationary, do ajoint test of the restriction β = α 1 = α 0 = 0. If the random walk withdrift is a reasonable null hypothesis, evidence of trending behavior willprobably be evident upon visual inspection. If this is the case, includinga trend in the test equation would make sense.In most empirical studies, researchers do the Dickey—Fuller test ofthe hypothesis β = 0 instead of the joint tests recommended by Bhargava.Nevertheless, the Bhargava formulation is useful for decidingwhether to include a trend or just a constant. To complicate mattersfurther, the asymptotic distribution of ρ and τ depend on whether aconstant or a trend is included in the test equation so a different setof critical values need to be computed for each speciÞcation of the testequation. Tables of critical values can be found in textbooks on timeserieseconometrics, such as Davidson and MacKinnon [35] or Hamilton[66].Parametric Adjustments for Higher-Ordered Serial CorrelationYou will need to make additional adjustments if ξ t in (2.51) exhibitshigher-order serially correlation. The augmented Dickey—Fuller test isa procedure that employs a parametric correction for such time dependence.To illustrate, suppose that ξ t follows the AR(2) processξ t = ρ 1 ξ t−1 + ρ 2 ξ t−2 + ² t , (2.55)where ² tiid∼ N(0, σ 2 ² ). Then by (2.50), ξ t−1 = q t−1 − γ 0 − γ 1 (t − 1), ⇐(24)

46 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(26)⇒and ξ t−2 = q t−2 − γ 0 − γ 1 (t − 2). Substitute these expressions into(2.55) and then substitute this result into (2.50) to get q t = α 0 + α 1 t + ⇐(25)ρ 1 q t−1 + ρ 2 q t−2 + ² t ,whereα 0 = γ 0 [1 − ρ 1 − ρ 2 ]+γ 1 [ρ 1 +2ρ 2 ], andα 1 = γ 1 [1 − ρ 1 − ρ 2 ]. Now subtract q t−1 from both sides of this result,add and subtract ρ 2 q t−1 to the right hand side, and you end up with∆q t = α 0 + α 1 t + βq t−1 + δ 1 ∆q t−1 + ² t , (2.56)where β =(ρ 1 + ρ 2 − 1), and δ 1 = −ρ 2 . (2.56) is called the augmentedDickey—Fuller (ADF) regression. Under the null hypothesis that q t hasaunitroot,β =0.As before, a test of the unit root null hypothesis can be conductedby estimating the regression (2.56) by OLS and comparing the studentizedcoefficient, τ on β (the t-ratio reported by standard regressionroutines) to the appropriate table of critical values. The distributionof τ, while dependent on the speciÞcation of the deterministic factors,is fortunately invariant to the number of lagged dependent variables inthe augmented Dickey—Fuller regression. 16Permanent-and-Transitory-Components RepresentationIt is often useful to model a unit root process as the sum of differentsub-processes. In section chapter 2.2.7 we will model the time-series asbeing the sum of ‘trend’ and ‘cyclical’ components. Here, we will thinkofaunitrootprocess{q t } as the sum of a random walk {ξ t } and anorthogonal stationary process, {z t }q t = ξ t + z t . (2.57)(27)⇒iidTo Þx ideas,letξ t = ξ t−1 + ² t be a driftless random walk with ² t ∼N(0, σ² 2 )andletz t = ρz t−1 + v t be a stationary AR(1) process withiid0 ≤ ρ < 1andv t ∼ N(0, σv). 2 17 Because the effect of the ² t shocks16 An alternative strategy for dealing with higher-order serial correlation is thePhillips and Perron [120] method. They suggest a test that employs a nonparametriccorrection of the OLS studentized coefficient for ˆβ so that its asymptoticdistribution is the same as that when there is no higher ordered serial correlation.We will not cover their method.

2.4. UNIT ROOTS 47on q t last forever, the random walk {ξ t } is called the permanent component.The stationary AR(1) part of the process, {z t },iscalledthetransitory component because the effect of v t shocks on z t and thereforeon q t eventually die out. This random walk—AR(1) model has anARIMA(1,1,1) representation. 18 To deduce the ARIMA formulation,take Þrst differences of (2.57) to get∆q t = ² t + ∆z t= ² t +(ρ∆z t−1 + ∆v t )+(ρ² t−1 − ρ² t−1 )= ρ[∆z t−1 + ² t−1 ]+(² t − ρ² t−1 + v t − v t−1 )= ρ∆q t−1 +(² t − ρ² t−1 + v t − v t−1 ), (2.58)| {z }(a)where ρ∆q t−1 is the autoregressive part. The term labeled (a) in thelast line of (2.58) is the moving-average part. To see the connection,write this term out as,² t + v t − (ρ² t−1 + v t−1 )=u t + θu t−1 , (2.59)where u t is an iid process with E(u t ) = 0 and E(u 2 t ) = σ 2 u. Nowyou want to choose θ and σ 2 u such that u t + θu t−1 is observationallyequivalent to ² t + v t − (ρ² t−1 + v t−1 ),whichyoucandobymatchingcorresponding moments. Let ζ t = ² t + v t − (ρ² t−1 + v t−1 )andη t = u t + θu t−1 . Then you have,⇐(28)E(ζt 2 ) = σ² 2 (1 + ρ 2 )+2σv,2E(ηt 2 ) = σ2 u (1 + θ2 ),E(ζ t ζ t−1 ) = −(σv 2 + ρσ2 ² ),E(η t η t−1 ) = θσ

48 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSσu 2 (1 + θ2 ) = σ² 2 (1 + ρ2 )+2σv 2 , (2.60)θσu 2 = −(σv 2 + ρσ² 2 ). (2.61)(30)⇒(31)⇒These are two equations in the unknowns, σu 2 and θ which can be solved.The equations are nonlinear in σu 2 and getting the exact solution ispretty messy. To sketch out what to do, Þrst get θ 2 =[σv 2 + ρσ² 2 ] 2 /(σu) 2 2from (2.61). Substitute it into (2.60) to get x 2 + bx + c =0wherex = σu 2,b= −[σ2 ² (1 + ρ2 )+2σv 2], and c =[σ2 v + ρσ2 ² ]2 . The solution forσu 2 can then be obtained by the quadratic formula.Variance Ratios(32)⇒The variance ratio statistic at horizon k is the variance of the k-periodchange of a variable divided by k times the one-period changeVR k = Var(q t − q t−k )kVar(∆q t )= Var(∆q t + ···+ ∆q t−k+1 ). (2.62)kVar(∆q t )The use of these statistics were popularized by Cochrane [29] who usedthem to conduct nonparametric tests of the unit root hypothesis inGNP and to measure the relative size of the random walk componentin a time-series.Denote the k-th autocovariance of the stationary time-series {x t } byγk x =Cov(x t ,x t−k ). The denominator of (2.62) is kγ ∆q0 , the numeratoris Var(q t −q t−k+1 )=k h γ ∆q0 + P k−1j=1(1 − j k )(γ∆q j + γ ∆q−j ) i , so the varianceratio statistic can be written asVR k = γ∆q 0 + P k−1j=1(1 − j k )(γ∆q j + γ ∆q−j )γ ∆q0= 1+ 2 P k−1j=1(1 − j k )γ∆q jγ ∆q(2.63)0k−1 X= 1+2 (1 − j k )ρ∆qj=1j ,where ρ ∆qj= γ ∆qj /γ ∆q0 is the j-th autocorrelation coefficient of ∆q t .Measuring the size of the random walk. Suppose that q t evolves accordingto the permanent—transitory components model of (2.57). If

2.4. UNIT ROOTS 49ρ = 1, the increments ∆q t are independent and the numerator of VR kis Var(q t − q t−k )=Var(∆q t + ∆q t−1 + ···∆q t−k+1 )=kVar(∆q t ), whereVar(∆q t )=σ² 2 + σv. 2 In the absence of transitory component dynamics,VR k = 1 for all k ≥ 1.If 0 < ρ < 1, {q t } is still a unit root process, but its dynamicsare driven in part by the transitory part, {z t }.ToevaluateVR k , Þrstnote that γ0 z = σv/(1 2 − ρ 2 ). The k-th autocovariance of the transitorycomponent is γk z =E(z t z t−k )=ρ k γ0, z γ0 ∆z =E[∆z t ][∆z t ]=2(1− ρ)γ0 z ⇐(33)and the k-th autocovariance of ∆z t isγk ∆z =E[∆z t ][∆z t−k ]=−(1 − ρ) 2 ρ k−1 γ0 z < 0. (2.64)By (2.64), ∆z t is negatively correlated with its past values and thereforeexhibits mean reverting behavior because a positive change today isexpected to be reversed in the future. You also see that γ ∆q0 = σ² 2 +γ0∆zand for k>1γ ∆qk= γ ∆zk < 0. (2.65)By (2.65), the serial correlation in {∆q t } is seen to be determined bythe dynamics in the transitory component {z t }. Interactions betweenchanges are referred to as the short run dynamics of the process. Thus,working on (2.63), the variance ratio statistic for the random walk—AR(1) model can be written asVR k = 1− 2(1 − ρ)2 γ z 0P k−1j=1γ ∆q0³ ´1 −j ρj−1 k→ 1 − 2(1 − ρ)γz 0γ ∆q as k →∞0= 1− γ∆z 0σ² 2 + . (2.66)γ∆z 0VR ∞ − 1 is the fraction of the short run variance of ∆q t generated bychanges in the the transitory component. VR ∞ is therefore increasingin the relative size of the random walk component σ² 2 /γ0 ∆z .Near Observational EquivalenceBlough [16],Faust [50], and Cochrane [30] point out that for a samplewith Þxed T any unit root process is observationally equivalent to a

50 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSvery persistent stationary process. As a result, the power of unit roottests whose null hypothesis is that there is a unit root can be no largerthan the size of the test. 19To see how the problem comes up, consider again the permanent—transitory representation of (2.57). Assume that σ²2 = 0 in (2.57), sothat {q t } is truly an AR(1) process. Now, for any Þxed sample sizeT < ∞, it would be possible to add to this AR(1) process a randomwalk with an inÞnitesimal σ²2 which leaves the essential properties of{q t } unaltered, even though when we drive T →∞, the random walkwill dominate the behavior of q t . The practical implication is that itmay be difficult or even impossible to distinguish between a persistentbut stationary process and a unit root process with any Þnite sample.So even though the AR(1) plus the very small random walk process isinfactaunitrootprocess,σ² 2 canalwaysbechosensufficiently small,regardless of how large we make T so long as it is Þnite, that its behavioris observationally equivalent to a stationary AR(1) process.Turning the argument around, suppose we begin with a true unitroot process but the random walk component, σ²2 is inÞnitesimallysmall. For any Þnite T , this process can be arbitrarily well approximatedby an AR(1) process with judicious choice of ρ and σu.22.5 Panel Unit-Root TestsUnivariate/single-equation econometric methods for testing unit rootscan have low power and can give imprecise point estimates when workingwith small sample sizes. Consider the popular Dickey—Fuller testforaunitrootinatime-series{q t } and assume that the time-series aregenerated by∆q t = α 0 + α 1 t +(ρ − 1)q t−1 + ² t , (2.67)where ² tiid∼ N(0, σ 2 ). If ρ =1, α 1 = α 0 =0,q t follows a driftless unitroot process. If ρ =1, α 1 =0, α 0 6=0,q t follows a unit root processwith drift If |ρ| < 1, y t is stationary. It is mean reverting if α 1 =0,andis stationary around a trend if α 1 6=0.19 Power is the probability of rejecting the null when it is false. The size of a testis the probability of rejecting the null when it is true.

2.5. PANEL UNIT-ROOT TESTS 51TodotheDickey—Fullertestforaunitrootinq t , run the regression(2.67) and compare the studentized coefficient for the slope to theDickey—Fuller distribution critical values. Table 2.1 shows the power ofthe Dickey—Fuller test when the truth is ρ =0.96. 20 With 100 observations,the test with 5 percent size rejects the unit root only 9.6 percentof the time when the truth is a mean reverting process.Table 2.1: Finite Sample Power of Dickey—Fuller test, ρ =0.96.T 5 percent 10 percentTest 25 5.885 11.895equation 50 6.330 12.975includes 75 7.300 14.460constant 100 9.570 18.7151000 99.995 100.000Test 25 5.715 10.720equation 50 5.420 10.455includes 75 5.690 11.405trend 100 7.650 14.6651000 99.960 100.000Notes: Table reports percentage of rejections at 5 percent or 10 percent criticalvalue when the alternative hypothesis is true with ρ =0.96. 20000 replications.Critical values are from Hamilton (1994) Table B.6.100 quarterly observations is about what is available for exchangerate studies over the post Bretton-Woods ßoating period, so low poweris a potential pitfall in unit-root tests for international economists. Butagain, from Table 2.1, if you had 1000 observations, you are almostguaranteed to reject the unit root when the truth is that q t is stationarywith ρ =0.96. How do you get 1000 observations without having towait 250 years? How about combining the 100 time-series observationsfrom 10 roughly similar countries. 21 This is the motivation for recently20 Power is the probability that the test correctly rejects the null hypothesis becausethe null happens to be false.21 It turns out that the 1000 cross-section—time-series observations contain less

52 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSproposed panel unit-root tests have by Levin and Lin [91], Im, Pesaranand Shin [78], and Maddala and Wu [99]. We begin with the popularLevin—Lin test.The Levin—Lin TestLet {q it } be a balanced panel 22 of N time-series with T observationswhich are generated by∆q it = δ i t + β i q it−1 + u it , (2.68)where −2 < β i ≤ 0, and u it has the error-components representationu it = α i + θ t + ² it . (2.69)α i is an individual—speciÞc effect, θ t is a single factor common time effect,and ² it is a stationary but possibly serially correlated idiosyncraticeffect that is independent across individuals. For each individual i, ² ithas the Wold moving-average representation² it =∞Xj=0θ ij ² it−j + u it . (2.70)q it isaunitrootprocessifβ i =0andδ i = 0. If there is no drift in theunit root process, then α i = 0. The common time effect θ t is a crudemodel of cross-sectional dependence.Levin—Lin propose to test the null hypothesis that all individualshave a unit rootH 0 : β 1 = ···= β N = β =0,against the alternative hypothesis that all individuals are stationaryH A : β 1 = ···= β N = β < 0.information than 1000 observations from a single time-series. In the time-series, ˆρconverges at rate T , but in the panel, ˆρ converges at rate T √ N where N is thenumber of cross-section units, so in terms of convergence toward the asymptoticdistribution, it’s better to get more time-series observations.22 A panel is balanced if every individual has the same number of T observations.

2.5. PANEL UNIT-ROOT TESTS 53The test imposes the homogeneity restrictions that β i are identicalacross individuals under both the null and under the alternative hypothesis.The test proceeds as follows. First, you need to decide if you wantto control for the common time effect θ t . If you do, you subtract offthe cross-sectional mean and the basic unit of analysis is˜q it = q it − 1 NNXj=1q jt . (2.71)Potential pitfalls of including common-time effect. Doing so howeverinvolves a potential pitfall. θ t , as part of the error-components model,is assumed to be iid. The problem is that there is no way to imposeindependence. SpeciÞcally, if it is the case that each q it is drivenin part by common unit root factor, θ t is a unit root process. Then˜q it = q it − 1 P Nj=1qN jt will be stationary. The transformation renders ⇐(34)all the deviations from the cross-sectional mean stationary. This mightcause you to reject the unit root hypothesis when it is true. Subtractingoff the cross-sectional average is not necessarily a fatal ßaw in theprocedure, however, because you are subtracting off only one potentialunit root from each of the N time-series. It is possible that the Nindividuals are driven by N distinct and independent unit roots. Theadjustment will cause all originally nonstationary observations to bestationary only if all N individuals are driven by the same unit root.An alternative strategy for modeling cross-sectional dependence is todo a bootstrap, which is discussed below. For now, we will proceedwith the transformed observations. The resulting test equations are∆˜q it = α i + δ i t + β i˜q it−1 +k iXj=1φ ij ∆˜q it−j + ² it . (2.72)The slope coefficient on ˜q it−1 is constrained to be equal across individuals,but no such homogeneity is imposed on the coefficients on thelagged differences nor on the number of lags k i . To allow for this speci-Þcation in estimation, regress ∆˜q it and ˜q it−1 on a constant (and possibly

54 CHAPTER 2. SOME USEFUL TIME-SERIES METHODStrend) and k i lags of ∆˜q it . 23∆˜q it = a i + b i t +˜q it−1 = a 0 i + b 0 it +k iXj=1k iXj=1c ij ∆˜q it−j +ê it , (2.73)c 0 ij∆˜q it−j +ˆv it , (2.74)where ê it and ˆv it are OLS residuals. Now run the regressionê it = δ iˆv it−1 +û it , (2.75)set ˆσ ei 2 = 1 P Tt=kiT −k i −1 +2 û2 it, and form the normalized observationsẽ it = êitˆσ ei,ṽ it = ˆv itˆσ ei. (2.76)Denote the long run variance of ∆q it by σ 2 qi = γ i 0 +2 P ∞j=0 γ i j,whereγ i 0 = E(∆q 2 it) andγ i j = E(∆q it ∆q it−j ). Let ¯k = 1 NP Ni=1k i and estimateσ 2 qi by Newey and West [114]ˆσ 2 qi =ˆγi 0 +2 kXj=1µ1 − j ˆγ j i k +1, (2.77)where ˆγ j i = 1 P Tt=2+j∆˜qT −1it ∆˜q it−j . Let s i = ˆσ qiˆσ ei, S N = 1 Nrun the pooled cross-section time-series regressionP Ni=1s i andẽ it = βṽ it−1 +˜² it . (2.78)The studentized coefficient is τ = ˆβ P N P Tt=1i=1 ṽ it−1 /ˆσ˜² where ˆσ˜² =1 P Ni=1 P Tt=1˜²NTit . As in the univariate case, τ is not asymptoticallystandard normally distributed. In fact, τ diverges as the number of23 To choose k i , one option is to use AIC or BIC. Another option is to use Hall’s [69]general-to-speciÞc method recommended by Campbell and Perron [19]. Start withsome maximal lag order ` and estimate the regression. If the absolute value of thet-ratio for ĉ i` is less than some appropriate critical value, c ∗ , reset k i to ` − 1 andrepeattheprocessuntilthet-ratiooftheestimatedcoefficient with the longest lagexceeds the critical value c ∗ .

2.5. PANEL UNIT-ROOT TESTS 55Table 2.2: Mean and Standard Deviation Adjustments for Levin—Lin τStatistic, reproduced from Levin and Lin [91]τNC ∗ τC ∗ τCT∗˜T K µ ∗˜T σ ∗˜T µ ∗˜T σ ∗˜T µ ∗˜T σ ∗˜T25 9 0.004 1.049 -0.554 0.919 -0.703 1.00330 10 0.003 1.035 -0.546 0.889 -0.674 0.94935 11 0.002 1.027 -0.541 0.867 -0.653 0.90640 11 0.002 1.021 -0.537 0.850 -0.637 0.87145 11 0.001 1.017 -0.533 0.837 -0.624 0.84250 12 0.001 1.014 -0.531 0.826 -0.614 0.81860 13 0.001 1.011 -0.527 0.810 -0.598 0.78070 13 0.000 1.008 -0.524 0.798 -0.587 0.75180 14 0.000 1.007 -0.521 0.789 -0.578 0.72890 14 0.000 1.006 -0.520 0.782 -0.571 0.710100 15 0.000 1.005 -0.518 0.776 -0.566 0.695250 20 0.000 1.001 -0.509 0.742 -0.533 0.603∞ — 0.000 1.000 -0.500 0.707 -0.500 0.500observations NT gets large, but Levin and Lin show that the adjustedstatisticτ ∗ = τ − N ˜TS N τµ ˆσ−2∗˜T ²ˆβ −1D→ N(0, 1), (2.79)σ ∗˜Tas ˜T →∞,N →∞where ˜T = T −¯k−1, and µ ∗˜T and σ ∗˜T are adjustmentfactors reproduced from Levin and Lin’s paper in Table 2.2.Performance of Levin and Lin’s adjustment factors in a controlled environment.Suppose the data generating process (the truth) is, thateach individual is the unit root process∆q it = α i +2Xj=1φ ij ∆q it−j + ² it , (2.80)where ² itiid∼ N(0, σ i ),andeachoftheσ i is drawn from a uniform distributionover the range 0.1 to 1.1. That is, σ i ∼ U[0.1, 1.1]. Also,

56 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSTable 2.3: How Well do Levin—Lin adjustments work? Percentiles froma Monte Carlo Experiment.Statistic N T trend 2.5% 5% 50% 95% 97.5%τ 20 100 no -7.282 -6.995 -5.474 -3.862 -3.54320 500 no -7.202 -6.924 -5.405 -3.869 -3.560τ ∗ 20 100 no -2.029 -1.732 -0.092 1.613 1.96520 500 no -1.879 -1.557 0.012 1.595 1.894τ 20 100 yes -10.337 -10.038 -8.642 -7.160 -6.89620 500 yes -10.126 -9.864 -8.480 -7.030 -6.752τ ∗ 20 100 yes -1.171 -0.825 0.906 2.997 3.50320 500 yes -1.028 -0.746 0.702 2.236 2.571φ ij ∼ U[−0.3, 0.3], and α i ∼ N(0, 1) if a drift is included, (otherwiseα =0). 24 Table 2.3 shows the Monte Carlo distribution of Levin andLin’s τ and τ ∗ generated from this process. Here are some things tonote from the table. First, the median value of τ is very far from 0. Itwould get bigger (in absolute value) if we let N get bigger. Second, τ ∗looks like a standard normal variate when there is no drift in the DGP(and no trend in the test equation). Third, the Monte Carlo distributionfor τ ∗ looks quite different from the asymptotic distribution whenthere is drift in the DGP and a trend is included in the test equation.This is what we call Þnite sample size distortion of the test. When thereis known size distortion, you might want to control for it by doing abootstrap, which is covered below.Another option is to try to correct for the size distortion. Thequestion here is, if you correct for size distortion, does the Levin—Lintest have good power? That is, will it reject the null hypothesis when itis false with high probability? The answer suggested in Table 2.4 is yes.It should be noted, that even though the Levin-Lin test is motivatedin terms of a homogeneous panel, it has moderate ability to reject thenull when the truth is a mixed panel in which some of the individuals24 Instead of me arbitrarily choosing values of these parameters for each of theindividual units, I let the computer pick out some numbers at random.

2.5. PANEL UNIT-ROOT TESTS 57Table 2.4: Size adjusted power of Levin—Lin test with T =100,N =20Proportion Constant Trendstationary a/ 5% 10% 5 % 10%0.2 0.141 0.275 0.124 0.2180.4 0.329 0.439 0.272 0.3970.6 0.678 0.761 0.577 0.6870.8 0.942 0.967 0.906 0.9441.0 1.000 1.000 1.000 1.000Notes: a/ Proportion of individuals in the panel that are stationary.components have root equal to 0.96. Source: Choi [26].Stationaryare unit root process and others are stationary.Bias AdjustmentThe OLS estimator ˆρ is biased downward in small samples. Kendall [85]showed that the bias of the least squares estimator is E(ˆρ) −ρ ' −(1 +3ρ)/T . A bias-adjusted estimate of ρ isT ˆρ +1ˆρ ∗ =T − 3 . (2.81)The panel estimator of the serial correlation coefficient is also biaseddownwards in small samples. A Þrst-order bias-adjustment of the panelestimate of ρ can be done using a result by Nickell [116] who showedthat(ˆρ − ρ) → A T B T, (2.82)C Tas T → ∞,N → ∞ where A TC T =1− 2ρ(1−B T )[(1−ρ)(T −1)] .Bootstrapping τ ∗= −(1+ρ)T −1 ,B T = 1 − 1 (1−ρ T )T (1−ρ) , andThe fact that τ diverges can be distressing. Rather than to rely onthe asymptotic adjustment factors that may not work well in some regionsof the parameter space, researchers often choose to test the unit

58 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSroot hypothesis using a bootstrap distribution of τ. 25 Furthermore,the bootstrap provides an alternative way to model cross-sectional dependencein the error terms, as discussed above. The method discussedhere is called the residual bootstrap because we will be resampling fromthe residuals.To build a bootstrap distribution under the null hypothesis that allindividuals follow a unit-root process, begin with the data generatingprocess (DGP)∆q it = µ i +k iXj=1φ ij ∆q i,t−j + ² it . (2.83)Since each q it isaunitrootprocess,itsÞrst difference follows an autoregression.While you may prefer to specify the DGP as an unrestrictedvector autoregression for all N individuals, the estimation such a systemturns out not to be feasible for even moderately sized N.The individual equations of the DGP can be Þtted by least squares.If a linear trend is included in the test equation a constant must be includedin (2.83). To account for dependence across cross-sectional units,estimate the joint error covariance matrix Σ = E(² t ² 0 t) byˆΣ = 1 P Tt=1ˆ²T t ˆ² 0 t where ˆ² t =(ˆ² 1t ,...,ˆ² Nt ) is the vector of OLS residuals.The parametric bootstrap distribution for τ is built as follows.1. Draw a sequence of length T + R innovation vectors from˜² t ∼ N(0, ˆΣ).2. Recursively build up pseudo—observations {ˆq it },i = 1,...,N,t =1,...,T + R according to (2.83) with the ˜² t and estimatedvalues of the coefficients ˆµ iand ˆφ ij .3. Drop the Þrst R pseudo-observations, then run the Levin—Lin teston the pseudo-data. Do not transform the data by subtractingoff the cross-sectional mean and do not make the τ ∗ adjustments.This yields a realization of τ generated in the presence of crosssectionaldependent errors.4. Repeat a large number (2000 or 5000) times and the collection of τand ¯t statistics form the bootstrap distribution of these statisticsunder the null hypothesis.25 For example, Wu [135] and Papell [118].

2.5. PANEL UNIT-ROOT TESTS 59This is called a parametric bootstrap because the error terms aredrawn from the parametric normal distribution. An alternative is to doa nonparametric bootstrap. Here, you resample the estimated residuals,which are in a sense, the data. To do a nonparametric bootstrap, do thefollowing. Estimate (2.83) using the data. Denote the OLS residualsby(ˆ² 11 , ˆ² 21 ,...,ˆ² N1 ) ← obs. 1(ˆ² 12 , ˆ² 22 ,...,ˆ² N2 ) ← obs. 2.(ˆ² 1T , ˆ² 2T ,...,ˆ² NT ).← obs. TNow resample the residual vectors with replacement. For each observationt =1,...,T, draw one of the T possible residual vectors withprobability 1 . Because the entire vector is being resampled, the crosssectionalcorrelation observed in the data is preserved. Let the resam-Tpled vectors be(² ∗ 11,² ∗ 21,...,² ∗ N1) ← obs. 1(² ∗ 12,² ∗ 22,...,² ∗ N2) ← obs. 2..(² ∗ 1T ,² ∗ 2T ,...,² ∗ NT) ← obs. Tand use these resampled residuals to build up values of ∆q it recursivelyusing (2.83) with ˆµ i and ˆφ ij , and run the Levin-Lin test on these observationsbut do not subtract off the cross-sectional mean, and do notmake the τ ∗ adjustments. This gives a realization of τ. Nowrepeatalarge number of times to get the nonparametric bootstrap distributionof τ.The Im, Pesaran and Shin TestIm, Pesaran and Shin suggest a very simple panel unit root test. Theybegin with the ADF representation (2.72) for individual i (reproducedhere for convenience) (eq. 2.84)(35)∆˜q it = α i + δ i t + β i˜q it−1 +k iXj=1φ ij ∆˜q it−j + ² it , (2.84)

60 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(36)⇒(37)⇒where E(² it ² js )=0,i 6= j for all t, s. A common time effect may beremoved in which case ˜q it = q it − (1/N ) P Nj=1 q jt is the deviation fromthe cross-sectional average as the basic unit of analysis.Let τ i be the studentized coefficient from the ith ADF regression.Since the ² it are assumed to be independent across individuals, the τ i arealso independent, and by the central limit theorem, τ NT = 1 P Ni=1τN iconverges to the standard normal distribution Þrst as T →∞then asN →∞.Thatis√N[¯τNT − E(τ it |β i =0)]qVar(τit |β =0)D→ N(0, 1), (2.85)as T →∞,N →∞. IPS report selected critical values for ¯τ NT withthe conditional mean and variance adjustments of the distribution. Aselected set of these critical values are reproduced in Table 2.5. Analternative to relying on the asymptotic distribution is to do a residualbootstrap of the τ NT statistic. As before, when doing the bootstrap,do not subtract off the cross-sectional mean.The Im, Pesaran and Shin test as well as the Maddala—Wu test (discussedbelow) relax the homogeneity restrictions under the alternativehypothesis. Here, the null hypothesisis tested against the alternativeH 0 : β 1 = ···= β N = β =0,H A : β 1 < 0 ∪ β 2 < 0 ···∪ ββ N < 0.The alternative hypothesis is not H 0 , which is less restrictive than theLevin—Lin alternative hypothesis.The Maddala and Wu TestMaddala and Wu [99] point out that the IPS strategy of combiningindependent tests to construct a joint test is an idea suggested by R.A.Fisher [53]. Maddala and Wu follow Fisher’s suggestion and proposefollowing test. Let the p-value of τ i from the augmented Dickey—Fullertestforaunitrootbep i =Prob(τ < τ i ) = R τ i−∞ f(x)dx be the p-value of τ i from the ADF test on (2.72), where f(τ) is the probability

2.5. PANEL UNIT-ROOT TESTS 61Table 2.5: Selected Exact Critical Values for the IPS ¯τ NT StatisticConstantTrendT 20 40 100 20 40 100A. 5 percent5 -2.19 -2.16 -2.15 -2.82 -2.77 -2.7510 -1.99 -1.98 -1.97 -2.63 -2.60 -2.58N 15 -1.91 -1.90 -1.89 -2.55 -2.52 -2.5120 -1.86 -1.85 -1.84 -2.49 -2.48 -2.4625 -1.82 -1.81 -1.81 -2.46 -2.44 -2.43B. 10 percent5 -2.04 -2.02 -2.01 -2.67 -2.63 -2.6210 -1.89 -1.88 -1.88 -2.52 -2.50 -2.49N 15 -1.82 -1.81 -1.81 -2.46 -2.44 -2.4420 -1.78 -1.78 -1.77 -2.42 -2.41 -2.4025 -1.75 -1.75 -1.75 -2.39 -2.38 -2.38Source: Im, Pesaran and Shin [78].density function of τ. Solve for g(p), the density of p i by the method oftransformations, g(p i )=f(τ i )|J| where J = dτ i /dp i is the Jacobian ofthe transformation, and |J| is its absolute value. Since dp i /dτ i = f(τ i ),the Jacobian is 1/f(τ i )andg(p i ) = 1 for 0 ≤ p i ≤ 1. That is, p i isuniformly distributed on the interval [0, 1] (p i ∼ U[0, 1]).Next, let y i = −2ln(p i ). Again, using the method of transformations,the probability density function of y i is h(y i )=g(p i )|dp i /dy i |.But g(p i )=1and|dp i /dy i | = p i /2=(1/2)e −y i/2 , so it follows thath(y i )=(1/2)e −y i/2 which is the chi-square distribution with 2 degreesof freedom. Under cross-sectional independence of the error terms ² it ,the joint test statistic also has a chi-square distributionλ = −2NXi=1ln(p i ) ∼ χ 2 2N . (2.86)The asymptotic distribution of the IPS test statistic was establishedby sequential T →∞,N →∞asymptotics, which some econometri-

62 CHAPTER 2. SOME USEFUL TIME-SERIES METHODScians view as being too restrictive. 26 Levin and Lin derive the asymptoticdistribution of their test statistic by allowing both N and T simultaneouslytogotoinÞnity.A remarkable feature of the Maddala—Wu—Fisher test is that it avoids issues of sequential or joint N,T asymptotics.(2.86) gives the exact distribution of the test statistic.The IPS test is based on the sum of τ i , whereas the Maddala—Wutest is based on the sum of the log p-values of τ i . Asymptotically, thetwo tests should be equivalent, but can differ in Þnite samples. Anotheradvantage of Maddala—Wu is that the test statistic distribution does notdepend on nuisance parameters, as does IPS and LL. The disadvantageis that p-values need to be calculated numerically.Potential Pitfalls of Panel Unit-Root TestsPanel unit-root tests need to be applied with care. One potential pitfallwith panel tests is that the rejection of the null hypothesis does notmean that all series are stationary. It is possible that out of N timeseries,only 1 is stationary and (N-1) are unit root processes. This isan example of a mixed panel. Whether we want the rejection of theunit root process to be driven by a single outlier or not depends on thepurpose the researcher uses the test. 27A second potential pitfall is that cross-sectional independence isa regularity condition for these tests. Transforming the observationsby subtracting off the cross-sectional means will leave some residualdependence across individuals if common time effects are generated bya multi-factor process. This residual cross-sectional dependence canpotentially generate errors in inference.A third potential pitfall concerns potential small sample size distortionof the tests. While most of the attention has been aimed at26 That is, they deduce the limiting behavior of the test statistic Þrst by lettingT → ∞ holding N Þxed, then letting N → ∞ and invoking the central limittheorem.27 Bowman [17] shows that both the LL and IPS tests have low power againstoutlier driven alternatives. He proposes a test that has maximal power. Taylor andSarno [131] propose a test based on Johansen’s [80] maximum likelihood approachthat can test for the number of unit-root series in the panel. Computational considerations,however, generally limit the number of time-series that can be analyzedto 5 or less.

2.6. COINTEGRATION 63improving the power of unit root tests, Schwert [126] shows that thereare regions of the parameter space under which the size of the augmentedDickey—Fuller test is wrong in small samples. Since the paneltestsarebasedontheaugmentedDickey—Fullertestinsomewayoranother, it is probably the case that this size distortion will get impoundedinto the panel test. To the extent that size distortion is anissue, however, it is not a problem that is speciÞc to the panel tests.2.6 CointegrationTheunitrootprocesses{q t } and {f t } will be cointegrated if there existsa linear combination of the two time-series that is stationary. Tounderstand the implications of cointegration, let’s Þrstlookatwhathappens when the observations are not cointegrated.No cointegration. Let ξ qt = ξ qt−1 + u qt and ξ ft = ξ ft−1 + u ft be ⇐(38)iidtwo independent random walk processes where u qt ∼ N(0, σq)and2 ⇐(39)iidu ft ∼ N(0, σf). 2 Let z t =(z qt ,z ft ) 0 follow a stationary bivariate pro- ⇐(40)cess such as a VAR. The exact process for z t doesn’t need to explicitlymodeled at this point. Now consider the two unit root series built upfrom these componentsq t = ξ qt + z qt ,f t = ξ ft + z ft . (2.87)Since q t and f t are driven by independent random walks, they will driftarbitrarily far apart from each other over time. If you try to Þnd avalue of β to form a stationary linear combination of q t and f t , you willfail becauseq t − βf t =(ξ qt − βξ ft )+(z qt − βz ft ). (2.88)For any value of β, ξ qt − βξ ft =(ũ 1 +ũ 2 + ···ũ t )whereũ t ≡ u qt − βu ftso the linear combination is itself a random walk. q t and f t clearly donot share a long run relationship. There may, however, be short-runinteractions between their Þrst differencesà ! à ! à !∆qt ∆zqt ²qt= + . (2.89)∆f t ∆z ft ² ft

64 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSBy analogy to the derivation of (2.58), if z t follows a Þrst-order VAR,you can show that (2.89) follows a vector ARMA process. Thus, whenboth {q t } and {f t } are unit root processes but are driven by independentrandom walks, they can be Þrst differenced to induce stationarity andtheir Þrst differences modeled as a stationary vector process.Cointegration. {q t } and {f t } will be cointegrated if they are driven byiidthe same random walk, ξ t = ξ t−1 +² t , where ² t ∼ N(0, σ 2 ). For exampleifand you look for a value of β that rendersq t = ξ t + z qt ,f t = φ(ξ t + z ft ), (2.90)q t − βf t =(1− βφ)ξ t + z qt − βφz ft , (2.91)stationary, you will succeed by choosing β = 1 φ since q t − ftφ = z qt − z ftis the difference between two stationary processes so it will itself bestationary. {q t } and {f t } share a long-run relationship. We say thatthey are cointegrated with cointegrating vector (1, − 1 ). Since randomφwalks are sometimes referred to as stochastic trend processes, whentwo series are cointegrated we sometimes say that they share a commontrend. 28The Vector Error-Correction RepresentationRecall that for the univariate AR(2) process, you can rewrite q t =ρ 1 q t−1 + ρ 2 q t−2 + u t in augmented Dickey—Fuller test equation form as∆q t =(ρ 1 + ρ 2 − 1)q t−1 − ρ 2 ∆q t−1 + u t , (2.92)where u tiid∼ N(0, σ 2 u). If q t isaunitrootprocess,then(ρ 1 + ρ 2 − 1) = 0,and (ρ 1 +ρ 2 −1) −1 clearly doesn’t exist. There is in a sense a singularity28 Suppose you are analyzing three variables (q 1t ,q 2t ,q 3t ). If they are cointegrated,there can be at most 2 independent random walks driving the series. If there are 2random walks, there can be only 1 cointegrating vector. If there is only 1 randomwalk, there can be as many as 2 cointegrating vectors.

2.6. COINTEGRATION 65in q t−1 because ∆q t is stationary and this can be true only if q t−1 dropsout from the right side of (2.92).By analogy, suppose that in the bivariate case the vector (q t ,f t )isgenerated according to"qtf t#="a11 a 12a 21#" #qt−1+a 22 f t−1"b11 b 12b 21#" #qt−2+b 22 f t−2" #uqtu ft, (2.93)where (u qt ,u ft ) 0 iid ∼ N(0, Σ u ). Rewrite (2.93) as the vector analog of theaugmented Dickey—Fuller test equation" #∆qt=∆f twhere"r11 r 12r 21#" #qt−1−r 22 f t−1"b11 b 12b 21#" #∆qt−1+b 22 ∆f t−1" # "r11 r 12 a11 + b=11 − 1 a 12 + b 12r 21 r 22 a 21 + b 21 a 22 + b 22 − 1#≡ R." #uqtu ft,(2.94)If {q t } and {f t } areunitrootprocesses,theirÞrst differences are stationary.This means the terms on the right hand side of (2.94) are stationary.Linear combinations of levels of the variables appear in the system.r 11 q t−1 + r 12 f t−1 appears in the equation for ∆q t and r 21 q t−1 + r 22 f t−1appears in the equation for ∆f t .If {q t } and {f t } do not cointegrate, there are no values of the r ijcoefficients that can be found to form stationary linear combinationsof q t and f t . The level terms must drop out. R is the null matrix, and(∆q t , ∆f t ) follows a vector autoregression.If {q t } and {f t } do cointegrate, then there is a unique combinationof the two variables that is stationary. The levels enter on the right sidebut do so in the same combination in both equations. This means thatthe columns of R are linearly dependent and the R, which is singular,can be written as" #r11 −βrR =11.r 21 −βr 21(2.94) can now be written as" #∆qt∆f t=" #r11r 21(q t−1 − βf t−1 ) −"b11 b 12b 21#" #∆qt−1+b 22 ∆f t−1" #uqtu ft

66 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS=" #r11r 21z t−1 −"b11 b 12b 21#" #∆qt−1+b 22 ∆f t−1" #uqtu ft, (2.95)(41)(eq.2.95)(42)⇒(43)(eq.2.96)where z t−1 ≡ q t−1 −βf t−1 is called the error-correction term, and(2.95)is the vector error correction representation (VECM). A VAR in Þrstdifferences would be misspeciÞed because it omits the error correctionterm.To express the dynamics governing z t , multiply the equation for ∆f tby β and subtract the result from the equation for ∆q t to getz t = (1+r 11 − βr 21 )z t−1 − (b 11 + βb 21 )∆q t−1−(b 12 + βb 22 )∆f t−1 + u qt − βu ft . (2.96)(44)(eq.2.97)(45)⇒Theentiresystemisthengivenby⎡⎢⎣∆q t∆f tz t⎤⎥⎦ =⎡⎢⎣+⎤b 11 b 12 r 11⎥b 21 b 22 r 12 ⎦−(b 11 + βb 21 ) −(b 12 + βb 22 ) 1+r 11 − βr 21⎡⎢⎣⎤u qt⎥u ftu qt − βu ft⎡⎢⎣∆q t−1∆f t−1z t−1⎦ . (2.97)(∆q t , ∆f t ,z t ) 0 is a stationary vector, and (2.97) looks like a VAR(1) inthese three variables, except that the columns of the coefficient matrixare linearly dependent. In many applications, the cointegration vector(1, −β) isgivenaprioriby economic theory and does not need to beestimated. In these situations, the linear dependence of the VAR (2.97)tells you that all of the information contained in the VECM is preservedin a bivariate VAR formed with z t and either ∆q t or ∆f t .Suppose you follow this strategy. To get the VAR for (∆q t ,z t ),substitute f t−1 =(q t−1 − z t−1 )/β into the equation for ∆q t to get⎤⎥⎦∆q t = b 11 ∆q t−1 + b 12 ∆f t−1 + r 11 z t−1 + u qt= a 11 ∆q t−1 + a 12 z t−1 + a 13 z t−2 + u qt ,(46)⇒where a 11 = b 11 + b 12β ,a 12 = r 11 − b 12β ,anda 13 = b 12. Similarly, substituteβf t−1 out of the equation for z t to getz t = a 21 ∆q t−1 + a 22 z t−1 + a 23 z t−2 +(u qt − βu ft ),

2.7. FILTERING 67where a 21 = − ³ b 11 + βb 21 + b 12+ b β 22´, a22 =1+r 11 − βr 21 + b 22 + b 12, βand a 23 = − ³ b 22 + b 12´.β Together you have the VAR(2)"∆qtz t#="a11 a 12+a 21"#" #∆qt−1+a 22 z t−1" #" #0 a13 ∆qt−20 a 23 z t−2#u qt. (2.98)u qt − βu ft(2.98) is easier to estimate than the VECM and the standard forecastingformulae for VARs can be employed without modiÞcation.2.7 FilteringMany international macroeconomic time-series contain a trend. Thetrend may be deterministic or stochastic (i.e., aunitrootprocess).Real business cycle (RBC) theories are designed to study the cyclicalfeatures of the data, not the trends. So in RBC research, the data thatis being studied is usually passed through a linear Þlter to remove thelow-frequency or trend component of the data. To understand whatÞltering does to the data you need to have some understanding of thefrequency or spectral representation of time series where we think ofthe observations as being built up from individual subprocesses thatexhibit cycles over different frequencies.Linear Þlters take a possibly two-sided moving average of an originalset of observations q t to create a new series ˜q t˜q t =∞Xj=−∞a j q t−j , (2.99)where the weights are summable, P ∞j=−∞ |a j | < ∞. One way to assesshow the Þlter transforms the properties of the original data is to seewhich frequency components from the original data that are allowed topass through and how these frequency components are weighted—thatis, are the particular frequency components that are allowed throughrelatively more or less important than they were in the original data.

68 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSThe Spectral Representation of a Time SeriesIn section 2.4, a unit-root time series was decomposed into the sum ofa random walk and a stationary AR(1) component. Here, we want tothink of the time-series observations as being built up of underlyingcyclical (cosine) functions each with different amplitudes and exhibitingcycles of different frequencies. A key question in spectral analysisis, which of these frequency components are relatively important indetermining the behavior of the observed time-series?To Þx ideas, begin with the deterministic time-series, q t = a cos(ωt),where time is measured in years. This function exhibits a cycle everyt = 2π years. By choosing values of ω between 0 and π, youcangetωthe process to exhibit cycles at any length that you desire. This isillustrated in Figure 2.1 where q 1t = a cos(t) exhibits a cycle every2π =6.28 years and q 2t = a cos(πt) displays a cycle every 2 years.1.510.5-0.500 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6-1-1.5Figure 2.1: Deterministic Cycles—q 1t = cos(t) (dashed)cyclesevery2π =6.28 years and q 2t =cos(πt) (solid) cycles every 2 years.Something is clearly missing at this point and it is randomness.We introduce uncertainty with a random phase shift. If you compareq 1t = a cos(t) toq 3t = a cos(t + π/2), q 3t is just q 1t with a phase shift(horizontal movement) of π . This phase shift is illustrated in Figure 2.22

2.7. FILTERING 69Now let ˜λ ∼ U[0, π] 29 . Imagine that we take a draw from this distribu-1.510.5-0.500 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6-1-1.5Figure 2.2: π/2Phase shift. Solid: cos(t), Dashed: cos(t + π/2).tion. Let the realization be λ, and form the time-seriesq t = a cos(ωt + λ). (2.100)Once λ is realized, q t is a deterministic function with periodicity 2π and ωphase shift λ but q t is a random function ex ante. We will need thefollowing two basic trigonometric relations.Two useful trigonometric relations. Let b and c be constants, and i bethe imaginary number where i 2 = −1. Thencos(b + c) = cos(b)cos(c) − sin(b)sin(c) (2.101)e ib = cos(b)+i sin(b) (2.102)(2.102) is known as de Moivre’s theorem. You can rearrange it to getcos(b) = (eib + e −ib )2, and sin(b) = (eib − e −ib ). (2.103)2i29 Youonlyneedtoworryabouttheinterval[0, π] because the cosine function issymmetric about zero—cos(x) =cos(−x) for0≤ x ≤ π

70 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSNow let b = ωt and c = λ and use (2.101) to represent (2.100) asq t = a cos(ωt + λ)= cos(ωt)[a cos(λ)] − sin(ωt)[a sin(λ)].Next, build the time-series q t = q 1t + q 2t from the two sub-series q 1t andq 2t ,whereforj =1, 2q jt =cos(ω j t)[a j cos(λ j )] − sin(ω j t)[a j sin(λ j )],and ω 1 < ω 2 . The result is a periodic function which is displayed ontheleftsideofFigure2.3.10.80.60.40.20-0.2-0.4-0.6-0.8-11 6 11 16 21 26 31 363020100-10-20-301 6 11 16 21 26 31 36Figure 2.3: For 0 ≤ ω 1 < ··· < ω N ≤ π, q t = P Nj=1 q jt ,whereq jt =cos(ω j t)[a j cos(λ j )] − sin(ω j t)[a j sin(λ j )]. Left panel: N =2. Rightpanel: N =1000The composite process with N = 2 is clearly deterministic but ifyou build up the analogous series with N = 100 of these components,as shown in the right panel of Figure 2.3, the series begins to look likea random process. It turns out that any stationary random process canbe arbitrarily well approximated in this fashion letting N →∞.

2.7. FILTERING 71To summarize at this point, for sufficiently large number N of theseunderlying periodic components, we can represent a time-series q t asq t =NXj=1cos(ω j t)u j − sin(ω j t)v j , (2.104)where u j = a j cos(λ j )andv j = a j sin(λ j ), E(u 2 i )=σ2 i ,E(u iu j )=0,i 6= j, E(vi 2 )=σi 2 ,E(v i v j )=0,i6= j.Now suppose that E(u i v j ) = 0 for all i, j and let N →∞. 30 Youare carving the interval into successively more subintervals and arecramming more ω j into the interval [0, π]. Since each u j and v j isassociated with an ω j , in the limit, write u(ω) andv(ω) as functionsof ω. For future reference, notice that because cos(−a) =cos(a), wehave u(−ω) = u(ω) whereas because sin(−a) = − sin(a), you havev(−ω) =−v(ω). The limit of sums of the areas in these intervals is theintegralZ πq t = cos(ωt)du(ω) − sin(ωt)dv(ω). (2.105)0Using (2.103), (2.105) can be represented asq t =Z π0e iωt + e −iωtdu(ω) −2Z πe iωt − e −iωtdv(ω) . (2.106)2i0| {z }(a)Let dz(ω) = 1 [du(ω)+idv(ω)]. The second integral labeled (a) canbe2simpliÞed asZ π0e iωt − e −iωtdv(ω) =2i==Z π0Z π0Z π0⇐(49)e iωt − e −iωt à !2dz(ω) − du(ω)2iie −iωt − e iωt(2dz(ω) − du(ω))2Z π(e −iωt − e iωt e iωt − e −iωt)dz(ω)+du(ω).0 2Substitute this last result back into (2.106) and cancel terms to get30 This is in fact not true because E(u i v i ) 6= 0, but as we let N → ∞, theimportance of these terms become negligible.⇐(50)

72 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSq t =Z πe −iωt du(ω)0| {z }(a)Z π+0e iωt dz(ω)| {z }(b)Z π−0e −iωt dz(ω) . (2.107)| {z }(c)Since u(−ω) =u(ω), the term labeled (a) in (2.107) can be written asR π0 e−iωt du(ω) = R 0−π eiωt du(ω). The third term labeled (c) in (2.107) isR π0 e−iωt dz(ω) = 1 R π2 0 e−iωt du(ω)+ 1 R π2 0 ie−iωt dv(ω) = 1 R 02 −π eiωt du(ω) −(51)⇒1 R 02 −π ieiωt dv(ω). Substituting these results back into (2.107) and cancelingterms you get, q t = 1 R 02 −π eiωt [du(ω) +idv(ω)] + R π0 eiωt dz(ω)= R π−π eiωt dz(ω). This is known as the Cramer Representation of q t ,which we restate asq t = limNXN→∞j=1a j cos(ω j t + λ j )=Z π−πe iωt dz(ω). (2.108)The point of all this is that any time-series can be thought of as beingbuilt up from a set of underlying subprocesses whose individualfrequency components exhibit cycles of varying frequency. The otherside of this argument is that you can, in principle, take any time-seriesq t and Þgure out what fraction of its variance is generated from thosesubprocesses that cycle within a given frequency range. The businesscycle frequency, which lies between 6 and 32 quarters is of key interestto, of all people, business cycle researchers.Notice that the process dz(ω) is built up from independent increments.Forcoincidentincrements,youcandeÞne the function s(ω)dωto be(s(ω)dω λ = ωE[dz(ω)dz(λ)] =0 otherwise , (2.109)where an overbar denotes the complex conjugate. 31 Sincee iωt e iωt = cos 2 (ωt) +sin 2 (ωt) = 1 at frequency ω, it follows thatE[e iωt e iωt dz(ω)dz(ω)] = s(ω)dω. That is, s(ω)dω is the variance ofthe ω−frequency component of q t , and is called the spectral densityfunction of q t . Since by (2.108), q t is built up from frequency componentsranging from [−π, π], the total variance of q t must be the integral31 If a and b are real numbers and z = a + bi is a complex number, the complexconjugate of z is ¯z = a − bi. The product z¯z = a 2 + b 2 is real.

2.7. FILTERING 73of s(ω). That is 32E(q 2 t ) = E[= E[==Z π−πZ π Z π−πZ π−πZ π−πZ πe iωt dz(ω)−π−πe iλt dz(λ)]e iωt e iλt dz(ω)dz(λ)]E[dz(ω)dz(λ)]s(ω)dω. (2.110)The spectral density and autocovariance generating functions. Theautocovariancegenerating function for a time series q t is deÞned to beg(z) =∞Xj=−∞γ j z j ,where γ j =E(q t q t−j ) is the j-th autocovariance of q t .Ifweletz = e −iω ,thenZ1 πg(e −iω )e iωk dω = 1 ∞XZ πγ j e iω(k−j) dω.2π −π2πj=−∞−πLet a = k−j. Thene iωa =cos(ωa)+i sin(ωa) and the integral becomes,R π−π cos(ωa)dω +i R π−π sin(ωa)dω =(1/a)sin(aω)|π −π −(i/a)cos(aω)| π −π.The second term is 0 because cos(−aπ) = cos(aπ). The Þrst termis 0 too because the sine of any nonzero integer multiple of π is 0and a is an integer. Therefore, the only value of a that matters isa = k − j = 0, which implies that γ k = 1 R π2π −π g(e−iω )e iωk dω. Settingk =0,youhaveγ 0 =Var(q t )= 1 R π2π −π g(e−iω )dω, but you know that ⇐(52)Var(q t )= R π−π s(ω)dω, so the spectral density function is proportionalto the autocovariance generating function with z = e −iω . Notice also,that when you set ω =0,thens(0) = P ∞j=−∞ γ j . The spectral densityfunction of q t at frequency 0 is the same thing as the long-run varianceof q t . It follows thatZ πVar(q t )= s(ω)dω = 1 Z πg(e −iω )dω, (2.111)−π 2π −πwhere g(z) = P ∞j=−∞ γ j z j .⇐(53)32 We obtain the last equality because dz(ω) is a process with independent incrementsso unless λ = ω, Edz(ω)dz(λ) =0.

74 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSLinear FiltersYou can see how a Þlter changes the character of a time series by comparingthe spectral density function of the original observations withthat of the Þltered data.Let the original data q t have the Wold moving-average representation,q t = b(L)² t where b(L) = P ∞j=0 b j L j and ² t ∼ iid with E(² t )=0and Var(² t )=σ 2 ² . The k-th autocovariance isγ k = E(q t q t−k )=E[b(L)² t b(L)² t−k ]⎛⎞∞X X ∞= E⎝b j ² t−j b s ² t−s−k⎠ = σ²2j=0s=0⎛∞X⎝j=0and the autocovariance generating function for q t isg(z) === σ 2 ²⎛∞X∞X ∞Xγ k z k = σ²2 ⎝k=−∞ k=−∞ j=0⎛∞X ∞X⎝σ ²2k=−∞ j=0⎛∞X⎝ b j z j X ∞j=0 k=jb j b j−k⎞⎠ z kb j b j−k⎞⎠ z k z j z −j = σ 2 ²∞X∞Xk=−∞ j=0b j−k z −(j−k) ⎞⎠ = σ 2 ² b(z)b(z −1 ).b j b j−k⎞⎠ ,b j z j b j−k z −(j−k)(54)⇒But from (2.111), you know that s(ω) = g(eiω ). To summarize, these2πresults, the spectral density of q t can be represented ass(ω) = 12π g(e−iω )= 12π σ2 ² b(e −iω )b(e iω ). (2.112)(55)⇒Let the transformed (Þltered) data be given by ˜q t = a(L)q t wherea(L) = P ∞j=−∞ a j L j . Then ˜q t = a(L)q t = a(L)b(L)² t = ˜b(L)² t ,where˜b(L) =a(L)b(L). Clearly, the autocovariance generating function ofthe Þltered data is ˜g(z) =σ²˜b(z)˜b(z 2 −1 )=σ² 2 a(z)b(z)b(z −1 )a(z −1 )=a(z)a(z −1 )g(z), and letting z = e −iω , the spectral density function ofthe Þltered data is˜s(ω) =a(e −iω )a(e iω )s(ω). (2.113)

2.7. FILTERING 75The Þlter has the effect of scaling the spectral density of the originalobservations by a(e −iω )a(e iω ). Depending on the properties of the Þlter,some frequencies will be magniÞed while others are downweighted.One way to classify Þlters is according to the frequencies that areallowed to pass through and those that are blocked. A high pass Þlterlets through only the high frequency components. A low pass Þlterallows through the trend or growth frequencies. A business cycle passÞlter allows through frequencies ranging from 6 to 32 quarters. Themost popular Þlter used in RBC research is the Hodrick—Prescott Þlter,which we discuss next.The Hodrick—Prescott FilterHodrick and Prescott [76] assume that the original series q t is generatedby the sum of a trend component (τ t ) and a cyclical (c t )component,q t = τ t + c t . The trend is a slow-moving low-frequency component andis in general not deterministic. The objective is to construct a Þlterto to get rid of τ t from the data. This leaves c t ,whichisthepartofthe data to be studied. The problem is that for each observation q t ,there are two unknowns (τ t and c t ). The question is how to identifythe separate components?The cyclical part is just the deviation of the original series from thelong-run trend, c t = q t − τ t . Suppose your identiÞcation scheme is tominimize the variance of the cyclical part. You would end up settingits variance to 0 which means setting τ t = q t . This doesn’t help atall—the trend is just as volatile as the original observations. It thereforemakes sense to attach a penalty for making τ t too volatile. Do this byminimizing the variance of c t subject to a given amount of prespeciÞed‘smoothness’ in τ t . Since ∆τ t is like the Þrst derivative of the trendand ∆ 2 τ t is like the second derivative of the trend, one way to get asmoothly evolving trend is to force the Þrst derivative of the trend toevolve smoothly over time by limiting the size of the second derivative.This is what Hodrick and Prescott suggest. Choose a sequence of points{τ t } to minimizeTXt=1(q t − τ t ) 2 + λTX−1t=1(∆ 2 τ t+1 ) 2 , (2.114)

76 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSwhere λ is the penalty attached to the volatility of the trend component.For quarterly data, researchers typically set λ = 1600. 33 Noting that∆ 2 τ t+1 = τ t+1 − 2τ t + τ t−1 ,differentiate (2.114) with respect to τ t andre-arrange the Þrst-order conditions to get the Euler equationsq 1 − τ 1 = λ[τ 3 − 2τ 2 + τ 1 ],q 2 − τ 2 = λ[τ 4 − 4τ 3 +5τ 2 − 2τ 1 ],. .q t − τ t = λ[τ t+2 − 4τ t+1 +6τ t − 4τ t−1 + τ t−2 ], t =3,...,T − 2. .q T −1 − τ T −1 = λ[−2τ T +5τ T −1 − 4τ T −2 + τ T −3 ],q T − τ T = λ[τ T − 2τ T −1 + τ T −2 ].Let c =(c 1 ,...,c T ) 0 ,q=(q 1 ,...,q T ) 0 ,andτ =(τ 1 ,...,τ T ) 0 ,andwritethe Euler equations in matrix formwhere the T × T matrix G is given byG =⎡⎢⎣q =(λG + I T )τ, (2.115)1 −2 1 0 ··· ··· 0−2 5 −4 1 0 ··· ··· 01 −4 6 −4 1 0 ··· ··· 00 1 −4 6 −4 1 0.. .. . .. .0 0 1 −4 6 −4 1 0. 0 1 −4 6 −4 1. 0 1 −4 5 −20 ··· ··· 0 1 −2 1Get the trend component by τ =(λG+I T ) −1 q. The cyclical componentfollows by subtracting the trend from the original observationsc = q − τ =[I T − (λG + I T ) −1 ]q.33 The following derivation of the Þlter follows Pederson [121].⎤⎥⎦.

2.7. FILTERING 77Properties of the Hodrick—Prescott FilterFor t = 3,...,T − 2, the Euler equations can be writtenq t − τ t = λu(L)τ t ,whereu(L) =(1− L) 2 (1 − L −1 ) 2 = P 2j=−2 u j L j ⇐(56)with u −2 = u 2 =1,u −1 = u 1 = −4, and u 0 = 6. We note for futurereference that c t = q t − τ t implies that c t = λu(L)τ t .You’ve already determined that q t =(λu(L)+1)τ t = v(L)τ t wherev(L) =1+λu(L) =1+λ(1 − L) 2 (1 − L −1 ) 2 , so it follows thatτ t = v(L) −1 q t =q t1+λ(1 − L) 2 (1 − L −1 ) 2 .v −1 (L) is the trend Þlter. Once you compute τ t , subtract the resultfrom the data, q t to get c t . This is equivalent to forming c t = δ(L)q twhereδ(L) =1− v −1 (L) = λ(1 − L)2 (1 − L −1 ) 21+λ(1 − L) 2 (1 − L −1 ) . 2Since (1 − L) 2 (1 − L −1 )=L −2 (1 − L) 4 ,theÞlter is equivalent to Þrst ⇐(57)applying (1 − L) 4 on q t , and then applying λL −2 v −1 (L) ontheresult. 34⇐(58)This means the Hodrick-Prescott Þlter can induce stationary into thecyclical component from a process that is I(4).The spectral density function of the cyclical component is s c (ω) =δ(e −iω )δ(e iω )s q (ω), whereδ(e −iω )= λ[(1 − e−iω )(1 − e iω )] 2λ[(1 − e −iω )(1 − e iω )] 2 +1 .From our trigonometric identities, (1−e −iω )(1−e iω )=2(1−cos(ω)), itfollows that δ(ω) =4λ[1−cos(ω)]2 . Each frequency of the original series4λ[1−cos(ω)] 2 +1is therefore scaled by |δ(ω)| 2 = h i4λ(1−cos(ω)) 2. 24λ(1−cos(ω)) 2 +1 This scaling factor isplottedinFigure2.4.34 This is shown in King and Rebelo (84).

78 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS1.210.80.60.40.20-3.1-2.8-2.4-2.1-1.7-1.4-1.0-0.7-0.30.00.40.71.11.41.82.12.52.8FrequencyFigure 2.4: Scale factor |δ(ω)| 2 for cyclical component in the Hodrick—Prescott Þlter.

Chapter 3The Monetary ModelThe monetary model is central to international macroeconomic analysisand is a recurrent theme in this book. The model identiÞes a set of underlyingeconomic fundamentals that determine the nominal exchangerate in the long run. The monetary model was originally developed asa framework to analyze balance of payments adjustments under Þxedexchange rates. After the breakdown of the Bretton Woods system themodel was modiÞed into a theory of nominal exchange rate determination.The monetary approach assumes that all prices are perfectly ßexibleand centers on conditions for stock equilibrium in the money market.Although it is an ad hoc model, we will see in chapters 4 and 9 thatmany predictions of the monetary model are implied by optimizingmodels both in ßexible price and in sticky price environments. Themonetary model also forms the basis for work on target zones (chapter10) and in the analysis of balance of payments crises (chapter 11).A note on notation: Throughout this chapter the level of a variablewill be denoted in upper case letters and the natural logarithm in lowercase. The only exception to this rule is that the level of the interestrate is always denoted in lower case. Thus i t is the nominal interestrate and in logs, s t is the nominal exchange rate in American terms,p t is the price level, y t is real income. Stars are used to denote foreigncountry variables.79

80 CHAPTER 3. THE MONETARY MODEL3.1 Purchasing-Power ParityA key building block of the monetary model is purchasing-power parity(PPP), which can be motivated according to the Casellian approach orby the commodity-arbitrage view.Cassel’s ApproachThe intellectual origins of PPP began in the early 1800s with the writingsof Wheatly and Ricardo. These ideas were subsequently revivedby Cassel [22]. The Casselian approach begins with the observationthat the exchange rate S is the relative price of two currencies. Sincethe purchasing power of the home currency is 1/P and the purchasingpower of the foreign currency is 1/P ∗ , in equilibrium, the relative valueof the two currencies should reßect their relative purchasing powers,S = P/P ∗ .What is the appropriate deÞnition of the price level? The Casselianview suggests using the general price level. Whether the general pricelevel samples prices of non-traded goods or not is irrelevant. As aresult, the consumer price index (CPI) is typically used in empiricalimplementations of this theory. The following passage from Cassel isused by Frenkel [60] to motivate the use of the CPI in PPP research.“Some people believe that Purchasing Power Paritiesshould be calculated exclusively on price indices for suchcommodities as for the subject of trade between the twocountries. This is a misinterpretation of the theory . . . Thewhole theory of purchasing power parity essentially refersto the internal value of the currencies concerned, and variationsin this value can be measured only by general indexÞgures representing as far as possible the whole mass ofcommodities marketed in the country.”The theory implies that the log real exchange rate q ≡ s + p ∗ − pis constant over time. However, even casual observation rejects thisprediction. Figure 3.1 displays foreign currency values of the US dollarand PPPs relative to four industrialized countries formed from CPIs

3.1. PURCHASING-POWER PARITY 81US-UK1.210.80.60.40.2073 75 77 79 81 83 85 87 89 91 93 95 97-0.2US-Germany0.80.70.60.50.40.30.20.10-0.1-0.273 75 77 79 81 83 85 87 89 91 93 95 97US-Japan1.41.210.80.60.40.20-0.2-0.473 75 77 79 81 83 85 87 89 91 93 95 97US-Switzerland1.210.80.60.40.20-0.273 75 77 79 81 83 85 87 89 91 93 95 97Figure 3.1: Log nominal exchange rates (boxes) and CPI-based PPPs(solid).expressed in logarithms over the ßoating period. Figure 3.2 shows theanalogous series for the US and UK over a long historical period extendingfrom 1871 to 1997. While there are protracted periods in whichthe nominal exchange rate deviates from the PPP, the two series tendto revert towards each other over time.As a result, international macroeconomists view Casselian PPP asa theory of the long-run determination of the exchange rate in whichthe PPP (p − p ∗ ) is a long-run attractor for the nominal exchange rate.

82 CHAPTER 3. THE MONETARY MODEL6040200-20-40-60-80Nominal Exchange Rate (solid)PPPs from CPIs (boxes)-100-1201871 1882 1893 1904 1915 1926 1937 1948 1959 1970 1981 1992Figure 3.2: US—UK log nominal exchange rates and CPI-based PPPsmultiplied by 100. 1871-1997.The Commodity-Arbitrage ApproachThe commodity-arbitrage view of PPP, articulated by Samuelson [124],simply holds that the law-of-one price holds for all internationally tradedgoods. Thus if the law-of-one price holds for the goods individually, itwill hold for the appropriate price index as well. Here, the appropriateprice index should cover only those goods that are traded internationally.It can be argued that the producer price index (PPI) is a betterchoice for studying PPP since it is more heavily weighted towardstraded goods than the CPI which includes items such as housing serviceswhich do not trade internationally. We will consider empiricalanalyses on PPP in chapter 7.PPP is clearly violated in the short run. Casual observation ofFigures 3.1 and 3.2 suggest however that PPP may hold in the longrun. There exists econometric evidence to support long-run PPP, butwe will defer discussion of these issues until chapter 7.In spite of the obvious short-run violations, PPP is one of the buildingblocks in the monetary model and as we will see in the Lucas model

3.2. THE MONETARY MODEL OF THE BALANCE OF PAYMENTS83(chapter 4) and in the Redux model (chapter 9) as well. Why is that? ⇐(60)One reason frequently given is that we don’t have a good theory for whyPPP doesn’t hold so there is no obvious alternative way to provide internationalprice level linkages. A second and perhaps more convincingreason is that all theories involve abstractions that are false at somelevel and as Friedman [64] argues, we should judge a theory not by therealism of its assumptions but by the quality of its predictions.3.2 The Monetary Model of the Balanceof PaymentsThe Frenkel and Johnson [62] collection develops the monetary approachto the balance of payments under Þxed exchange rates. Toillustrate the main idea, consider a small open economy that maintainsa perfectly credible Þxed exchange rate ¯s. 1 i t is the domestic nominalinterest rate, B t is the monetary base, R t is the stock of foreignexchange reserves held by the central bank, D t is domestic credit extendedby the central bank. In logarithms, m t is the money stock,y t is national income, and p t is the price level. The money supply isM t = µB t = µ(R t +D t )whereµ is the money multiplier. A logarithmicexpansion of the money supply and its components about their meanvalues allows us to writem t = θr t +(1− θ)d t (3.1)where θ =E(R t )/E(B t ), r t =ln(R t ), and d t =ln(D t ). 2A transactions motive gives rise to the demand for money in whichlog real money demand m d t −p t depends positively on y t and negativelyon the opportunity cost of holding money i tm d t − p t = φy t − λi t + ² t . (3.2)1 A small open economy takes world prices and world interest rates as given.2 A Þrst-order expansion about mean values givesM t − E(M t ) = µ[R t − E(R t )] + µ[D t − E(D t )]. But µ = E(M t )/E(B t ) whereB t = R t + D t is the monetary base. Now divide both sides by E(M t )toget[M t − E(M t )]/E(M t )=θ[R t − E(R t )]/E(R t )+(1 − θ)[D t − E(D t )]/E(D t ). Notingthat for a random variable X t ,[X t − E(X t )]/E(X t ) ' ln(X t ) − ln(E(X t )), apartfrom an arbitrary constant, we get (3.1) inthetext.

84 CHAPTER 3. THE MONETARY MODEL0 < φ < 1 is the income elasticity of money demand, 0 < λ is theiidinterest semi-elasticity of money demand, and ² t ∼ (0, σ² 2 ).Assume that purchasing-power parity (PPP) and uncovered interestparity (UIP) hold. Since the exchange rate is Þxed, PPP implies thatthe price level p t =¯s + p ∗ t is determined by the exogenous foreign pricelevel. Because the Þx is perfectly credible, market participants expectno change in the exchange rate and UIP implies that the interest ratei t = i ∗ t is given by the exogenous foreign interest rate. Assume that themoney market is continuously in equilibrium by equating m d t in (3.2)to m t in (3.1) and rearranging to getθr t =¯s + p ∗ t + φy t − λi ∗ t − (1 − θ)d t + ² t . (3.3)(3.3) embodies the central insights of the monetary approach to thebalance of payments. If the home country experiences any one or acombination of the following: a high rate of income growth, declininginterest rates, or rising prices, the demand for nominal money balanceswill grow. If money demand growth is not satisÞed by an accommodatingincrease in domestic credit d t , the public will obtain theadditional money by running a balance of payments surplus and accumulatinginternational reserves. If, on the other hand, the central bankengages in excessive domestic credit expansion that exceeds money demandgrowth, the public will eliminate the excess supply of money byrunning a balance of payments deÞcit.We will meet this model again in chapters 10 and 11 in the study oftarget zones and balance of payments crises. In the remainder of thischapter, we develop the model as a theory of exchange rate determinationin a ßexible exchange rate environment.3.3 The Monetary Model under FlexibleExchange RatesThe monetary model of exchange rate determination consists of a pairof stable money demand functions, continuous stock equilibrium in themoney market, uncovered interest parity, and purchasing-power parity.

3.3. THE MONETARY MODEL UNDER FLEXIBLE EXCHANGE RATES85Under ßexible exchange rates, the money stock is exogenous. Equilibriumin the domestic and foreign money markets are given bym t − p t = φy t − λi t , (3.4)m ∗ t − p∗ t = φyt ∗ − λi∗ t , (3.5)where 0 < φ < 1 is the income elasticity of money demand, and λ > 0is the interest rate semi-elasticity of money demand. Money demandparameters are identical across countries.International capital market equilibrium is given by uncovered interestparityi t − i ∗ t =E ts t+1 − s t , (3.6)where E t s t+1 ≡ E(s t+1 |I t ) is the expectation of the exchange rate atdate t+1 conditioned on all public information I t , available to economic ⇐(61)agents at date t.Price levels and the exchange rate are related through purchasingpowerparitys t = p t − p ∗ t . (3.7)To simplify the notation, callf t ≡ (m t − m ∗ t ) − φ(y t − y ∗ t )the economic fundamentals. Now substitute (3.4), (3.5), and (3.6) into(3.7) to gets t = f t + λ(E t s t+1 − s t ), (3.8)and solving for s t giveswheres t = γf t + ψE t s t+1 , (3.9)γ ≡ 1/(1 + λ),ψ ≡ λγ = λ/(1 + λ).(3.9) is the basic Þrst-order stochastic difference equation of the monetarymodel and serves the same function as an ‘Euler equation’ inoptimizing models. It says that expectations of future values of the

86 CHAPTER 3. THE MONETARY MODELexchange rate are embodied in the current exchange rate. High relativemoney growth at home leads to a weakening of the home currencywhile high relative income growth leads to a strengthening of the homecurrency.Next, advance time by one period in (3.9) to gets t+1 = γf t+1 +ψE t+1 s t+2 . Take expectations conditional on time t informationand use the law of iterated expectations to getE t s t+1 = γE t f t+1 + ψE t s t+2 and substitute back into (3.9). Now dothis again for s t+2 ,s t+3 ,...,s t+k ,andyougets t = γkXj=0(ψ) j E t f t+j +(ψ) k+1 E t s t+k+1 . (3.10)Eventually, you’ll want to drive k →∞butindoingsoyouneedtospecify the behavior the term (ψ) k E t s t+k .The fundamentals (no bubbles) solution. Since ψ < 1, you obtain theunique fundamentals (no bubbles) solution by restricting the rate atwhich the exchange rate grows by imposing the transversality conditionlimk→∞ (ψ)k E t s t+k =0, (3.11)which limits the rate at which the exchange rate can grow asymptotically.If the transversality condition holds, let k →∞in (3.10) to getthe present-value formulas t = γ∞Xj=0(ψ) j E t f t+j (3.12)The exchange rate is the discounted present value of expected futurevalues of the fundamentals. In Þnance, the present value model is apopular theory of asset pricing. There, s is the stock price and f is theÞrm’s dividends. Since the exchange rate is given by the same basicformula as stock prices, the monetary approach is sometimes referredto as the ‘asset’ approach to the exchange rate. According to thisapproach, we should expect the exchange rate to behave just like theprices of other assets such as stocks and bonds. From this perspectiveit will come as no surprise that the exchange rate more volatile than

3.3. THE MONETARY MODEL UNDER FLEXIBLE EXCHANGE RATES87the fundamentals, just as stock prices are much more volatile thandividends. Before exploring further the relation between the exchangerate and the fundamentals, consider what happens if the transversalitycondition is violated.Rational bubbles. If the transversality condition does not hold, it ispossible for the exchange rate to be governed in part by an explosivebubble {b t } that will eventually dominate its behavior. To see why, letthe bubble evolve according tob t =(1/ψ)b t−1 + η t , (3.13)where η tiid∼ N(0, σ 2 η). The coefficient (1/ψ) exceeds 1 so the bubbleprocess is explosive. Now add the bubble to the fundamental solution(3.12) and call the resultŝ t = s t + b t . (3.14)You can see that ŝ t violates the transversality condition by substituting(3.14) into (3.11) to getψ t+k E t ŝ t+k = ψ t+k E t s t+k +ψ t+k E t b t+k = b t .| {z }0However, ŝ t is a solution to the model, because it solves (3.9). You cancheck this out by substituting (3.14) into (3.9) to gets t + b t =(ψ/λ)f t + ψ[E t S t+1 +(1/ψ)b t ].The b t terms on either side of the equality cancel out so ŝ t is indeed isanother solution to (3.9) but the bubble will eventually dominate andwill drive the exchange rate arbitrarily far away from the fundamentalsf t . The bubble arises in a model where people have rational expectationsso it is referred to as a rational bubble. What does a rationalbubble look like? Figure 3.3 displays a realization of a ŝ t for 200 timeperiods where ψ =0.99 and the fundamentals follow a driftless randomwalk with innovation variance 0.035 2 . Early on, the exchange rateseems to return to the fundamentals but the exchange rate diverges astime goes on.

88 CHAPTER 3. THE MONETARY MODEL25201510exchange ratewith bubble50-5fundamentals-101 26 51 76 101 126 151 176Figure 3.3: A realization of a rational bubble where ψ =0.99, and thefundamentals follow a random walk. The stable line is the realization of thefundamentals.Now it may be the case that the foreign exchange market is occasionallydriven by bubbles but real-world experience suggests that suchbubbles eventually pop. It is unlikely that foreign exchange marketsare characterized by rational bubbles which do not pop. As a result,we will focus on the no-bubbles solution from this point on.3.4 Fundamentals and Exchange Rate VolatilityA major challenge to international economic theory is to understandthe volatility of the exchange rate in relation to the volatility of theeconomic fundamentals. Let’s Þrst take a look at the stylized factsconcerning volatility. Then we’ll examine how the monetary model isable to explain these facts.

3.4. FUNDAMENTALS AND EXCHANGE RATE VOLATILITY 89Table 3.1: Descriptive statistics for exchange-rate and equity returns,and their fundamentals.AutocorrelationsMean Std.Dev. Min. Max. ρ 1 ρ 4 ρ 8 ρ 16ReturnsS&P 2.75 5.92 -13.34 18.31 0.24 -0.10 0.15 0.09UKP 0.41 5.50 -13.83 16.47 0.12 0.03 0.01 -0.29DEM 0.46 6.35 -13.91 15.74 0.09 0.23 0.04 -0.07YEN 0.73 6.08 -15.00 16.97 0.13 0.18 0.06 -0.29Deviation from fundamentalsDiv. 1.31 0.30 0.49 1.82 1.01 1.03 1.05 0.94UKP 0 0.18 -0.46 0.47 0.89 0.61 0.25 -0.12DEM 0 0.31 -0.61 0.59 0.98 0.91 0.77 0.55YEN 0 0.38 -0.85 0.50 0.98 0.88 0.76 0.68Notes: Quarterly observations from 1973.1 to 1997.4. Percentage returns on theStandard and Poors composite index (S&P) and its log dividend yield (Div.) arefrom Datastream. Percentage exchange rate returns and deviation of exchange ratefrom fundamentals (s t −f t )withf t =(m t −m ∗ t )−(y t−yt ∗)arefromtheInternationalFinancial Statistics CD-ROM. (s t − f t ) are normalized to have zero mean. The USdollar is the numeraire currency. UKP is the UK pound, DEM is the deutschemark,and YEN is the Japanese yen.Stylized Facts on Volatility and Dynamics.Some descriptive statistics for dollar quarterly returns on the pound,deutsche-mark, yen are shown in the Þrst panel of Table 3.1. To underscorethe similarity between the exchange rate and equity prices,the table also includes statistics for the Standard and Poors compositestock price index. The second panel displays descriptive statistics forthe deviation of the respective asset prices from their fundamentals.For equities, this is the S&P log dividend yield. For currency values, itis the deviation of the exchange rate from the monetary fundamentals, ⇐(62)f t − s t have been normalized to have mean 0. The volatility of a timeseries is measured by its sample standard deviation.

90 CHAPTER 3. THE MONETARY MODELThe main points that can be drawn from the table are1. The volatility of exchange rate returns ∆s t is virtually indistinguishablefrom stock return volatility.2. Returns for both stocks and exchange rates have low Þrst-orderserial correlation.3. From our discussion about the properties of the variance ratiostatistic in chapter 2.4, the negative autocorrelations in exchangerate returns at 16 quarters suggest the possibility of mean reversion.4. The deviation of the price from the fundamentals display substantialpersistence, and much less volatility than returns. Thebehavior of the dividend yield, while similar to the behavior of theexchange rate deviations from the monetary fundamentals, displaysslightly more persistence and appears to be nonstationaryoverthesampleperiod.The data on returns and deviations from the fundamentals are showninFigure3.4whereyouclearlyseehowtheexchangerateisexcessivelyvolatile in comparison to its fundamentals.Excess Volatility and the Monetary ModelThe monetary model can be made consistent with the excess volatilityin the exchange rate if the growth rate of the fundamentals is apersistent stationary process.(63)⇒iid∆f t = ρ∆f t−1 + ² t . (3.15)with ² t ∼ N(0, σ² 2 ). The implied k−step ahead prediction formulaeare E t (∆f t+k )=ρ k ∆f t . Converting to levels, you get E t (f t+k )=f t +P ki=1ρ i ∆f t = f t +[(1−ρ k )/(1−ρ)]ρ∆f t . Using these prediction formulaein (3.12) givess t = γ∞Xj=0ψ j f t + γ∞Xj=0ψ j1 − ρ ρ∆f t − γ∞Xj=0(ρψ) j1 − ρ ρ∆f t= f t + ρψ1 − ρψ ∆f t, (3.16)

3.5. TESTING MONETARY MODEL PREDICTIONS 91wherewehaveusedthefactthatγ =1− ψ. Some additional algebrarevealsVar(∆s t )= (1 − ρψ)2 +2ρψ(1 − ρ)Var(∆f(1 − ρψ) 2 t ) > Var(∆f t ).This is not very encouraging since the levels of the fundamentals areexplosive. The end-of-chapter problems show that neither an AR(1) nora permanent—transitory components representation (chapter 2.4) forthe fundamentals allows the monetary model to explain why exchangerate returns are more volatile than the growth rate of the fundamentals.3.5 Testing Monetary Model PredictionsThis section looks at two empirical strategies for evaluating the monetarymodel of exchange rates.MacDonald and Taylor’s TestThe Þrst strategy that we look at is based on MacDonald and Taylor’s[96] adaptation of Campbell and Shiller’s [20] tests of the presentvalue model. 3 This section draws on material on cointegration presentedin chapter 2.6.Let I t be the time t information set available to market participants.Subtracting f t from both sides of (3.8) givess t − f t = λE(s t+1 − s t |I t )=λ(i t − i ∗ t ). (3.17)s t is by all indications a unit-root process, whereas ∆s t and E(∆s t+1 |I t )are clearly stationary. It follows from the Þrst equality in (3.17) thats t and f t must be cointegrated. Using (3.12) and noting that ψ = λγgives⎛λE t (∆s t+1 ) = λ ⎝γ∞Xj=0ψ j E t f t+1+j − γ∞Xj=0ψ j E t f t+j⎞⎠3 The seminal contributions to this literature are Leroy and Porter [90] andShiller [127].

92 CHAPTER 3. THE MONETARY MODEL=∞Xj=1ψ j E t ∆f t+j . (3.18)(3.17) and (3.18) allow you to represent the deviation of the exchangerate from the fundamental as the present value of future fundamentalsgrowthζ t = s t − f t =∞Xj=1ψ j E t ∆f t+j . (3.19)Since s t and f t are cointegrated they can be represented by a vectorerror correction model (VECM) that describes the evolution of(∆s t , ∆f t , ζ t ), where ζ t ≡ s t − f t . As shown in chapter 2.6, the lineardependence among (∆s t , ∆f t , ζ t ) induced by cointegration impliesthat the information contained in the VECM is preserved in a bivariatevector autoregression (VAR) that consists of ζ t and either ∆s t or ∆f t .Thus we will drop ∆s t and work with the p−th order VAR for (∆f t , ζ t )Ã∆ftζ t!=à pX a11,j a 12,jj=1a 21,j!à !∆ft−j+a 22,j ζ t−jòtv t!. (3.20)The information set available to the econometrician consists of currentand lagged values of ∆f t and ζ t . We will call this informationH t = {∆f t , ∆f t−1 ,...,ζ t , ζ t−1 ,...}. Presumably H t is a subset of economicagent’s information set, I t . Take expectations on both sides of(3.19) conditional on H t and use the law of iterated expectations toget 4 ∞Xζ t = ψ j E(∆f t+j |H t ). (3.21)j=1What is the point of deriving (3.21)? The point is to show that youcan use the prediction formulae implied the data-generating process(3.20) to compute the necessary expectations. Expectations of marketparticipants E(∆f t+j |I t ) are unobservable but you can still test thetheory by substituting the true expectations with your estimate of theseexpectations, E(∆f t+j |H t ).4 Let X, Y, and Z be random variables. The law of iterated expectations saysE[E(X|Y,Z)|Y ]=E(X|Y ).

3.5. TESTING MONETARY MODEL PREDICTIONS 93To simplify computations of the conditional expectations of futurefundamentals growth, reformulate the VAR in (3.20) in the VAR(1)companion formY t = BY t−1 + u t , (3.22)whereB =⎛⎜⎝Y t =⎛⎜⎝∆f t∆f t−1.∆f t−p+1ζ tζ t−1.ζ t−p+1⎞⎟⎠, u t =⎛⎜⎝² t0.0v t0.0⎞⎟⎠a 11,1 a 11,2 ··· a 11,p a 12,1 a 12,2 ··· a 12,p1 0 ··· 0 0 0 ··· 00 1 0··· 0 0 0 ··· 0. ··· ··· . . . . .0 ··· ···1 0 0 ··· ··· 0a 21,1 a 21,2 ··· a 21,p a 22,1 a 22,2 ··· a 22,p0 ··· ··· 0 1 0 ··· 00 ··· ··· 0 0 1 0··· 0. ··· ··· . . . . .0 ··· ··· 0 0 ··· ···1 0Now let e 1 be a (1 × 2p) row vector with a 1 in the Þrst element andzeros elsewhere and let e 2 be a (1 × 2p) rowvectorwitha1asthep +1−th element and zeros elsewheree 1 =(1, 0,...,0), e 2 =(0,...,0, 1, 0,...,0).,⎞⎟⎠These are selection vectors that give⇐(65)e 1 Y t = ∆f t , e 2 Y t = ζ t .Now the k-step ahead forecast of f t is conveniently expressed asE(∆f t+j |H t )=e 1 E(Y t+j |H t )=e 1 B j Y t . (3.23)

94 CHAPTER 3. THE MONETARY MODELSubstitute (3.23) into (3.21) to getζ t = e 2 Y t =∞Xj=1= e 1⎛⎝ψ j e 1 B j Y t∞Xj=1ψ j B j ⎞⎠ Y t (3.24)= e 1 ψB(I − ψB) −1 Y t .Equating coefficients on elements of Y t yields a set of nontrivial restrictionspredicted by the theory which can be subjected to statisticalhypothesis testse 2 (I − ψB) =e 1 ψB. (3.25)Estimating and Testing the Present-Value ModelWe use quarterly US and German observations on the exchange rate,money supplies and industrial production indices from the InternationalFinancial Statistics CD-ROM from 1973.1 to 1997.4, to re-estimate theMacDonald and Taylor formulation and test the restrictions (3.25). Weview the US as the home country. The bivariate VAR is run on (∆f t , ζ t )with observations demeaned prior to estimation. The fundamentals aregiven by f t =(m t −m ∗ t )−(y t −y ∗ t ) where the income elasticity of moneydemand is Þxed at φ =1.The BIC (chapter 2.1) tells us that a VAR(4) is the appropriate.Estimation proceeds by letting x 0 t =(∆f t−1 ,...,∆f t−4 , ζ t−1 ,...,ζ t−4 )and running least squares on∆f t = x 0 t β + ² t,ζ t = x 0 t δ + v t.Expanding (3.25) and making the correspondence between the coefficientsin the matrix B and the regressions, we write out the testablerestrictions explicitly asβ 1 + δ 1 =0, β 5 + δ 5 =1/ψ,β 2 + δ 2 =0, β 6 + δ 6 =0,β 3 + δ 3 =0, β 7 + δ 7 =0,β 4 + δ 4 =0, β 8 + δ 8 =0.

3.5. TESTING MONETARY MODEL PREDICTIONS 95These restrictions are tested for a given value of the interest semielasticityof money demand, λ = ψ/(1 − ψ). To set up the Wald test,let π 0 =(β 0 , δ 0 ) be the grand coefficient vector from the OLS regressions, ⇐(70)R =(I 8 : I 8 ) be the restriction matrix and r 0 =(0, 0, 0, 0, (1/ψ), 0, 0, 0),Ω T = Σ T ⊗ Q −1T ,whereΣ T = 1 P²t ² 0 T t, Q T = 1 Pxt x 0 T t. Then asT →∞, the Wald statisticW =(Rπ − r) 0 [RΩ T R 0 ] −1 (Rπ − r) D ∼ χ 2 8 .Here are the results. The Wald statistics and their associated valuesof λ are W = 284, 160(λ = 0.02), W = 113, 872(λ = 0.10), W =44, 584(λ =0.16), and W =18, 291(λ =0.25). The restrictions arestrongly rejected for reasonable values of λ.One reason why the model fares poorly can be seen by comparing thetheoretically implied deviation of the spot rate from the fundamentals˜ζ t = e 1 ψB(I − ψB) −1 Y t ,which is referred to as the ‘spread’ with the actual deviation, ζ t = s t −f t .ThesearedisplayedinFigure3.5whereyoucanseethattheimpliedspread is much too smooth.Long-Run Evidence for the Monetary Model fromPanel DataThe statistical evidence against the rational expectations monetarymodel is pretty strong. One of the potential weak points of the modelis that PPP is assumed to hold as an exact relationship when it isprobably more realistic to think that it holds in the long run.Mark and Sul [101] investigate the empirical link between the monetarymodel fundamentals and the exchange rate using quarterly observationsfor 19 industrialized countries from 1973.1 to 1997.4 and thepanel exchange rate predictive regression⇐(72)s it+k − s it = βζ it + η it+k , (3.26)where η it+k = γ i + θ t+k + u it+k has an error-components representa- ⇐(73)tion with individual effect γ i ,commontimeeffect θ t and idiosyncratic

96 CHAPTER 3. THE MONETARY MODELTable 3.2: Monetary fundamentals out-of-sample forecasts of US dollarreturns with nonparametric bootstrapped p-values under cointegration.1-quarter ahead 16-quarters aheadCountry U-statistic p-value U-statistic p-valueAustralia 1.024 0.904 0.864 0.222Austria 0.984 0.013 0.837 0.131Belgium 0.999 0.424 0.405 0.001Canada 0.985 0.074 0.552 0.009Denmark 1.014 0.912 0.858 0.174Finland 1.001 0.527 0.859 0.164France 0.994 0.155 0.583 0.004Germany 0.986 0.056 0.518 0.003Great Britain 0.983 0.077 0.570 0.012Greece 1.016 0.909 1.046 0.594Italy 0.997 0.269 0.745 0.016Japan 1.003 0.579 0.996 0.433Korea 0.912 0.002 0.486 0.012Netherlands 0.986 0.041 0.703 0.032Norway 0.998 0.380 0.537 0.002Spain 0.996 0.341 0.672 0.028Sweden 0.975 0.034 0.372 0.001Switzerland 0.982 0.008 0.751 0.049Mean 0.991 0.010 0.686 0.001Median 0.995 0.163 0.688 0.001Notes: Bold face indicate statistical signiÞcance at the 10 percent level.

3.5. TESTING MONETARY MODEL PREDICTIONS 97effect u it+k . A panel combines the time-series observations of severalcross-sectional units. The individuals in the cross section are differentcountries which are indexed by i =1,...,N.Out-of-Sample Fit and PredictionMark and Sul’s primary objective is to use the regression to generateout-of-sample forecasts of the depreciation. They base their methodologyon the work of Meese and Rogoff [104] who sought to evaluate theempirical performance of alternative exchange rate models that werepopular in the 1970s by conducting a monthly postsample Þt analysis.Suppose there are j =1,...J models under consideration. Let x j tbe a vector of exchange rate determinants implied by ‘model j,’ ands t = x 0 jt β j + e j t be regression representation of model j. What Meeseand Rogoff did was to divide the complete size T (time-series) samplein two. Sample 1 consists of observations t =1,...t 1 and sample 2consists of observations t = t 1 +1,...,T,wheret 1

98 CHAPTER 3. THE MONETARY MODELthey have a tendency to fall apart out of sample. There are many possibleexplanations for the instability, but ultimately, the reason boilsdown to the failure to Þnd a time-invariant relationship between the exchangerate and the fundamentals. Although their conclusions regardingthe importance of macroeconomic fundamentals for the exchangerate were nihilistic, Meese and Rogoff established a rigorous tradition ininternational macroeconomics of using out-of-sample Þt or forecastingperformance as model evaluation criteria.Panel Long-Horizon RegressionLet’s return to Mark and Sul’s analysis. They evaluate the predictivecontent of the monetary model fundamentals by initially estimating theregression on observations through 1983.1. Note that the regressand in(3.26) are past (not contemporaneous) deviations of the exchange ratefrom the fundamentals. It is a predictive regression that generates actualout-of-sample forecasts. The k = 1 regression is used to forecast 1-quarter ahead, and the k = 16 regression is used to forecast 16 quartersahead. The sample is then updated by one observation and a new setof forecasts are generated. This recursive updating of the sample andforecast generation is repeated until the end of the data set is reached.β = 0 if the monetary fundamentals contain no predictive content or iftheexchangerateandthefundamentalsdonotcointegrate.Let observations T − T 0 to T be sample reserved for forecast evaluation.If ŝ it+k − s it is the k−step ahead regression forecast formed at t,the root-mean-square prediction error (RMSPE) of the regression isvuT XR 1 = t 1 (ŝ it − s it−k )T 2 .0 t=T 0The monetary fundamentals regression is compared to the random walkiidwith drift, s it+1 = µ i + s it + ² it where ² it ∼ (0, σi 2 ). The k−step aheadforecasted change from the random walk is ŝ it+k − s it = kµ i .LetR 2 bethe random walk model’s RMSPE. Theil’s [134] statistic U = R 1 /R 2 isthe ratio of the RMSPE of the two models. The regression outperformsthe random walk in prediction accuracy when U

3.5. TESTING MONETARY MODEL PREDICTIONS 99for a test of the hypothesis that the regression and the random walkmodels give equally accurate predictions. There is a preponderanceof statistically superior predictive performance by the monetary modelexchange rate regression.

100 CHAPTER 3. THE MONETARY MODEL20Stock Returns and Log Dividend Yield20Dollar/Pound15151050-5-101050-5-10-15-1573 75 77 79 81 83 85 87 89 91 93 95 97-2073 75 77 79 81 83 85 87 89 91 93 95 97Dollar/Deutschemark20151050-5-10-15-2073 75 77 79 81 83 85 87 89 91 93 95 97Dollar/Yen20151050-5-10-15-2073 75 77 79 81 83 85 87 89 91 93 95 97Figure 3.4: Quarterly stock and exchange rate returns (jagged line),1973.1 through 1997.4, with price deviations from the fundamentals(smooth line).

3.5. TESTING MONETARY MODEL PREDICTIONS 1010.80.60.4Actual0.20Theoretical-0.2-0.4-0.6-0.874 76 78 80 82 84 86 88 90 92 94 96Figure 3.5: Theoretical and actual spread, s t − f t .

102 CHAPTER 3. THE MONETARY MODELMonetary Model Summary1. The monetary model builds on purchasing-power parity, uncoveredinterest parity, and stable transactions-based money demandfunctions.2. Domestic and foreign money and real income levels are the fundamentaldeterminants of the nominal exchange rate.3. The exchange rate is viewed as the relative price of two monies,which are assets. Since asset prices are in general more volatilethan their fundamentals, it comes as no surprise that exchangerates exhibit excess volatility. The present value form of thesolution underscores the concept that the exchange rate is anasset price.4. The monetary model is a useful Þrst approximation in Þxingour intuition about exchange rate dynamics even though it failsto explain the data on many dimensions. Because purchasingpower parity is assumed to hold as an exact relationship, themodel cannot explain the dynamics of the real exchange rate.Indeed, the main reason to study nominal exchange rate behavioris if we think that nominal exchange rate movements arecorrelated with real exchange rate changes so that they havereal consequences.

3.5. TESTING MONETARY MODEL PREDICTIONS 103ProblemsLet the fundamentals have the permanent—transitory components representationf t = ¯f t + z t , (3.27)where ¯f t = ¯f iidt−1 + ² t is the permanent part with ² t ∼ N(0, σ² 2 )andz t =iidρz t−1 +u t is the transitory part with u t ∼ N(0, σu 2 ), and 0 < ρ < 1. Notethatthe time-t expectation of a random walk k periods ahead is E t ( ¯f t+k )= ¯f t ,and the time-t expectation of the AR(1) partk periods ahead is E t z t+k =ρ k z t .(3.27)impliesthek-step ahead prediction formula E t (f t+k )= ¯f t +ρ k z t .1. Show thats t = ¯f t +11 + λ(1 − ρ) z t. (3.28)2. Suppose that the fundamentals are stationary by setting σ ² =0. Thenthe permanent part ¯f t drops out and the fundamentals are governedby a stationary AR(1) process. Show thatµ1 2Var(s t )=Var(f t ), (3.29)1 + λ(1 − ρ)3. Let’s restore the unit root component in the fundamentals by settingσ² 2 > 0butturnoff the transitory part by setting σ2 u =0. Nowthefundamentals follow a random walk and the exchange rate is givenexactlybythefundamentalss t = f t . (3.30)Theexchangerateinheritstheunitrootfromf t . Since unit rootprocesses have inÞnite variances, we should take Þrst differences toinduce stationarity. Doing so and taking the variance, (3.30) predictsthat the variance of the exchange rate is exactly equal to the varianceof the fundamentals.Now re-introduce the transitory part σu 2 > 0. Show that depreciationof the home currency is∆s t = ² t + (ρ − 1)z t−1 + u t. (3.31)1 + λ(1 − ρ)

104 CHAPTER 3. THE MONETARY MODELwhereVar(∆s t )=σ² 2 + 2(1 − ρ)[1 + λ(1 − ρ)] 2 Var(z t).Why does the variance of the depreciation still not exceed the varianceof the fundamentals growth?

Chapter 4The Lucas ModelThe present-value interpretation of the monetary model underscores theideathatweshouldexpecttheexchangeratetobehavelikethepricesof other assets–such as stocks and bonds. This is one of that model’sattractive features. One of its unattractive features is that the modelis ad hoc in the sense that the money demand functions upon whichit rests were not shown to arise explicitly from decisions of optimizingagents. Lucas’s [95] neoclassical model of exchange rate determinationgives a rigorous theoretical framework for pricing foreign exchange andother assets in a ßexible price environment and is not subject to thiscriticism. It is a dynamic general equilibrium model of an endowmenteconomy with complete markets where the fundamental determinantsoftheexchangeratearethesameasthoseinthemonetarymodel.The economic environment for dynamic general equilibrium analysisneeds to be speciÞed in some detail. To make this task manageable,we will begin by modeling the real part of the economy that operatesunder a barter system. We will obtain a solution for the real exchangerate and real stock-pricing formulae. This perfect-markets real generalequilibrium model is sometimes referred to as an Arrow [3]—Debreu [34]model because it can be mapped into their static general equilibriumframework. We know that the Arrow—Debreu competitive equilibriumyields a Pareto Optimum. Why is this connection useful? Because ittells us that we can understand the behavior of the market economy bysolving for the social optimum and it is typically more straightforwardto obtain the social optimum than to directly solve for the market105

106 CHAPTER 4. THE LUCAS MODEL(76)⇒equilibrium.In order to study the exchange rate, we need to have a monetaryeconomy. The problem is that there is no role for Þat money in theArrow—Debreu environment. The way that Lucas gets around thisproblem is to require people to use money when they buy goods. Thisrequirement is called a ‘cash-in-advance’ constraint and is a popularstrategy for introducing money in general equilibrium along the linesof the transactions motive for holding money. A second popular strategythat puts money in the utility function will be developed in chapter9.The models we will study in this chapter and in chapter 5 haveno market imperfections and exhibit no nominal rigidities. Marketparticipants have complete information and rational expectations. Whystudy such a perfect world? First, we have a better idea for solvingfrictionless and perfect-markets models so it is a good idea to start infamiliar territory. Naturally, these models of idealized economies willnot fully explain the real world. So we want to view these models asproviding a benchmark against which to measure progress. If and whenthe data ‘reject’ these models, take one should note the manner in whichthey are rejected to guide the appropriate extensions and reÞnementsto the theory.There is a good deal of notation for the model which is summarizedin Table 4.1.4.1 The Barter EconomyConsider two countries each inhabited by a large number of individualswho have identical utility functions and identical wealth. People maybelieve that they are individuals but the respond in the same way tochanges in incentives. Because people are so similar you can normalizethe constant populations of each country to 1 and model the people ineach country by the actions of a single representative agent (household)in Lucas model. This is the simplest way to aggregate across individualsso that we can model macroeconomic behavior.‘Firms’ in each country are pure endowment streams that generatea homogeneous nonstorable country-speciÞc good using no labor or

4.1. THE BARTER ECONOMY 107capital inputs. Some people like to think of these Þrms as fruit trees.You can also normalize the number of Þrms in each country to 1. x tis the exogenous domestic output and y t is the exogenous foreign output.The evolution of output is given by x t = g t x t−1 at home and byy t = gt ∗ y t−1 abroad where g t and gt∗ are random gross rates of changethat evolve according to a stochastic process that is known by agents.Each Þrm issues one perfectly divisible share of common stock whichis traded in a competitive stock market. The Þrms pay out all of theiroutput as dividends to shareholders. Dividends form the sole source ofsupport for individuals. We will let x t be the numeraire good and q tbe the price of y t in terms of x t . e t is the ex-dividend market value ofthe domestic Þrm and e ∗ t is the ex-dividend market value of the foreignÞrm.The domestic agent consumes c xt units of the home good, c yt unitsof the foreign good and holds ω xt shares of the domestic Þrm and ω ytshares of the foreign Þrm. Similarly, the foreign agent consumes c ∗ xt,units of the home good, c ∗ yt units of the foreign good and holds ωxt∗shares of the domestic Þrm and ωyt ∗ shares of the foreign Þrm.The domestic agent brings into period t wealth valued atW t = ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ), (4.1)where x t + e t and q t y t + e ∗ t arethewith-dividendvalueofthehomeandforeign Þrms. The individual then allocates current wealth towards newshare purchases e t ω xt + e ∗ t ω yt , and consumption c xt + q t c ytW t = e t ω xt + e ∗ t ω y t+ c xt + q t c yt . (4.2)Equating (4.1) to (4.2) gives the consolidated budget constraintc xt + q t c yt + e t ω xt + e ∗ t ω y t= ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ). (4.3)Let u(c xt ,c yt ) be current period utility and 0 < β < 1 be the subjectivediscount factor. The domestic agent’s problem then is to choose sequencesof consumption and stock purchases, {c xt+j ,c yt+j, ω xt+j , ω yt+j } ∞ j=0,to maximize expected lifetime utility⎛∞XE t⎝j=0⎞β j u(c xt+j ,c yt+j ) ⎠ , (4.4)

108 CHAPTER 4. THE LUCAS MODELsubject to (4.3).You can transform the constrained optimum problem into an unconstrainedoptimum problem by substituting c xt from (4.3) into (4.4).Theobjectivefunctionbecomesu(ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ) − e t ω xt − e ∗ t ω yt − q t c yt ,c yt )+E t [βu(ω xt (x t+1 + e t+1 )+ω yt (q t+1 y t+1 + e ∗ t+1)(4.5)−e t+1 ω xt+1 − e ∗ t+1ω yt+1 − q t+1 c yt+1 ,c yt+1 )] + ···(77)⇒Let u 1 (c xt ,c yt )=∂u(c xt ,c yt )/∂c xt be the marginal utility of x-consumptionand u 2 (c xt ,c yt )=∂u(c xt ,c yt )/∂c yt be the marginal utility of y-consumption.Differentiating (4.5) with respect to c yt , ω xt ,andω yt , setting the resultto zero and rearranging yields the Euler equationsc yt : q t u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.6)ω xt : e t u 1 (c xt ,c yt )=βE t [u 1 (c xt+1 ,c yt+1 )(x t+1 + e t+1 )], (4.7)ω yt : e ∗ t u 1 (c xt ,c yt )=βE t [u 1 (c xt+1 ,c yt+1 )(q t+1 y t+1 + e ∗ t+1)]. (4.8)These equations must hold if the agent is behaving optimally. (4.6)is the standard intratemporal optimality condition that equates therelative price between x and y to their marginal rate of substitution.Reallocating consumption by adding a unit of c y increases utility byu 2 (·). This is Þnanced by giving up q t units of c x , each unit of whichcosts u 1 (·) units of utility for a total utility cost of q t u 1 (·). If the individualis behaving optimally, no such reallocations of the consumptionplan yields a net gain in utility.(4.7) is the intertemporal Euler equation for purchases of the domesticequity. The left side is the utility cost of the marginal purchaseof domestic equity. To buy incremental shares of the domestic Þrm, itcosts the individual e t units of c x , each unit of which lowers utility byu 1 (c xt ,c yt ). The right hand side of (4.7) is the utility expected to bederived from the payoff of the marginal investment. If the individualis behaving optimally, no such reallocations between consumption andsaving can yield a net increase in utility. An analogous interpretationholds for intertemporal reallocations of consumption and purchases ofthe foreign equity in (4.8).

4.1. THE BARTER ECONOMY 109The foreign agent has the same utility function and faces the analogousproblem to maximizesubject to⎛∞XE t⎝j=0β j u(c ∗ xt+j ,c∗ yt+j ) ⎞⎠ , (4.9)c ∗ xt + q tc ∗ y t+ e t ωxt ∗ + e∗ t ω∗ y t= ωxt−1 ∗ (x t + e t )+ωyt−1 ∗ (q ty t + e ∗ t ). (4.10)The analogous set of Euler equations for the foreign individual arec ∗ yt : q tu 1 (c ∗ xt ,c∗ yt )=u 2(c ∗ xt ,c∗ yt ), (4.11)ωxt ∗ : e tu 1 (c ∗ xt ,c∗ yt )=βE t[u 1 (c ∗ xt+1 ,c∗ yt+1 )(x t+1 + e t+1 )], (4.12)ωyt ∗ : e ∗ t u 1 (c ∗ xt,c ∗ yt) =βE t [u 1 (c ∗ xt+1,c ∗ yt+1)(q t+1 y t+1 + e ∗ t+1)].(4.13)A set of four adding up constraints on outstanding equity shares andthe exhaustion of output in home and foreign consumption completethe speciÞcation of the barter modelω xt + ωxt ∗ = 1, (4.14)ω yt + ωyt ∗ = 1, (4.15)c xt + c ∗ xt = x t , (4.16)c yt + c ∗ yt = y t . (4.17)Digression on the social optimum. You can solve the model by grindingout the equilibrium, but the complete markets and competitive settingmakes available a ‘backdoor’ solution strategy of solving the problemconfronting a Þctitious social planner. The stochastic dynamic bartereconomy can conceptually be reformulated in terms of a static competitivegeneral equilibrium model—the properties of which are well known.The reformulation goes like this.We want to narrow the deÞnition of a ‘good’ so that it is deÞnedprecisely by its characteristics (whether it is an x−good or a y−good),the date of its delivery (t), and the state of the world when it is delivered(x t ,y t ). Suppose that there are only two possible values for x t (y t )in

110 CHAPTER 4. THE LUCAS MODELeach period–a high value x h (y h )andalowvaluex`(y`). Then thereare 4 possible states of the world (x h ,y h ), (x h ,y`), (x`,y h ), and (x`,y`).‘Good 1’ is x delivered at t = 0 in state 1. ‘Good 2’ is x deliveredat t = 0 in state 2, ‘good 8’ is y delivered at t =1instate4,andso on. In this way, all possible future outcomes are completely spelledout. The reformulation of what constitutes a good corresponds to acomplete system of forward markets. Instead of waiting for nature toreveal itself over time, we can have people meet and contract for allfuture trades today (Domestic agents agree to sell so many units of xto foreign agents at t = 2 if state 3 occurs in exchange for q 2 units of y,and so on.) After trades in future contingencies have been contracted,we allow time to evolve. People in the economy simply fulÞll theircontractual obligations and make no further decisions. The point isthat the dynamic economy has been reformulated as a static generalequilibrium model.Since the solution to the social planner’s problem is a Pareto optimalallocation and you know by the fundamental theorems of welfareeconomics that the Pareto Optimum supports a competitive equilibrium,it follows that the solution to the planner’s problem will alsodescribe the equilibrium for the market economy. 1Wewillletthesocialplannerattachaweightofφ to the homeindividual and 1 − φ to the foreign individual. The planner’s problemis to allocate the x and y endowments optimally between the domesticand foreign individuals each period by maximizingE t ∞ Xj=0β j h φu(c xt+j ,c yt+j )+(1− φ)u(c ∗ xt+j ,c∗ yt+j )i , (4.18)subject to the resource constraints (4.16) and (4.17). Since the goodsare not storable, the planner’s problem reduces to the timeless problemof maximizingφu(c xt ,c yt )+(1− φ)u(c ∗ xt,c ∗ yt),1 Under certain regularity conditions that are satisÞed in the relatively simpleenvironments considered here, the results from welfare economics that we need are,i) A competitive equilibrium yields a Pareto Optimum, and ii) Any Pareto Optimumcan be replicated by a competitive equilibrium.

4.1. THE BARTER ECONOMY 111subject to (4.16) and (4.17). The Euler equations for this problem areφu 1 (c xt ,c yt ) = (1− φ)u 1 (c ∗ xt,c ∗ yt), (4.19)φu 2 (c xt ,c yt ) = (1− φ)u 2 (c ∗ xt ,c∗ yt ). (4.20)(4.19) and (4.20) are the optimal or efficient risk-sharing conditions.Risk-sharing is efficient when consumption is allocated so that themarginal utility of the home individual is proportional, and thereforeperfectly correlated, to the marginal utility of the foreign individual.Because individuals enjoy consuming both goods and the utility functionis concave, it is optimal for the planner to split the available x andy between the home and foreign individuals according to the relativeimportance of the individuals to the planner.The weight φ can be interpreted as a measure of the size of the homecountry in the market version of the world economy. Since we assumedat the outset that agents have equal wealth, we will let both agents beequally important to the planner and set φ =1/2. Then the Paretooptimal allocation is to split the available output of x and y equallyc xt = c ∗ xt = x t2 , and c yt = c ∗ yt = y t2 .Having determined the optimal quantities, to get the market solutionwe look for the competitive equilibrium that supports this Pareto optimum.The market equilibrium. If agents owned only their own country’s Þrms,individuals would be exposed to idiosyncratic country-speciÞc riskthatthey would prefer to avoid. The risk facing the home agent is that thehome Þrm experiences a bad year with low output of x when the foreignÞrm experiences a good year with high output of y. One way to insureagainst this risk is to hold a diversiÞed portfolio of assets.AdiversiÞcation plan that perfectly insures against country-speciÞcrisk and which replicates the social optimum is for each agent to holdstock in half of each country’s output. 2 The stock portfolio that achieves2 Agents cannot insure against world-wide macroeconomic risk (simultaneouslylow x t and y t ).

112 CHAPTER 4. THE LUCAS MODELcomplete insurance of idiosyncratic risk is for each individual to ownhalf of the domestic Þrm and half of the foreign Þrm 3ω xt = ω ∗ xt = ω yt = ω ∗ yt = 1 2 . (4.21)We call this a ‘pooling’ equilibrium because the implicit insurancescheme at work is that agents agree in advance that they will pooltheir risk by sharing the realized output equally.The solution under constant relative-risk aversion utility. Let’s adopta particular functional form for the utility function to get explicit solutions.We’ll let the period utility function be constant relative-riskaversion in C t = c θ xtcyt1−θ , a Cobb-Douglas index of the two goodsThenu(c x ,c y )= C1−γ t1 − γ . (4.22)u 1 (c xt ,c yt )= θC1−γ t,c xt(1 − θ)C1−γ tu 2 (c xt ,c yt )= .c ytand the Euler equations (4.6)—(4.13) becomeq t = 1 − θ x t, (4.23)θ y t" µCt+1 e (1−γ)Ãt= βE t 1+ e !#t+1, (4.24)x t x t+1C te ∗ " µCt+1t= βE tq t y t C t (1−γ)Ã!#1+ e∗ t+1. (4.25)q t+1 y t+1(79)⇒From (4.23) the real exchange rate q t is determined by relative outputlevels. (4.24) and (4.25) are stochastic difference equations in the ‘pricedividend’ratios e t /x t and e ∗ t /(q t y t ). If you iterate forward on them as3 Actually, Cole and Obstfeld [31]) showed that trade in goods alone are sufficientto achieve efficient risk sharing in the present model. These issues are dealt with inthe end-of-chapter problems.

4.2. THE ONE-MONEY MONETARY ECONOMY 113you did in (3.9) for the monetary model, the equity price—dividend ratiocan be expressed as the present discounted value of future consumptiongrowth raised to the power 1 − γ. You can then get an explicit solutiononce you make an assumption about the stochastic process governingoutput. This will be covered in section 4.5 below.An important point to note is that there is no actual asset tradingin the Lucas model. Agents hold their investments forever and neverrebalance their portfolios. The asset prices produced by the model areshadow prices that must be respected in order for agents to willingly tohold the outstanding equity shares according to (4.21).4.2 The One-Money Monetary EconomyIn this section we introduce a single world currency. The economicenvironment can be thought of as a two-sector closed economy. Theidea is to introduce money without changing the real equilibrium thatwe characterized above. One of the difficulties in getting money intothe model is that the people in the barter economy get along just Þnewithout it. An unbacked currency in the Arrow—Debreu world that generatesno consumption payoffs will not have any value in equilibrium.To get around this problem, Lucas prohibits barter in the monetaryeconomy and imposes a ‘cash-in-advance’ constraint that requires peopleto use money to buy goods. As we enter period t the followingspeciÞc cash-in-advance transactions technology must be adhered to.1. x t and y t are revealed.2. λ t , the exogenous stochastic gross rate of change in money is revealed.The total money supply M t , evolves according toM t = λ t M t−1 . The economy-wide increment ∆M t =(λ t −1)M t−1 ,is distributed evenly to the home and foreign individuals whereeach agent receives the lump-sum transfer ∆Mt =(λ2 t − 1) M t−1.23. A centralized securities market opens where agents allocate theirwealth towards stock purchases and the cash that they will need topurchase goods for consumption. To distinguish between the aggregatemoney stock M t and the cash holdings selected by agents,

114 CHAPTER 4. THE LUCAS MODELdenote individual’s choice variables by lower case letters, m t andm ∗ t . Securities market closes.4. Decentralized goods trading now takes place in the ‘shoppingmall.’ Each household is split into ‘worker—shopper’ pairs. Theshopper takes the cash from security markets trading and buysx and y−goods from other stores in the mall (shoppers are notallowed to buy from their own stores). The home-country workercollects the x− endowment and offers it for sale in an x−goodstore in the ‘mall.’ The y−goods come from the foreign country‘worker’ in the foreign country who collects and sells they−endowment in the mall. The goods market closes.5. The cash value of goods sales are distributed to stockholders asdividends. Stockholders carry these nominal dividend paymentsinto the next period.The state of the world is the gross growth rate of home output, foreignoutput, and money (g t ,g ∗ t , λ t ), and is revealed prior to trading.Because the within-period uncertainty is revealed before any tradingtakes place, the household can determine the precise amount of moneyit needs to Þnance the current period consumption plan. As a result,it is not necessary to carry extra cash from one period to the next. Ifthe (shadow) nominal interest rate is always positive, households willmake sure that all the cash is spent each period. 4To formally derive the domestic agent’s problem, let P t be the nominalprice of x t . Current-period wealth is comprised of dividends fromlast period’s goods sales, the market value of ex-dividend equity shares4 It may seem strange to talk about the interest rate and bonds since individualsdo not hold nor trade bonds. That is because bonds are redundant assets in thecurrent environment and consequently are in zero net supply. But we can computethe shadow interest rate to keep the bonds in zero net supply. The equilibriuminterest rate is such that individuals have no incentive either to issue or to buynominal debt contracts. We will use the model to price nominal bonds at the endof this section.

4.2. THE ONE-MONEY MONETARY ECONOMY 115and the lump-sum monetary transferW t = P t−1(ω xt−1 x t−1 + ω yt−1 q t−1 y t−1 )P t| {z }Dividends+ ω xt−1 e t + ω yt −1e ∗ t| {z }+Ex-dividend share values∆M t2P t| {z }Money transfer. (4.26)In the securities market, the domestic household allocates W t towardscash m t to Þnance shopping plans and to equitiesW t = m t+ ω xt e t + ω yt e ∗ tP . (4.27)tThe household knows that the amount of cash required to Þnance thecurrent period consumption plan ism t = P t (c xt + q t c yt ). (4.28)The cash-in-advance constraint is said to bind. Substituting (4.28) into(4.27), and equating the result to (4.26) eliminates m t and gives thesimpler consolidated budget constraintc xt + q t c yt + ω xt e t + ω yt e ∗ t = P t−1P t[ω xt−1 x t−1 + ω yt−1 q t−1 y t−1 ]+ ∆M t2P t+ ω xt−1 e t + ω yt−1 e ∗ t . (4.29)The domestic household’s problem is therefore to maximize⎛∞XE t⎝j=0⎞β j u(c xt+j ,c yt+j ) ⎠ , (4.30)subject to (4.29). As before, the terms that matter at date t areu(c xt ,c yt )+βE t u(c xt+1 ,c yt+1 ),so you can substitute (4.29) into the utility function to eliminate c xt andc xt+1 and to transform the problem into one of unconstrained optimization.The Euler equations characterizing optimal household behaviorare⇐(81-83)

116 CHAPTER 4. THE LUCAS MODELc yt : q t u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.31)Ã !#Ptω xt : e t u 1 (c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) x t + e t+1 , (4.32)P t+1Ã !#ω yt : e ∗ t u Pt1(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) q t y t + e ∗ t+1 .(4.33)P t+1The foreign household solves an analogous problem. Using the foreigncash-in-advance constraintm ∗ t = P t(c ∗ t + q tc ∗ yt ). (4.34)the consolidated budget constraint for the foreign household is(84-86)⇒c ∗ xt + q t c ∗ yt + ω ∗ xte t + ω ∗ yte ∗ t = P t−1P t[ω ∗ xt−1x t−1 + ω ∗ yt−1q t−1 y t−1 ]The job is to maximize+ ∆M t2P t+ ω ∗ xt−1e t + ω ∗ yt−1e ∗ t . (4.35)⎛∞XE t⎝j=0β j u(c ∗ xt+j,c ∗ yt+j)subject to (4.35).The foreign household’s problem generates a symmetric set of Eulerequationsc ∗ yt : q tu 1 (c ∗ xt ,c∗ yt )=u 2(c ∗ xt ,c∗ yt ),ω ∗ xt : e t u 1 (c ∗ xt,c ∗ yt) =βE t"u 1 (c ∗ xt+1,c ∗ yt+1)"⎞⎠ ,ÃPtP t+1x t + e t+1!#,Ãωyt ∗ : e∗ t u 1(c ∗ xt ,c∗ yt )=βE t u 1 (c ∗ xt+1 ,c∗ yt+1 ) Ptq t y t + e ∗ t+1P t+1The adding-up constraints that complete the model are1 = ω xt + ωxt,∗1 = ω yt + ωyt,∗M t = m t + m ∗ t ,x t = c xt + c ∗ xt ,y t = c yt + c ∗ yt.!#.

4.2. THE ONE-MONEY MONETARY ECONOMY 117To solve the model, aggregate the cash-in-advance constraints over thehome and foreign agents and use the adding-up constraints to getM t = P t (x t + q t y t ). (4.36)This is the quantity equation for the world economy where velocity isalways 1. The single money generates no new idiosyncratic countryspeciÞcrisk. The equilibrium established for the barter economy (constantand equal portfolio shares) is still the perfect risk-pooling equilibriumω xt = ω ∗ xt = ω yt = ω ∗ yt = 1 2 ,c xt = c ∗ xt = x t2 ,c yt = c ∗ yt = y t2 .The only thing that has changed are the equity pricing formulae, whichnow incorporate an ‘inßation premium.’ The inßation premium arisesbecause the nominal dividends of the current period must be carriedover into the next period at which time their real value can potentiallybe eroded by an inßation shock.Solution under constant relative risk aversion utility. Under the utilityfunction (4.22), the real exchange rate is q t = h i³1−θ xtθ y t´. Substituting ⇐(87)this into (4.36), the inverse of the gross inßation rate is PtP t+1= Mt x t+1M t+1 x t.Together, these expressions can be used to rewrite the equity pricingequations as" µCt+1e t= βE tx tC te ∗ " µCt+1t= βE tq t y t C t (1−γ)ÃMtM t+1+ e t+1x t+1!#, (4.37) (1−γ)Ã !#Mt+ e∗ t+1. (4.38)M t+1 q t+1 y t+1To price nominal bonds, you are looking for the shadow price of a hypotheticalnominal bond such that the public willingly keeps it in zero netsupply. Let b t be the nominal price of a bond that pays one dollar at theend of the period. The utility cost of buying the bond is u 1 (c xt ,c yt )b t /P t .

118 CHAPTER 4. THE LUCAS MODELIn equilibrium, this is offset by the discounted expected marginal utilityof the one-dollar payoff, βE t [u 1 (c xt+1 ,c yt+1 )/P t+1 ]. Under the constantrelative risk aversion utility function (4.22) we have" µCt+1 (1−γ)#M tb t = βE t . (4.39)C t M t+1If i t is the nominal interest rate, then b t =(1+i t ) −1 . Nominal interestrates will be positive in all states of nature if b t < 1 and is likely to betrue when the endowment growth rate and monetary growth rates arepositive.4.3 The Two-Money Monetary EconomyTo address exchange rate issues, you need to introduce a second nationalcurrency. Let the home country money be the ‘dollar’ and theforeign country money be the ‘euro.’ We now amend the transactionstechnology to require that the home country’s x—goods can only bepurchased with dollars and the foreign country’s y—goods can only bepurchased with euros. In addition, x−dividends are paid out in dollarsand y−dividends are paid out in euros. Agents can acquire the foreigncurrency required to Þnance consumption plans during securitiesmarket trading.Let P t be the dollar price of x, Pt∗ be the euro price of y, andS tbetheexchangerateexpressedasthedollarpriceofeuros. M t is theoutstanding stock of dollars, N t is the outstanding stock of euros andthey evolve over time according toM t = λ t M t−1 , and N t = λ ∗ t N t−1,where (λ t , λ ∗ t ) are exogenous random gross rates of change in M andN.If the domestic household received transfers only of M, it faces foreignpurchasing-power risk because it it also needs N to buy y-goods.Introducing the second currency creates a new country-speciÞcriskthathouseholds will want to hedge. The complete markets paradigm allowsmarkets to develop whenever there is a demand for a product. The

4.3. THE TWO-MONEY MONETARY ECONOMY 119products that individuals desire are claims to future dollar and eurotransfers. 5 So to develop this idea, let r t be the price of a claim toall future dollar transfers in terms of x and rt∗ be the price to all futureeuro transfers in terms of x. Let there be one perfectly divisibleclaim outstanding for each of these monetary transfer streams. Let thedomestic agent hold ψ Mt claims on the dollar streams and ψ Nt claimson the euro streams whereas the foreign agent holds ψMt ∗ claims onthe dollar stream and ψNt ∗ claims on the euro stream. Initially, thehome agent is endowed with ψ M =1, ψ N = 0 and the foreign agent hasψN ∗ =1, ψM ∗ = 0 which they are free to trade.Now to develop the problem confronting the domestic household,note that current-period wealth consists of nominal dividends paid fromequity ownership carried over from last period, current period monetarytransfers the market value of equity and monetary transfer claimsW t = P t−1ω xt−1 x t−1 + S tPt−1∗ω yt−1 y t−1P tP| {z t}Dividends+ ψ Mt−1∆M t+ ψ Nt−1S t ∆N tP tP| {z t}Monetary Transfers+ ω xt−1 e t + ω yt−1 e ∗ t + ψ Mt−1r t + ψ Nt−1 rt∗ . (4.40)| {z }Market value of securitiesThis wealth is then allocated to stocks, claims to future monetary transfers,dollars and euros for shopping in securities market trading accordingtoW t = ω xt e t + ω yt e ∗ t + ψ Mt r t + ψ Nt rt ∗ + m t+ n tS t. (4.41)P t P tThe current values of x t ,y t ,M t , and N t are revealed before trading occursso domestic households acquire the exact amount of dollars andeuros required to Þnance current period consumption plans. In equilibrium,we have the binding cash-in-advance constraintsm t = P t c xt , (4.42)5 In the real world, this type of hedge might be constructed by taking appropriatepositions in futures contracts for foreign currencies.

120 CHAPTER 4. THE LUCAS MODELn t = Pt ∗ c yt, (4.43)which you can use to eliminate m t and n t from the allocation of currentperiod wealth to rewrite (4.41) asW t = c xt + S tPt∗P tc yt| {z }Goods+ ω xt e t + ω yt e ∗ t| {z }Equity+ ψ Mt r t + ψ Nt rt∗ | {z }. (4.44)Money transfersThe consolidated budget constraint of the home individual is thereforec xt + S tPt∗P t+ S tP ∗c yt + ω xt e t + ω yt e ∗ t + ψ Mtr t + ψ Nt rt ∗ = P t−1ω xt−1 x t−1P tt−1ω yt−1 y t−1 + ψ Mt−1∆M t+ ψ Nt−1S t ∆N tP tP tP t+ω xt−1 e t + ω yt−1 e ∗ t + ψ xt−1r t + ψ yt−1 rt ∗ . (4.45)The domestic household’s problem is to maximize⎛∞XE t⎝j=0⎞β j u(c xt+j ,c yt+j ) ⎠ (4.46)(88-92)⇒subject to (4.45). The associated Euler equations areS t Pt∗c yt : u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.47)P tà !#Ptω xt : e t u 1 (c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) x t + e t+1 , (4.48)P t+1ÃSt+1 P ∗tω yt : e ∗ t u 1(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) y t + e ∗ t+1 , (4.49)P t+1à !#∆Mt+1ψ Mt : r t u 1 (c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 ) + r t+1 , (4.50)P t+1à !#ψ Nt : rt ∗ u ∆Nt+1 S t+11(c xt ,c yt )=βE t"u 1 (c xt+1 ,c yt+1 )+ rt+1∗ .(4.51)P t+1The foreign agent solves the analogous problem which generate a set ofsymmetric Euler equations, do not need to be stated here.!#

4.3. THE TWO-MONEY MONETARY ECONOMY 121We know that in equilibrium, the cash-in-advance constraints bind.The cash-in-advance constraints for the foreign agent arem ∗ t = P t c ∗ xt, (4.52)n ∗ t = P ∗t c∗ yt (4.53)In addition, we have the adding-up constraints1 = ψ Mt + ψMt,∗1 = ψ Nt + ψNt ∗ ,x t = c xt + c ∗ xt ,y t = c yt + c ∗ yt,M t = m t + m ∗ t ,N t = n t + n ∗ t .Together, the adding-up constraints and the cash-in-advance constraintsgive a unit-velocity quantity equation for each countryM t = P t x tN t = P ∗t y t,which can be used to eliminate the endogenous nominal price levelsfrom the Euler equations.The equilibrium where people are able to pool and insure againsttheir country-speciÞc risks is given by⇐(93)ω xt = ω ∗ xt = ω yt = ω ∗ yt = ψ Mt = ψ ∗ Mt = ψ Nt = ψ ∗ Nt = 1 2 .Both the domestic and foreign representative households own half of thedomestic endowment stream, half of the foreign endowment stream,half of all future domestic monetary transfers and half of all futureforeign monetary transfers. In short, they split the world’s resourcesin half so the pooling equilibrium supports the symmetric allocationc xt = c ∗ xt = x tand c 2 yt = c ∗ yt = yt. 2To solve for the nominal exchange rate S t , we know by (4.47) thatthe real exchange rate isu 2 (c xt ,c yt )u 1 (c xt ,c yt ) = S tPt∗P t= S tN t x tM t y t. (4.54)

122 CHAPTER 4. THE LUCAS MODELRearranging (4.54) gives the nominal exchange rateS t = u 2(c xt ,c yt ) M t y t. (4.55)u 1 (c xt ,c yt ) N t x tAs in the monetary approach, the fundamental determinants of thenominal exchange rate are relative money supplies and relative GDPs.The two major differences are Þrst that in the Lucas model the exchangerate depends on preferences (utility), and second that it doesnot depend explicitly on expectations of the future.(94)⇒The solution under constant relative risk aversion utility. Using theutility function (4.22), the equilibrium real exchange rate is q t = ((1 −θ)/θ)(x t /y t ). The income terms cancel out and the exchange rate isS t =The Euler equations are" µCt+1e t= βE tx tC te ∗ " µCt+1t= βE tq t y t C t" µCt+1r t= βE tx tC t(1 − θ) M t. (4.56)θ N t (1−γ)ÃMt (1−γ)ÃNt (1−γ)Ã∆Mt+1rt∗ " µCt+1 (1−γ)Ã1 − θ= βE tx t C t θ+ e !#t+1, (4.57)M t+1 x t+1!#+ e∗ t+1, (4.58)N t+1 q t+1 y t+1+ r !#t+1, (4.59)M t+1 x t+1!#∆N t+1+ r∗ t+1. (4.60)N t+1 x t+1Just as you can calculate the equilibrium price of nominal bondseven though they are not traded in equilibrium, you can compute theequilibrium forward exchange rate even though there is no explicit forwardmarket. To do this, let b t be the date t dollar price of a 1-periodnominal discount bond that pays one dollar at the beginning of periodt+1, and let b ∗ t be the date t euro price of a 1-period nominal discountbond that pays one euro at the beginning of period t+1. By coveredinterest parity (1.2 ), the one-period ahead forward exchange rate is,F t = S tb ∗ tb t. (4.61)

4.3. THE TWO-MONEY MONETARY ECONOMY 123The equilibrium bond prices are" µCt+1 1−γ#M tb t = βE t , (4.62)C t M t+1b ∗ t = βE t" µCt+1C t 1−γN tN t+1#. (4.63)

124 CHAPTER 4. THE LUCAS MODELTable 4.1: Notation for the Lucas Modelx Thedomesticgood.y The foreign good.q Relative price of y in terms of x.c x Home consumption of home good.c y Home consumption of foreign good.C Domestic Cobb-Douglas consumption index, c θ x c(1−θ) y .C ∗ Foreign Cobb-Douglas consumption index, c ∗θx cy ∗(1−θ) .c ∗ x Foreign consumption of home good.c ∗ y Foreign consumption of foreign good.ω x Shares of home Þrm held by home agent.ω y Shares of foreign Þrm held by home agent.ωx∗ Shares of home Þrm held by foreign agent.ωy∗ Shares of foreign Þrm held by foreign agent.s Nominal exchange rate. Dollar price of euro.e Price of home Þrm equity in terms of x.e ∗ Price of foreign Þrm equity in terms of x.P Nominal Price of x in dollars.P ∗ Nominal Price of y in euros.M Dollars in circulation.N Euros in circulation.λ t Rate of growth of M.λ ∗ t Rate of growth of N.m Dollars held by domestic household.m ∗ Dollars held by foreign household.n Euros held by domestic household.n ∗ Euros held by foreign household.r t Price of claim to future dollar transfers in terms of x.rt ∗ Price of claim to future euro transfers in terms of x.ψ Mt Shares of dollar transfer stream held by home agent.ψ Nt Shares of euro transfer stream held by home agent.ψMt∗ Shares of dollar transfer stream held by foreign agent.ψNt∗ Shares of euro transfer stream held by foreign agent.Price of one-period nominal bond with one-dollar payoff.b t

4.4. INTRODUCTION TO THE CALIBRATION METHOD 1254.4 Introduction to the Calibration MethodThe Lucas model plays a central role in asset-pricing research. Chapter6 covers some tests of its predictions using time-series econometricmethods. At this point we introduce an alternative and popularmethodology called calibration. In the calibration method, the researchersimulates the model given ‘reasonable’ values to the underlyingtaste and technology parameters and looks to see whether thesimulated observations match various features of the real-world data.Because there is no capital accumulation or production, the technologyin the Lucas model is a stochastic process governing the evolutionof x t and y t . The reasonably simple mechanics underlying the modelmakes its calibration relatively straightforward. Our work here will setthe stage for the next chapter as real business cycle researchers relyheavily on the calibration method to evaluate the performance of theirmodels.Cooley and Prescott [33] set out the ingredients of the calibrationmethod proceeds as follows.1. Obtain a set of measurements from real-world data that we wantto explain. These are typically a set of sample moments suchas the mean, the standard deviation, and autocorrelations ofa time-series. Special emphasis is often placed on the crosscorrelationsbetween two series which measure the extent of theirco-movements.2. Solve and calibrate a candidate model. That is, assign values tothe deep parameters of tastes (the utility function) and technology(the production function) that make sense or that have beenestimated by others.3. Run (simulate) the model by computer and generate time-seriesof the variables that we want to explain.4. Decide whether the computer generated time-series implied bythe model ‘look like’ the observations that you want to explain. 66 The standard analysis is not based on classical statistical inference, although

126 CHAPTER 4. THE LUCAS MODEL4.5 Calibrating the Lucas ModelMeasurement. The measurements that we ask the Lucas model tomatch are the volatility (standard deviation) and Þrst-order autocorrelationof the gross rate of depreciation, S t+1 /S t , the forward premiumF t /S t , the realized forward proÞt (F t − S t+1 )/S t ,andtheslopecoefficientfrom regressing the gross depreciation on the forward premium.Using quarterly data for the U.S. and Germany from 1973.1 to 1997.1,the measurements are given in the row labeled ‘data’ in Table 4.2.Table 4.2: Measured and Implied Moments, US-GermanyS t+1S tVolatilityF t (F t−S t+1 ) S t+1S t S t S tAutocorrelationSlopeData -0.293 0.060 0.008 0.061 0.007 0.888 0.026Model -1.444 0.014 0.006 0.029 0.105 0.006 0.628Note: Model values generated with γ = 10, θ =0.5.F t (F t−S t+1 )S t S tThe implied forward and spot exchange rates exhibit the so-calledforward premium puzzle–that the forward premium predicts the futuredepreciation, but with a negative sign. Recall that the uncoveredinterest parity condition implies that the forward premium predicts thefuture depreciation with a coefficient of 1. The depreciation and therealized proÞt exhibit volatility of similar magnitude which is muchlarger than the volatility of the forward premium. All three series exhibitsubstantial serial dependence.Calibration. Let random variables be denoted with a ‘tilde.’ The ‘technology’that underlies the model are the exogenous monetary growthrates ˜λ, ˜λ ∗ , and the exogenous output growth rates ˜g, ˜g ∗ . Let the statevector be ˜φ =(˜λ, ˜λ ∗ , ˜g, ˜g ∗ ). The process governing the state vector is aÞnite-state Markov chain with stationary probabilities (see the chapterCecchetti et.al. [24], Burnside [18], Gregory and Smith [67] show how calibrationmethods can be combined with classical statistical inference, but the practice hasnot caught on.

4.5. CALIBRATING THE LUCAS MODEL 127appendix). Each element of the state vector is allowed to be in eitherof one of two possible states—high and low. A ‘1’ subscript indicatesthat the variable is in the high growth state and a ‘2’ subscript indicatesthat the variable is in the low growth state. Therefore, λ = λ 1indicates high domestic money growth, λ = λ 2 indicates low domesticmoney growth. Analogous designations hold for the other variables.The 16 possible states of the world areφ 1=(λ 1 , λ ∗ 1,g 1 ,g1) ∗ φ 9=(λ 2 , λ ∗ 1,g 1 ,g1)∗φ 2=(λ 1 , λ ∗ 1,g 1 ,g2) ∗ φ 10=(λ 2 , λ ∗ 1,g 1 ,g2)∗φ 3=(λ 1 , λ ∗ 1,g 2 ,g1) ∗ φ 11=(λ 2 , λ ∗ 1,g 2 ,g1)∗φ 4=(λ 1 , λ ∗ 1,g 2 ,g2) ∗ φ 12=(λ 2 , λ ∗ 1,g 2 ,g2)∗φ 5=(λ 1 , λ ∗ 2,g 1 ,g1) ∗ φ 13=(λ 2 , λ ∗ 2,g 1 ,g1)∗φ 6=(λ 1 , λ ∗ 2,g 1 ,g2) ∗ φ 14=(λ 2 , λ ∗ 2,g 1 ,g2)∗φ 7=(λ 1 , λ ∗ 2,g 2 ,g1) ∗ φ 15=(λ 2 , λ ∗ 2,g 2 ,g1)∗φ 8=(λ 1 , λ ∗ 2,g 2 ,g2) ∗ φ 16=(λ 2 , λ ∗ 2,g 2 ,g2).∗We will denote the 16 × 16 probability transition matrix for the stateby P, wherep ij =P[˜φ t+1= φ j|˜φ t= φ i]theij−th element.The price of the domestic and foreign currency bonds are,b t = βE t [(gt+1g θ ∗(1−θ)t+1 ) 1−γ ]/λ t+1 , and b ∗ t = βE t [(gt+1g θ ∗(1−θ)t+1 ) 1−γ ]/λ ∗ t+1,under the constant relative risk aversion utility function (4.22). Sincetheir values depend on the state of the world, we say that these arestate-contingent bond prices. Next, deÞne G =[(g θ g ∗(1−θ) ) 1−γ ]/λ andG ∗ =[(g θ g ∗(1−θ) ) 1−γ ]/λ ∗ ,andletd = λ/λ ∗ be the gross rate of depreciationof the home currency. The possible values of G and G ∗ and dare given in Table 4.3,Suppose the current state is φ k. By (4.56), the spot exchange rateis given by (1 − θ)d k /θ. The domestic bond price isb k = β P 16i=1 p k,i G i , the foreign bond price is b ∗ k = β P 16i=1 p k,i G ∗ i , theexpected gross change in the nominal exchange rate is P 16i=1 p k,i d i ,andthe state-k contingent risk premium isrp k =X16i=1p k,i d i − (P 16i=1 p k,i G ∗ i )( P 16i=1 p k,i G i ) .Next, we must estimate the probability transition matrix. The Þrstquestion is whether we should use consumption data or GDP? In the

128 CHAPTER 4. THE LUCAS MODELTable 4.3: Possible State ValuesG 1 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ 1 G ∗ 1 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ ∗ 1 d 1 = λ 1 /λ ∗ 1G 2 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ 1 G ∗ 2 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ ∗ 1 d 2 = λ 1 /λ ∗ 1G 3 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ 1 G ∗ 3 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ ∗ 1 d 3 = λ 1 /λ ∗ 1G 4 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ 1 G ∗ 4 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ ∗ 1 d 4 = λ 1 /λ ∗ 1G 5 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ 1 G ∗ 5 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ ∗ 2 d 5 = λ 1 /λ ∗ 2G 6 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ 1 G ∗ 6 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ ∗ 2 d 6 = λ 1 /λ ∗ 2G 7 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ 1 G ∗ 7 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ ∗ 2 d 7 = λ 1 /λ ∗ 2G 8 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ 1 G ∗ 8 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ ∗ 2 d 8 = λ 1 /λ ∗ 2G 9 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ 2 G ∗ 9 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ ∗ 1 d 9 = λ 2 /λ ∗ 1G 10 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ 2 G ∗ 10 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ ∗ 1 d 10 = λ 2 /λ ∗ 1G 11 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ 2 G ∗ 11 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ ∗ 1 d 11 = λ 2 /λ ∗ 1G 12 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ 2 G ∗ 12 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ ∗ 1 d 12 = λ 2 /λ ∗ 1G 13 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ 2 G ∗ 13 =[(g θ 1g ∗(1−θ)1 ) 1−γ ]/λ ∗ 2 d 13 = λ 2 /λ ∗ 2G 14 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ 2 G ∗ 14 =[(g θ 1g ∗(1−θ)2 ) 1−γ ]/λ ∗ 2 d 14 = λ 2 /λ ∗ 2G 15 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ 2 G ∗ 15 =[(g θ 2g ∗(1−θ)1 ) 1−γ ]/λ ∗ 2 d 15 = λ 2 /λ ∗ 2G 16 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ 2 G ∗ 16 =[(g θ 2g ∗(1−θ)2 ) 1−γ ]/λ ∗ 2 d 16 = λ 2 /λ ∗ 2Lucas model, consumption equals GDP so there is no theoretical presumptionas to which series we should use. Since prices depend onutility which depends on consumption. From this perspective, it makessense to use consumption data which is what we do. The consumptionand money data are from the International Financial Statistics and arein per capita terms.The next question is what estimation technique to use? Using generalizedmethod of moments or simulated method of moments (see chapter2.2.2 and chapter 2.2.3) to estimate the transition matrix might begood choices if the dimensionality of the problem were smaller. Sincewe don’t have a very long time span of data, it turns out that estimatingthe transition probability matrix P by GMM or by the SMMdoes not work well. Instead, we ‘estimate’ the transition probabilitiesby counting the relative frequency of the transition events.Let’s classify the growth rate of a variable as being high-growth

4.5. CALIBRATING THE LUCAS MODEL 129whenever it lies above its sample mean and in the low-growth stateotherwise. Then set high-growth states λ 1 , λ ∗ 1, g 1 ,andg1 ∗ to the averageof the high-growth rates found in the data. Similarly, assign the lowgrowthstates λ 2 , λ ∗ 2, g 2 ,andg2 ∗ to the average of the low-growth ratesfound in the data. Using per capita consumption and money data forthe US and Germany, and viewing the US as the home country, theestimates of the high and low state values areλ 1 =1.010—average US money growth good state,λ 2 =0.990—average US money growth bad state,λ ∗ 1 =1.011—average German money growth good state,λ ∗ 2 =0.991—average German money growth bad state,g 1 =1.009—average US consumption growth good state,g 2 =0.998—average US consumption growth bad state,g1 ∗ =1.012—average German consumption growth good state,g2 ∗ =0.993—average German consumption growth bad state.Now classify the data into the φ states according to whether the observationslie above or below the mean then set the transition probabilitiesp jk equal to the relative frequency of transitions from state φ j toφ k found in the data. The P estimated in this fashion, rounded to 2signiÞcant digits, is⎡⎢⎣.00 .00 .20 .00 .40 .00 .00 .00 .20 .00 .00 .00 .20 .00 .00 .00.20 .20 .20 .20 .00 .20 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00.17 .17 .00 .17 .17 .00 .00 .00 .00 .00 .00 .00 .00 .00 .17 .17.00 .00 .00 .00 .17 .00 .00 .00 .00 .17 .33 .17 .00 .00 .17 .00.08 .08 .08 .08 .15 .08 .08 .08 .15 .08 .08 .00 .00 .00 .00 .00.20 .00 .00 .00 .20 .00 .00 .00 .00 .00 .20 .00 .00 .20 .20 .00.00 .00 .00 .20 .40 .00 .00 .20 .00 .00 .00 .00 .20 .00 .00 .00.25 .00 .00 .00 .00 .50 .00 .00 .00 .00 .00 .00 .00 .00 .00 .25.00 .14 .00 .00 .00 .00 .14 .00 .14 .14 .00 .00 .00 .14 .14 .14.00 .00 .00 .00 .00 .00 .25 .00 .25 .00 .00 .25 .25 .00 .00 .00.00 .00 .20 .00 .20 .00 .00 .00 .20 .20 .00 .20 .00 .00 .00 .00.00 .25 .00 .25 .25 .00 .00 .00 .00 .00 .00 .00 .00 .25 .00 .00.00 .00 .00 .00 .13 .00 .00 .13 .13 .00 .13 .13 .25 .00 .13 .00.00 .00 .20 .00 .00 .00 .00 .00 .00 .00 .00 .00 .20 .00 .40 .20.00 .00 .00 .00 .25 .00 .25 .13 .00 .00 .00 .00 .13 .13 .00 .13.00 .00 .00 .20 .00 .20 .00 .00 .00 .00 .00 .00 .20 .20 .20 .00Results. We set the share of home goods in consumption to be θ =1/2,the coefficient of relative risk aversion to be γ =10,andthesubjective⎤⎥⎦

130 CHAPTER 4. THE LUCAS MODELdiscount factor to be β =0.99 and simulate the model as follows.Draw a sequence of T realizations of the gross change in the exchangerate, the forward premium, and the risk premium with theinitial state vector drawn from probabilities of the initial probabilityvector, v. Let u t be a iid uniform random variable on [0, 1]. The rulefor determining the initial state is,φ 1if u t

4.5. CALIBRATING THE LUCAS MODEL 131is much too small. The implied persistence of the depreciation and theforward premium is also too low to be consistent with the data.The model does predict that the forward rate is a biased predictorof the future spot rate due to the presence of a risk premium. However,the size of the implied risk premium appears to be too small to providean adequate explanation for the data. We study the forward premiumpuzzle in greater detail in chapter 6.

132 CHAPTER 4. THE LUCAS MODEL1.02A. Depreciation and Forward Premium1.011.000.990.980.9773 75 77 79 81 83 85 87 89 91 93 950.03B. Ex Post Profit and Risk Premium0.020.010.00-0.01-0.02-0.03-0.0473 75 77 79 81 83 85 87 89 91 93 95Figure 4.1: From the Lucas Model. A: Implied gross one-period aheadchange in nominal exchange rate S t+1 /S t and current forward premiumF t /S t (in boxes). B. Implied ex post forward payoff (S t+1 − F t )/S t(jagged line) and risk premium E t (S t+1 − F t )/S t (smooth line).

4.5. CALIBRATING THE LUCAS MODEL 133Lucas Model Summary1. It is a ßexible-price, complete markets, dynamic general equilibriummodel with optimizing agents. It is logically consistent andprovides the micro-foundations for international asset pricing.2. The Lucas model provides a framework for pricing assets, includingthe exchange rate, in an international setting. The exchangerate depends on the same set of fundamental variables as predictedby the monetary model. The empirical predictions of themodel will be developed more fully in chapter 6.3. Thereisnotradingvolumeforanyoftheassets. Thepricesderived in the model are shadow values under which the existingstock of assets are willingly held by the agents.4. Output is taken to be exogenous so the model not well equippedto explain quantities such as the current account.5. The Lucas model is designed to help us understand the determinationof the prices of assets–exchange rates, bonds, andstocks–that are consistent with equilibrium choices of consumption.Because it is an endowment model, the dynamics of consumption(or alternatively output) are taken exogeneously. Thisis actually a virtue of the model since a model with production,while perhaps more ‘realistic,’ does not change the underlyingasset pricing formulae which are based on the Euler equationsfor the consumer’s problem but complicates the job by forcingus to write down a model where equilibrium decisions of theÞrm generate not only realistic asset price movements but alsorealistic output dynamics. It is therefore not necessary or evendesirable to introduce production in order to understand equilibriumasset pricing issues.

134 CHAPTER 4. THE LUCAS MODELAppendix—Markov ChainsLet X t be a random variable and x t be a particular realization of X t . AMarkov chain is a stochastic process {X t } ∞ t=0 with the property that theinformation in the current realized value of X t = x t summarizes the entirepast history of the process. That is,P[X t+1 = x t+1 |X t = x t ,X t−1 = x t−1 ,...,X 0 = x 0 ]=P[X t+1 = x t+1 |X t = x t ].(4.64)AkeyresultthatsimpliÞes probability calculations of Markov chains is,(98)⇒ Property 1 If {X t } ∞ t=0P[X t = x t ∩ X t−1 = x t−1 ∩ ···∩ X 0 = x 0 ]=is a Markov chain, thenP[X t = x t |X t−1 = x t−1 ] ···P[X 1 = x 1 |X 0 = x 0 ]P[X 0 = x 0 ]. (4.65)Proof: Let A j be the event (X j = x j ). You can write the left side of(4.65) as,t−1 \ t−1 \P(A t ∩ A t−1 ∩ ···∩ A 0 ) = P(A t | A j )P( A j ) (multiplication rule)j=0j=0j=0t−1 \= P(A t |A t−1 )P( A j ) (Markov chain property)t−2 \ t−2 \= P(A t |A t−1 )P(A t−1 | A j )P( A j ) (mult. rule)j=0j=0t−2 \= P(A t |A t−1 )P(A t−1 |A t−2 )P( ) (Markov chain)j=0.= P(A t |A t−1 )P(A t−1 |A t−2 ) ···P(A 1 |A 0 )P(A 0 )Let λ j , j = 1,...,N denote the possible states for X t .AMarkovchainhas stationary probabilities if the transition probabilities from state λ i to λ jare time-invariant. That is,P[X t+1 = λ j |X t = λ i ]=p ij

4.5. CALIBRATING THE LUCAS MODEL 135Notice that in Markov chain analysis the Þrst subscript denotes the statethat you condition on. For concreteness, consider a Markov chain with twopossible states, λ 1 and λ 2 , with transition matrix," #p11 pP =12,p 21 p 22where the rows of P sum to 1.Property 2 The transition matrix over k steps isP k = PP ···P | {z }kProof. For the two state process, deÞnep (2)ij = P[X t+2 = λ j |X t = λ i ]= P[X t+2 = λ j ∩ X t+1 = λ 1 |X t = λ i ]+P[X t+1 = λ j ∩ X t+1 = λ 2 |X t = λ i ]=2XP[X t+1 = λ j ∩ X t+1 = λ k |X t = λ i ]k=1= P[X t+1 = λ j ∩ X t+1 = λ k ∩ X t = λ i ]P(X t = λ i )Now by property 1, the numerator in last equality can be decomposed as,P[X t+2 = λ j |X t+1 = λ k ]P[X t+1 = λ k |X t = λ i ]P[X t = λ i ] (4.67)Substituting (4.67) into (4.66) gives,(4.66)p (2)ij ==2XP[X t+1 = λ j |X t+1 = λ k ]P[X t+1 = λ k |X t = λ i ]k=12Xk=1p kj p ikwhich is seen to be the ij−th element of the matrix PP. The extension toany arbitrary number of steps forward is straightforward.⇐(99)

136 CHAPTER 4. THE LUCAS MODELProblems1. Risk sharing in the Lucas model [Cole-Obstfeld (1991)]. Let theperiod utility function be u(c x ,c y ) = θ ln c x +(1 − θ)lnc y for thehome agent and u(c ∗ x,c ∗ y)=θ ln c ∗ x +(1 − θ)lnc ∗ y for the foreign agent.Suppose That capital is internationally immobile. The home agentowns all of the x−endowment (φ x = 1), the foreign agent owns allof the y−endowment (φ ∗ y = 1). Show that in the equilibrium underportfolio autarchy, trade in goods alone is sufficient to achieve efficientrisk sharing.2. Consider now the single-good model. Let x t bethehomeendowmentand x ∗ t be the foreign endowment of the same good. The planner’sproblem is to maximizeφ ln c t +(1 − φ)lnc ∗ tsubject to c t + c ∗ t = x t + x ∗ t .Under zero capital mobility, the home agent’s problem is to maximizeln(c t )subjecttoc t = x t . The foreign agent maximizes ln(c ∗ t )subjectto c ∗ t = x∗ t . Show that asset trade is necessary in this case to achieveefficient risk sharing.(100)⇒(101)⇒(102)⇒3. Nontraded goods. Letx and y be traded as in the model of this chapter.In addition, let N be a nonstorable nontraded domestic goodgenerated by an exogenous endowment, and let N ∗ be a nonstorablenontraded foreign good also generated by exogenous endowment. Letthe domestic agent’s utility function be u(c xt ,c yt ,c N )=(C 1−γ )/(1−γ)where C = c θ 1x cθ 2y cθ 3N with θ 1 + θ 2 + θ 3 = 1. The foreign agent has thesame utility function. Show that trade in goods under zero capitalmobility does not achieve efficient risk sharing.4. Derive the exchange rate in the Lucas model under log utility, U(c xt ,c yt )=θ ln(c xt )+(1 − θ)ln(c yt ) and compare it with the solution under constantrelative risk aversion utility.5. Use the high and low growth states and the transition matrix givenin section 4.5 to solve for the price-dividend ratios for equities. Whatdoes the Lucas model have to say about the volatility of stock prices?How does the behavior of equity prices in the monetary economy differfrom the behavior of equity prices in the barter economy?

Chapter 5InternationalRealBusinessCyclesIn this chapter, we continue our study of models with perfect marketsin the absence of nominal rigidities but turn our attention understandinghow business cycles originate and how they are propagated andtransmitted from one country to another through current account imbalances.For this purpose, we will study real business cycle models.These are stochastic growth models that have been employed to addressbusiness cycle ßuctuations. As their name suggests, real business cyclemodels deal with the real side of the economy. They are Arrow—Debreumodels in which there is no role for money and their solution typicallyfocuses on solving the social planner’s problem.Analytic solutions to the stochastic growth model are available onlyunder special speciÞcations—for example when utility is time-separableand logarithmic and when capital fully depreciates each period. Complicationsbeyond these very simple structures require that the modelbe solved and evaluated numerically. We will work with durable capitalalong with the log utility speciÞcation. The resulting models aresimple enough for us to retain our intuition for what is going on butcomplicated enough so that we must solve them using numerical andapproximation methods.Real business cycle researchers evaluate their models using the calibrationmethod, which was outlined in chapter 4.4.4.137

138 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES5.1 Calibrating the One-Sector Growth ModelWe begin simply enough, with the closed economy stochastic growthmodel with log utility and durable capital. Then we will constructan international real business cycle model by piecing two one-countrymodels together.MeasurementThe job of real business cycle models is to explain business cycles butthe data typically contains both trend and cyclical components. 1 Wewill think of a macroeconomic time series such as GDP, as being builtup of the two components, y t = y τt + y ct ,wherey τt is the long-runtrend component and y ct is the cyclical component. Since businesscycletheory is typically not well equipped to explain the trend, theÞrst thing that real business cycle theorists do is to remove the noncyclicalcomponents by Þltering the data.There are many ways to Þlter out the trend component. Two verycrude methods are either to work with Þrst-differenced data or to useleast-squares residuals from a linear or quadratic trend. Most real businesscycle theorists, however choose to work with Hodrick—Prescott [76]Þltered data. This technique, along with background information onthe spectral representation of time-series is covered in chapter 2.Our measurements are based on quarterly log real output, consumptionof nondurables plus services, and gross business Þxed investment inper capita terms for the US from 1973.1 to 1996.4. The output measureis GDP minus government expenditures. The raw data and Hodrick-Prescott trends are displayed in Figure 5.1. The Hodrick-Prescott cyclicalcomponents are displayed in Figure 5.2. Investment is the mostvolatile of the series and consumption is the smoothest but all threeare evidently highly correlated. That is, they display a high degree of‘co-movement.’Table 5.1 displays some descriptive statistics of the Þltered (cyclicalpart) data. Each series displays substantial persistence and a highdegree of co-movement with output.1 The data also contains seasonal and irregular components which we will ignore.

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 1390.20.1GDP0Consumption-0.1-0.2-0.3Investment-0.4-0.573 75 77 79 81 83 85 87 89 91 93 95Figure 5.1: US data (symbols) and trend (no symbols) from Hodrick-Prescott Þlter. Observations are quarterly per capita logarithms ofGDP, consumption, and investment from 1973.1 to 1996.4.The modelWe will work with a version of the King, Plosser, and Rebelo [83]model that abstracts from the labor-leisure choice. The consumer haslogarithmic period utility deÞned over the single consumption goodu(C) =ln(C). Lifetime utility is P ∞j=0 β j u(C t+j ), where 0 < β < 1isthe subjective discount factor.The representative Þrm produces output Y t , by combining laborN t , and capital K t , according to a Cobb-Douglas production function.Individuals are compelled to provide a Þxed amount of N hours of laborto the Þrm each period. Permanent changes to technology take placethrough changes in labor productivity, X t . The number of effectivelabor units is NX t . This part of technical change is assumed to evolveexogeneously and deterministically at a gross rate of γ = X t+1 /X t .A second component that governs technology is a transient stochastic

140 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES0.150.1Investment0.05GDP0-0.05-0.1-0.15Consumption73 75 77 79 81 83 85 87 89 91 93 95Figure 5.2: Hodrick-Prescott Þltered cyclical observations.shock, A t . The production function isY t = A t K α t (NX t ) 1−α .α is capital’s share. Most estimates for the US place 0.33 ≤ α ≤ 0.40.Output can be consumed or saved. Savings (or investment I t )areused to replace worn capital and to augment the current capital stock.Capital depreciates at a rate δ and evolves according toK t+1 = I t +(1− δ)K t .There is no government and no foreign sector. There are also nomarket imperfections so we can work with Þctitious social planner’sproblem as we did with the Lucas model.Problem 1. The social planner wants to maximizeE t ∞ Xj=0β j U(C t+j ), (5.1)

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 141Table 5.1: Closed-Economy MeasurementsStd. AutocorrelationsDev. 1 2 3 4 6y t 0.022 0.86 0.66 0.46 0.27 0.02c t 0.013 0.85 0.72 0.57 0.38 0.14i t 0.056 0.89 0.73 0.56 0.40 0.08Cross correlation with y t−k at k6 4 1 0 -1 -4 -6c t 0.09 0.20 0.72 0.87 0.87 0.46 0.14i t 0.01 0.43 0.91 0.94 0.81 0.20 0.10Notes: All variables are logarithms of real per capita data for the US from 1973.1to 1996.4 and have been passed through the Hodrick—Prescott Þlter with λ = 1600.y t is gross domestic product less government spending, c t is consumption of nondurablesplusservices,andit is gross business Þxed investment. Source: InternationalFinancial Statistics.subject to Y t = A t Kt α (NX t) 1−α , (5.2)K t+1 = I t +(1− δ)K t , (5.3)Y t = C t + I t , (5.4)U(C) =ln(C). (5.5)The model allows for one normalization so you can set N =1.In the steady state, you will want the economy to evolve along abalanced growth path in which all quantities except for N grow at thesame gross rateγ = X t+1X t= Y t+1Y t= C t+1C t= I t+1I t= K t+1K t.The steady state is reasonably straightforward to obtain. However, ifcapital lasts more than one period, δ < 1, the dynamics of the modelmust be solved by approximation methods. We’ll Þrst solve for thesteady state and then take a linear approximation of the model aroundits steady state. The exogenous growth factor γ gives the model a

142 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESmoving steady state which is inconvenient. To Þx this,youcanÞrsttransform the model to get a Þxed steady state by normalizing all thevariables by labor efficiency units. Let lower case letters denote thesenormalized valuesy t = Y t, k t = K t, i t = I t, c t = C t.X t X t X t X tDividing (5.2) by X t gives y t = A t kt α. Dividing (5.3) by X t givesγk t+1 = i t +(1− δ)k t . To normalize lifetime utility (5.1), note thatP ∞j=0β j ln X t+j = P ∞j=0 β j ln(γ j X t )=ln(X t )/(1 − β)+ln(γ) P ∞j=0 jβ j(ch.5-1)⇒= ln(X t )/(1 − β) +ln(γ)β/(1 − β) 2 < ∞. Using this fact, addingand subtracting P ∞j=0 β j Pln(X t+j ) to (5.1) gives E ∞j=0t β j U(C t+j )=PΩ t +E ∞j=0t β j U(c t+j )whereU(c) =ln(c) andΩ t =ln(X t )/(1 − β)+β ln(γ)/(1 − β) 2 .SinceΩ t is exogenous, we can ignore it when solvingthe planner’s problem. We will call the transformed problem, problem2. This is the one we will solve.Problem 2. It will be useful to use the notation f(A t ,k t )=A t k α .SinceΩ is a constant, the social planner’s growth problem normalized bylabor efficiency units is to maximizeE t ∞ Xj=0β j U(c t+j ), (5.6)subject to y t = f(A t ,k t )=A t kt α , (5.7)γk t+1 = i t +(1− δ)k t , (5.8)y t = c t + i t , (5.9)U(c) =ln(c). (5.10)It will be useful to compactify the notation. Let λ t =(k t+1 ,k t ,A t ) 0 andcombine the constraints (5.7)—(5.9) to form the consolidated budgetconstraintc t = g(λ t ) = f(A t ,k t ) − γk t+1 +(1− δ)k t= A t k α t − γk t+1 +(1− δ)k t . (5.11)Under Cobb-Douglas production and log utility, you havef k = αyk , f kk = α(α − 1) y k 2 , u c = 1 c , u cc = −1c 2 .

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 143Letting g j = ∂c t /∂λ jt be the partial derivative of g(λ t ) with respectto the j−th element of λ t and g ij = ∂ 2 c t /(∂λ it ∂λ jt ) be the secondcross-partial derivative, for future reference you haveg 1 = −γ,g 2 = f k (A, k)+(1− δ),g 3 = y/A,g 11 = g 12 = g 21 = g 13 = g 31 = g 33 =0,g 22 = f kk (A, k),g 23 = g 32 = αk α−1 .Now substitute (5.11) into (5.6) to transform the constrained optimizationproblem into an unconstrained problem. You want to maximizeE t ∞ Xj=0β j u[g(λ t+j )], (5.12)where g(λ t ) is given in (5.11). At date t, k t is pre-determined and theonly choice variable is i t and choosing i t is equivalent to choosing k t+1 .The Þrst-order conditions for all t are−γu c (c t )+βE t u c (c t+1 )[f k (A t+1 ,k t+1 )+(1− δ)] = 0. (5.13)Notice that c t must obey the consolidated budget constraint (5.11). Itfollows that (5.13) is a nonlinear stochastic difference equation in k t .Analytic solutions to such equations are not easy to obtain so we resortto approximation methods.The Steady StateWe will compute the approximate solution around the model’s steadystate. InordertodothatweneedÞrst to Þnd the steady state. Denotesteady state values of output, consumption, investment and capitaly, c, i, k without the time subscript and let the steady state value ofA =1.Since f k = αk α−1 = α(y/k), (5.13) becomes γ = β[α(y/k) +(1 − δ)] from which we obtain the steady state output to capital ratio

144 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESy/k =(γ/β + δ − 1)/α. Now divide the production function (5.7) by kand re-arrange to get k =(y/k) 1/(α−1) =[(γ/β + δ − 1)/α] 1/(α−1) .Nowthat we know k, we can get y. From the accumulation equation (5.8),we have i/k = γ + δ − 1, and in turn, c/k = y/k − i/k. Again, given k,we can solve for c. To summarize, in the steady state we havey/k = (γ/β + δ − 1)/α, (5.14)i/k = γ + δ − 1, (5.15)c/k = y/k − i/k, (5.16)k = (y/k) 1/(α−1) . (5.17)Calibrating the ModelEach time period corresponds to a quarter. We set α = 0.33,β = 0.99, δ =0.10, γ =1.0038. 2 The transient technology shockevolves according to the Þrst-order autoregressionwhere ρ =0.93, and ² tiid∼ N(0, 0.010224 2 ).A t =(1− ρ)+ρA t−1 + ² t , (5.18)Approximate Solution Near the Steady State.Many methods have been applied to solve real business cycle models.One option for solving the model is to take a Þrst—order Taylor expansionof the nonlinear Þrst—order condition (5.13) in the neighborhoodaround the steady state. 3 . This yields the second—order stochasticdifference equation in k t − ka 0 +a 1 (k t+2 −k)+a 2 (k t+1 −k)+a 3 (k t −k)+a 4 (A t+1 −1)+a 5 (A t −1) = 0,(5.19)2 This is the depreciation rate used by Backus et. al. [5]. Cooley andPrescott [33] recommend δ =0.048. γ is the value used by Cooley and Prescottand King et. al..3 This is the method of King, Plosser, and Rebelo [83]

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 145where a 0 = U c g 1 + βU c g 2 =0,a 1 = βU cc g 1 g 2 ,a 2 = U cc g1 2 + βU cc g2 2 + βU c g 22 ,a 3 = U cc g 1 g 2 ,a 4 = βU c g 32 + βU cc g 2 g 3 ,a 5 = U cc g 1 g 3 .The derivatives are evaluated at steady state values.A second but equivalent option is to take a second—order Taylorapproximation to the objective function around the steady state and tosolve the resulting quadratic optimization problem. The second optionis equivalent to the Þrst because it yields linear Þrst—order conditionsaround the steady state. To pursue the second option, recall that λ t =(k t+1 ,k t ,A t ) 0 . Write the period utility function in the unconstrainedoptimization problem asR(λ t )=U[g(λ t )]. (5.20)Let R j = ∂R(λ t )/∂λ jt be the partial derivative of R(λ t ) with respectto the j−th element of λ t and R ij = ∂ 2 R(λ t )/(∂λ it ∂λ jt ) be the secondcross-partial derivative. Since R ij = R ji the relevant derivatives are,R 1 = U c g 1 ,R 2 = U c g 2 ,R 3 = U c g 3 ,R 11 = U cc g 2 1,R 22 = U cc g 2 2, +U c g 22R 33 = U cc g 2 3,R 12 = U cc g 1 g 2 ,R 13 = U cc g 1 g 3 ,R 23 = U cc g 2 g 3 + U c g 23 .The second-order Taylor expansion of the period utility function isR(λ t ) = R(λ)+R 1 (k t+1 − k)+R 2 (k t − k)+R 3 (A t − A)+ 1 2 R 11(k t+1 − k) 2

146 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES+ 1 2 R 22(k t − k) 2 + 1 2 R 33(A t − A) 2 + R 12 (k t+1 − k)(k t − k)+R 13 (k t+1 − k)(A t − A)+R 23 (k t − k)(A t − A).Suppose we let q =(R 1 ,R 2 ,R 3 ) 0 be the 3 × 1 row vector of partialderivatives (the gradient) of R, andQ be the 3 × 3matrixofsecondpartial derivatives (the Hessian) multiplied by 1/2 whereQ ij = R ij /2.Then the approximate period utility function can be compactly writtenin matrix form asR(λ t )=R(λ)+[q +(λ t − λ) 0 Q](λ t − λ). (5.21)The problem is now to maximizeE t ∞ Xj=0The Þrst order conditions are for all tβ j R(λ t+j ). (5.22)0 = (βR 2 + R 1 )+βR 12 (k t+2 − k)+(R 11 + βR 22 )(k t+1 − k)+R 12 (k t − k)+βR 23 (A t+1 − 1) + R 13 (A t − 1). (5.23)If you compare (5.23) to (5.19), you’ll see that a 0 = βR 2 + R 1 ,a 1 = βR 12 ,a 2 = R 11 + βR 22 ,a 3 = R 12 ,a 4 = βR 23 ,a 5 = R 13 . ThisveriÞes that the two approaches are indeed equivalent.Now to solve the linearized Þrst-order conditions, work with (5.19).Since the data that we want to explain are in logarithms, you can convertthe Þrst-order conditions into near logarithmic form. Letã i = ka i for i =1, 2, 3, and let a “hat” denote the approximate logdifference from the steady state so that ˆk t =(k t − k)/k ' ln(k t /k)and Ât = A t − 1 (since the steady state value of A =1). Nowletb 1 = −ã 2 /ã 1 , b 2 = −ã 3 /ã 1 , b 3 = −a 4 /ã 1 , and b 4 = −a 4 /ã 1 .The second—order stochastic difference equation (5.19) can be writtenas(1 − b 1 L − b 2 L 2 )ˆk t+1 = W t , (5.24)whereW t = b 3 Â t+1 + b 4 Â t .

5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 147The roots of the polynomial (1 − b 1 z − b 2 z 2 )=(1− ω 1 L)(1 − ω 2 L)satisfy b 1 = ω 1 + ω 2 and b 2 = −ω 1 ω 2 . Using the quadratic formulaand evaluating at the parameter values thatqwe used to calibrate themodel, the roots are, z 1 =(1/ω 1 )=[−b 1 − b 2 1 +4b 2 ]/(2b 2 ) ' 1.23, andqz 2 =(1/ω 2 )=[−b 1 + b 2 1 +4b 2 ]/(2b 2 ) ' 0.81. There is a stable root,|z 1 | > 1 which lies outside the unit circle, and an unstable root, |z 2 | < 1which lies inside the unit circle. The presence of an unstable root meansthat the solution is a saddle path. If you try to simulate (5.24) directly,the capital stock will diverge.To solve the difference equation, exploit the certainty equivalenceproperty of quadratic optimization problems. That is, you Þrst getthe perfect foresight solution to the problem by solving the stable rootbackwards and the unstable root forwards. Then, replace future randomvariables with their expected values conditional upon the time-tinformation set. Begin by rewriting (5.24) asW t = (1− ω 1 L)(1 − ω 2 L)ˆk t+1= (−ω 2 L)(−ω −12 L −1 )(1 − ω 2 L)(1 − ω 1 L)ˆk t+1= (−ω 2 L)(1 − ω −12 L−1 )(1 − ω 1 L)ˆk t+1 ,and rearrange to get(1 − ω 1 L)ˆk t+1 = −ω−1 2 L −11 − ω2 −1 L W −1 tµ 1= − L −1 X ∞ω 2= −∞Xj=1j=0µ 1ω 2 jW t+jµ 1ω 2 jW t+j . (5.25)The autoregressive speciÞcation (5.18) implies the prediction formulaej=1E t W t+j = b 3 E t  t+j+1 + b 4 E t  t+j =[b 3 ρ + b 4 ]ρ j  t .Use this forecasting rule in (5.25) to get∞Xµ 1 j ∞Xµ " #ρjρE t W t+j =[b 3 ρ + b 4 ]Ât = (b 3 ρ + b 4 )Ât.ω 2 ω 2 ω 2 − ρj=1

148 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESIt follows that the solution for the capital stock isˆk t+1 = ω 1ˆkt −"ρω 2 − ρ#[b 3 ρ + b 4 ]Ât. (5.26)To recover ŷ t , note that the Þrst-order expansion of the productionfunction gives y t = f(A, k) +f A Â t + f k kˆk t ,wheref A =1, andf k =(αy)/k. Rearrangement gives ŷ t = Ât + ˆk t .Torecoverî t , subtractthe steady state value γk = i +(1− δ)k from (5.8) and rearrange to getî t =(k/i)[γˆk t+1 −(1−δ)ˆk t ]. Finally, get ĉ t =ŷ t −î t from the adding-upconstraint (5.9). The log levels of the variables can be recovered byln(Y t ) = ŷ t +ln(X t )+ln(y),ln(C t ) = ĉ t +ln(X t )+ln(c),ln(I t ) = î t +ln(X t )+ln(i),ln(X t ) = ln(X 0 )+t ln(γ).Simulating the ModelWe’ll use the calibrated model to generate 96 time-series observationscorresponding to the number of observations in the data. From thesepseudo-observations, recover the implied log-levels and pass them throughthe Hodrick-Prescott Þlter. The steady state values arey =1.717, k =5.147, c =1.201, i/k =0.10.Plots of the Þltered log income, consumption, and investment observationsare given in Figure 5.3 and the associated descriptive statistics aregiven in Table 5.2. The autoregressive coefficient and the error varianceof the technology shock were selected to match the volatility of outputexactly. From the Þgure, you can see that both consumption and investmentexhibit high co-movements with output, and all three seriesdisplay persistence. However from Table 5.2 the implied investmentseries is seen to be more volatile than output but is less volatile thanthat found in the data. Consumption implied by the model is morevolatile than output, which is counterfactual.

5.2. CALIBRATING A TWO-COUNTRY MODEL 1490.050-0.05GDP (broken)Consumption-0.1-0.15Investment-0.273 75 77 79 81 83 85 87 89 91 93 95Figure 5.3: Hodrick-Prescott Þltered cyclical observations from themodel. Investment has been shifted down by 0.10 for visual clarity.This coarse overview of the one sector real business cycle modelshows that there are some aspects of the data that the model does notexplain. This is not surprising. Perhaps it is more surprising is howwell it actually does in generating ‘realistic’ time series dynamics of thedata. In any event, this perfect markets—no nominal rigidities Arrow-Debreu model serves as a useful benchmark against which reÞnementscan be judged.5.2 Calibrating a Two-Country ModelWe now add a second country. This two-country model is a specialcase of Backus et. al. [5]. Each county produces the same good so wewill not be able to study terms of trade or real exchange rate issues.The presence of country-speciÞc idiosyncratic shocks give an incentiveto individuals in the two countries to trade as a means to insure each

150 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESTable 5.2: Calibrated Closed-Economy ModelStd. AutocorrelationsDev. 1 2 3 4 6y t 0.022 0.90 0.79 0.67 0.53 0.23c t 0.023 0.97 0.89 0.77 0.63 0.31i t 0.034 0.70 0.50 0.36 0.19 -0.04Cross correlation with y t−k at k6 4 1 0 -1 -4 -6c t 0.49 0.77 0.96 0.90 0.79 0.33 0.04i t 0.29 0.11 0.41 0.74 0.73 0.61 0.44other against a bad relative technology shock so we can examine thebehavior of the current account.MeasurementWe will call the Þrst country the ‘US,’ and second country ‘Europe.’The data for European output, government spending, investment, andconsumption are the aggregate of observations for the UK, France, Germany,and Italy. The aggregate of their current account balances sufferfrom double counting and does not make sense because of intra-European trade. Therefore, we examine only the US current account,which is measured as a fraction of real GDP.Table 5.3 displays the features of the data that we will attempt toexplain–their volatility, persistence (characterized by their autocorrelations)and their co-movements (characterized by cross correlations).Notice that US and European consumption correlation is lower thanthe their output correlation.The Two-Country ModelBoth countries experience identical rates of depreciation of physicalcapital, long-run technological growth X t+1 /X t = Xt+1/X ∗ t∗ = γ, have

5.2. CALIBRATING A TWO-COUNTRY MODEL 151Table 5.3: Open-Economy MeasurementsStd. AutocorrelationsDev. 1 2 3 4 6ex t 0.01 0.61 0.50 0.40 0.40 0.12yt ∗ 0.014 0.84 0.62 0.36 0.15 -0.15c ∗ t 0.010 0.68 0.47 0.30 0.04 -0.15i ∗ t 0.030 0.89 0.75 0.57 0.40 0.07Cross correlations at lag k6 4 1 0 -1 -4 6y t ex t−k 0.43 0.42 0.41 0.41 0.37 0.03 0.32y t yt−k ∗ 0.28 0.22 0.21 0.36 0.43 0.36 0.22c t c ∗ t−k 0.26 0.39 0.28 0.25 0.05 0.15 0.26Notes: ex t is US net exports divided by GDP. Foreign country aggregates data fromFrance, Germany, Italy, and the UK. All variables are real per capita from 1973.1to 1996.4 and have been passed through the Hodrick—Prescott Þlter with λ = 1600.the same capital shares and Cobb-Douglas form of the production function,and identical utility. Let the social planner attach a weight of ω tothe domestic agent and a weight of 1 − ω to the foreign agent. In termsof efficiency units, the social planner’s problem is now to maximizesubject to,E t ∞ Xj=0β j [ωU(c t+j )+(1− ω)U(c ∗ t+j )], (5.27)y t = f(A t ,k t )=A t kt α , (5.28)yt ∗ = f(A ∗ t ,kt ∗ )=A ∗ t kt ∗α , (5.29)γk t+1 = i t +(1− δ)k t , (5.30)γkt+1 ∗ = i ∗ t +(1− δ)k∗ t , (5.31)y t + yt ∗ = c t + c ∗ t +(i t + i ∗ t ). (5.32)In the market economy interpretation, we can view ω to indicate thesize of the home country in the world economy. (5.28) and (5.29) are the

152 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESCobb—Douglas production functions for the home and foreign counties,with normalized labor input N = N ∗ = 1. (5.30) and (5.31) are thedomestic and foreign capital accumulation equations, and (5.31) is thenew form of the resource constraint. Both countries have the sametechnology but are subject to heterogeneous transient shocks to totalproductivity according to" # " # " #" # " #At 1 − ρ − δ ρ δ At−1 ²t=++ , (5.33)1 − ρ − δ δ ρA ∗ tA ∗ t−1where (² t ,² ∗ t ) 0 iid∼ N(0, Σ). We set ρ =0.906, δ =0.088, Σ 11 = Σ 22 =2.40e−4, and Σ 12 = Σ 21 =6.17e−5. The contemporaneous correlationof the innovations is 0.26.Apart from the objective function, the main difference between thetwo-county and one-country models is the resource constraint (5.32).World output can either be consumed or saved but a country’s net saving,which is the current account balance, can be non—zero(y t − c t − i t = −(y ∗ t − c ∗ t − i ∗ t ) 6= 0).Let λ t =(k t+1 ,k ∗ t+1,k t ,k ∗ t ,A t ,A ∗ t ,c ∗ t ) be the state vector, and indicatethe dependence of consumption on the state by c t = g(λ t ), andc ∗ t = h(λ t )(whichequalsc ∗ t trivially). Substitute (5.28)—(5.31) into(5.32) and re-arrange to getc t = g(λ t )=f(A t ,k t )+f(A ∗ t ,kt ∗ ) − γ(k t+1 + kt+1),∗+(1 − δ)(k t + kt ∗ ) − c ∗ t (5.34)c ∗ t = h(λ t )=c ∗ t . (5.35)For future reference, the derivatives of g and h are,² ∗ tg 1 = g 2 = −γ,g 3 = f k (A, k)+(1− δ),g 4 = f k (A ∗ ,k ∗ )+(1− δ),g 5 = f(A, k)/A,g 6 = f(A ∗ ,k ∗ )/A ∗ ,g 7 = −1,h 1 = h 2 = ···= h 6 =0,h 7 =1.

5.2. CALIBRATING A TWO-COUNTRY MODEL 153Next, transform the constrained optimization problem into an unconstrainedproblem by substituting (5.34) and (5.35) into (5.27). Theproblem is now to maximize³ωE t u[g(λt )] + βU[g(λ t+1 )] + β 2 U[g(λ t+2 )] + ···´(5.36)³+(1 − ω)E t u[h(λt )] + βU[h(λ t+1 )] + β 2 U[h(λ t+2 )] + ···´.At date t, the choice variables available to the planner are k t+1 ,k ∗ t+1,and c ∗ t . Differentiating (5.36) with respect to these variables and rearrangingresults in the Euler equationsγU c (c t ) = βE t U c (c t+1 )[g 3 (λ t+1 )], (5.37)γU c (c t ) = βE t U c (c t+1 )[g 4 (λ t+1 )], (5.38)U c (c t ) = [(1− ω)/ω]U c (c ∗ t ). (5.39)(5.39) is the Pareto—Optimal risk sharing rule which sets home marginalutility proportional to foreign marginal utility. Under log utility, homeand foreign per capita consumption are perfectly correlated,c t =[ω/(1 − ω)]c ∗ t .The Two-Country Steady StateFrom (5.37) and (5.38) we obtain y/k = y ∗ /k ∗ =(γ/β+δ−1)/α. We’vealready determined that c =[ω/(1 − ω)]c ∗ = ωc w where c w = c + c ∗is world consumption. From the production functions (5.28)—(5.29) weget k =(y/k) 1/(α−1) and k ∗ =(y ∗ /k ∗ ) 1/(α−1) . From (5.30)—(5.31) weget i = i ∗ =(γ + δ − 1)k. It follows that c = ωc w = ω[y + y ∗ − (i + i ∗ )]=2ω[y − i].Thus y − c − i =(1− 2ω)(y − i) and unless ω =1/2, the currentaccount will not be balanced in the steady state. If ω > 1/2 thehomecountry spends in excess of GDP and runs a current account deÞcit.How can this be? In the market (competitive equilibrium) interpretation,the excess absorption is Þnanced by interest income earned on pastlending to the foreign country. Foreigners need to produce in excess oftheir consumption and investment to service the debt. In a sense, theyhave ‘over-invested’ in physical capital.In the planning problem, the social planner simply takes away someof the foreign output and gives it to domestic agents. Due to the

154 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESconcavity of the production function, optimality requires that the worldcapital stock be split up between the two countries so as to equate themarginal product of capital at home and abroad. Since technology isidentical in the 2 countries, this implies equalization of national capitalstocks, k = k ∗ , and income levels y = y ∗ , even if consumption differs,c 6= c ∗ .Quadratic ApproximationYou can solve the model by taking the quadratic approximation of theunconstrained objective function about the steady state. Let R be theperiod weighted average of home and foreign utilityR(λ t )=ωU[g(λ t )] + (1 − ω)U[h(λ t )].Let R j = ωU c (c)g j +(1− ω)U c (c ∗ )h j , j =1,...,7betheÞrst partialderivative of R with respect to the j−the element of λ t . Denote thesecond partial derivative of R byR jk = ∂R(λ) = ω[U c (c)g jk +U cc g j g k ]+(1−ω)[U c (c ∗ )h jk +U cc (c ∗ )h j h k ].∂λ j ∂λ k(5.40)Let q =(R 1 ,...,R 7 ) 0 be the gradient vector, Q be the Hessian matrixof second partial derivatives whose j, k−th element is Q jk =(1/2)R j,k .Then the second-order Taylor approximation to the period utility functionisR(λ t )=[q +(λ t − λ) 0 Q](λ t − λ),and you can rewrite (5.36) asX ∞max E t β j [q +(λ t+j − λ) 0 Q](λ t+j − λ). (5.41)j=0Let Q j• be the j−th row of the matrix Q. TheÞrst-order conditionsare(k t+1 ): 0=R 1 + βR 3 + Q 1• (λ t − λ)+βQ 3• (λ t+1 − λ), (5.42)(kt+1) ∗ : 0=R 2 + βR 4 + Q 2• (λ t − λ)+βQ 4• (λ t+1 λ), (5.43)(c ∗ t ): 0=R 7 + Q 7• (λ t − λ). (5.44)

5.2. CALIBRATING A TWO-COUNTRY MODEL 155Now let a ‘tilde’ denote the deviation of a variable from its steady statevalue so that ˜k t = k t − k and write these equations out as0 = a 1˜kt+2 + a 2˜k∗ t+2 + a 3˜kt+1 + a 4˜k∗ t+1 + a 5˜kt + a 6˜k∗ t + a 7 Ã t+1+a 8 Ã ∗ t+1 + a 9Ãt + a 10 Ã ∗ t + a 11˜c ∗ t+1 + a 12˜c ∗ t + a 13, (5.45)0 = b 1˜kt+2 + b 2˜k∗ t+2 + b 3˜kt+1 + b 4˜k∗ t+1 + b 5˜kt + b 6˜k∗ t + b 7 Ã t+1+b 8 Ã ∗ t+1 + b 9Ãt + b 10 Ã ∗ t + b 11˜c ∗ t+1 + b 12˜c ∗ t + b 13, (5.46)0 = d 3˜kt+1 + d 4˜k∗ t+1 + d 5˜kt + d 6˜k∗ t + d 9 Ã t + d 10 Ã ∗ t+d 12˜c ∗ t + d 13 , (5.47)where the coefficients are given byj a j b j d j1 βQ 31 βQ 41 02 βQ 32 βQ 42 03 βQ 33 + Q 11 βQ 43 + Q 21 Q 714 βQ 34 + Q 12 βQ 44 + Q 22 Q 725 Q 13 Q 23 Q 736 Q 14 Q 24 Q 747 βQ 35 βQ 45 08 βQ 36 βQ 46 09 Q 15 Q 25 Q 7510 Q 16 Q 26 Q 7611 Q 37 Q 47 012 Q 17 Q 27 Q 7713 R 1 + βR 3 R 2 + βR 4 R 7Mimicking the algorithm developed for the one-country model andusing (5.47) to substitute out c ∗ t and c ∗ t+1 in (5.45) and (5.46) gives0 = ã 1˜kt+2 +ã 2˜k∗ t+2 +ã 3˜kt+1 +ã 4˜k∗ t+1 +ã 5˜kt +ã 6˜k∗ t +ã 7 Ã t+1+ã 8 Ã ∗ t+1 +ã 9 Ã t +ã 10 Ã ∗ t +ã 11 , (5.48)0 = ˜b 1˜kt+2 + ˜b 2˜k∗ t+2 + ˜b 3˜kt+1 + ˜b 4˜k∗ t+1 + ˜b 5˜kt + ˜b 6˜k∗ t + ˜b 7 Ã t+1+˜b 8 Ã ∗ t+1 + ˜b 9 Ã t + ˜b 10 Ã ∗ t + ˜b 11 . (5.49)

156 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESAt this point, the marginal beneÞt from looking at analytic expressionsfor the coefficients is probably negative. For the speciÞc calibration ofthe model the numerical values of the coefficients are,ã 1 =0.105, ˜b1 =0.105,ã 2 =0.105, ˜b2 =0.105,ã 3 = −0.218, ˜b3 = −0.212,ã 4 = −0.212, ˜b4 = −0.218,ã 5 =0.107, ˜b5 =0.107,ã 6 =0.107, ˜b6 =0.107,ã 7 = −0.128, ˜b7 = −0.161,ã 8 = −0.159, ˜b8 = −0.130,ã 9 =0.158, ˜b9 =0.158,ã 10 =0.158, ˜b10 =0.158,ã 11 =0.007, ˜b11 =0.007.You can see that ã 3 +ã 4 = ˜b 3 +˜b 4 and ã 7 +˜b 7 =ã 8 +˜b 8 which meansthat there is a singularity in this system. To deal with this singularity,let Ãw t = Ãt + Ã∗ t denote the ‘world’ technology shock and add (5.48)to (5.49) to getã 1˜kw t+2 +ã3 +ã 42˜k t+1+ã w + ˜b 75˜kw t +ã7 Ã w2t+1+ã 9 Ã w + ˜b 11t +ã11 2=0. (5.50)(5.50) is a second—order stochastic difference equation in ˜k t w = ˜k t + ˜k t ∗ ,which can be rewritten compactly as 4˜k t+2 w − m 1˜k t+1 w − m 2˜k tw = Wt+1 w (5.51)where Wt+1 w = m 3 Ã w t+1 + m 4 Ã w t ,andm 1 = −(ã 3 +ã 4 )/(2ã 1 ),m 2 = −ã 5 /ã 1 ,m 3 = −(ã 7 + ˜b 7 )/(2ã 1 ),m 4 = −ã 9 /ã 1 ,m 5 = −ã11 + ˜b 112ã 11.4 Unlike the one-country model, we don’t want to write the model in logs becausewe have to be able to recover ˜k and ˜k ∗ separately.

5.2. CALIBRATING A TWO-COUNTRY MODEL 157You can write second—order stochastic difference equation (5.51) as(1 − m 1 L − m 2 L 2 )ˆk t+1 w = Wt w . The roots of the polynomial(1 − m 1 z − m 2 z 2 )=(1− ω 1 L)(1 − ω 2 L) satisfy m 1 = ω 1 + ω 2 and m 2 =−ω 1 ω 2 . Under the parameter values used to calibrate the model and usingtheqquadratic formula, the roots are, z 1 = (1/ω 1 ) =[−m 1 − m 2 1 +4m 2 ]/(2m 2 ) ' 1.17, and z 2 = (1/ω 2 ) =q[−m 1 + m 2 1 +4m 2 ]/(2m 2 ) ' 0.84. The stable root |z 1 | > 1 lies outsidethe unit circle, and the unstable root |z 2 | < 1 lies inside the unit circle.From the law of motion governing the technology shocks (5.33), youhaveà w t+1 =(ρ + δ)Ãw t + ² w t , (5.52)where ² w t = ² t + ² ∗ t . Now E t W t+k = m 3 à w t+1 + m 4 à w t + m 5 =[m 3 (ρ + δ) +m 4 ](ρ + δ) k à w t + m 5 . As in the one-country model, usethese forecasting formulae to solve the unstable root forwards and thestable root backwards. The solution for the world capital stock is˜k t+1 w = ω 1˜k t w (ρ + δ) ³− [m3 (ρ + δ)+m 4ω 2 − (ρ + δ)t + m 5´. (5.53)Now you need to recover the domestic and foreign components ofthe world capital stock. Subtract (5.49) from (5.48) to get!Øb7˜k t+1 − ˜k t+1 ∗ = − ã 7ã 3 − ã 4!Øb8 − ã 8à t+1 + à ∗ t+1ã 3 − ã 4 (5.54)Add (5.53) to (5.54) to get˜k t+1 = 1 2 [˜k t+1 w +(˜k t+1 − ˜k t+1 ∗ )]. (5.55)The date t+1 world capital stock is predetermined at date t. Howthatcapital is allocated between the home and foreign country depends onthe realization of the idiosyncratic shocks Ãt+1 and Ã∗ t+1.Given ˜k t ,and˜k ∗ t , it follows from the production functions that theoutputs areỹ t = f A à t + f k˜kt = yÃt + α y k ˜k t , (5.56)ỹ ∗ t = f ∗ AÃ∗ t + f ∗ k ˜k ∗ t = y ∗ à ∗ t + α y∗k ∗ ˜k ∗ t , (5.57)

158 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESand investment rates areĩ t = γ˜k t+1 − (1 − δ)˜k t , (5.58)ĩ ∗ t = γ˜k ∗ t+1 − (1 − δ)˜k ∗ t . (5.59)Let world consumption be ˜c w t =˜c t +˜c ∗ t =ỹ t +ỹ ∗ t − (ĩ t + ĩ ∗ t ). By theoptimal risk-sharing rule (5.39) ˜c ∗ t =[(1− ω)/ω]˜c t,whichcanbeusedto determine˜c t = ω˜c w t . (5.60)It follows that ˜c ∗ t =˜c w t − ˜c t . The log-level of consumption is recoveredbyln(C t )=ln(X t )+ln(˜c t + c).Log levels of the other variables can be obtained in an analogous manner.Simulating the Two-Country ModelThe steady state values arey = y ∗ =1.53, k = k ∗ =3.66, i = i ∗ =0.42, c = c ∗ =1.11.The model is used to generate 96 time-series observations. Descriptivestatistics calculated using the Hodrick—Prescott Þltered cyclical parts ofthe log-levels of the simulated observations and are displayed in Table5.4 and Figure 5.4 shows the simulated current account balance.The simple model of this chapter makes many realistic predictions.It produces time-series that are persistent and that display coarse comovementsthat are broadly consistent with the data. But there arealso several features of the model that are inconsistent with the data.First, consumption in the two-country model is smoother than output.Second, domestic and foreign consumption are perfectly correlated dueto the perfect risk-sharing whereas the correlation in the data is muchlower than 1. A related point is that home and foreign output arepredicted to display a lower degree of co-movement than home andforeign consumption which also is not borne out in the data.

5.2. CALIBRATING A TWO-COUNTRY MODEL 1590.10.080.060.040.020-0.02-0.04-0.06-0.08-0.173 75 77 79 81 83 85 87 89 91 93 95 97Figure 5.4: Simulated current account to GDP ratio.Table 5.4: Calibrated Open-Economy ModelStd. AutocorrelationsDev. 1 2 3 4 6y t 0.022 0.66 0.40 0.15 0.07 0.04c t 0.017 0.63 0.42 0.18 0.12 -0.04i t 0.114 0.05 -0.13 -0.09 -0.10 0.03ex t 0.038 0.09 -0.09 -0.09 -0.10 -0.00yt ∗ 0.021 0.65 0.32 0.07 -0.15 -0.27c ∗ t 0.017 0.63 0.42 0.18 0.12 -0.04i ∗ t 0.116 0.03 -0.15 -0.07 -0.08 0.00Cross correlations at k6 4 1 0 -1 -4 -6ex t y t−k 0.00 0.18 0.41 0.44 0.21 0.15 0.15yt ∗ y t−k 0.10 0.06 0.27 0.18 0.06 0.28 0.05

160 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESInternational Real Business Cycles Summary1. The workhorse of real business cycle research is the dynamicstochastic general equilibrium model. These can be viewed asArrow-Debreu models and solved by exploiting the social planner’sproblem. They feature perfect markets and completelyfully ßexible prices. The models are fully articulated and arehave solidly grounded micro foundations.2. Real business cycle researchers employ the calibration method toquantitatively evaluate their models. Typically, the researchertakes a set of moments such as correlations between actual timeseries, and asks if the theory is capable of replicating these comovements.The calibration style of research stands in contrastwith econometric methodology as articulated in the Cowles commissiontradition. In standard econometric practice one beginsby achieving model identiÞcation, progressing to estimation ofthe structural parameters, and Þnally by conducting hypothesistests of the model’s overidentifying restrictions but how one determineswhether the model is successful or not in the calibrationtradition is not entirely clear.

Chapter 6Foreign Exchange MarketEfficiencyIn his second review article on efficient capital markets, Fama [49]writes,“I take the market efficiency hypothesis to be the simplestatement that security prices fully reßect all availableinformation.”He goes on to say,“. . . , market efficiency per se is not testable. It mustbe tested jointly with some model of equilibrium, an assetpricingmodel.”Market efficiency does not mean that asset returns are serially uncorrelated,nordoesitmeanthattheÞnancialmarkets present zeroexpected proÞts. The crux of market efficiency is that there are nounexploited excess proÞt opportunities. What is considered to be excessivedepends on the model of market equilibrium.This chapter is an introduction to the economics of foreign exchangemarket efficiency. We begin with an evaluation of the simplest model ofinternational currency and money-market equilibrium–uncovered interestparity. Econometric analyses show that it is strongly rejected by161

162CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYthe data. The ensuing challenge is then to understand why uncoveredinterest parity fails.We cover three possible explanations. The Þrst is that the forwardforeign exchange rate contains a risk premium. This argumentis developed using the Lucas model of chapter 4. The second explanationis that the true underlying structure of the economy is subject tochange occasionally but economic agents only learn about these structuralchanges over time. During this transitional learning period inwhich market participants have an incomplete understanding of theeconomy and make systematic prediction errors even though they arebehaving rationally. This is called the ‘peso-problem’ approach. Thethird explanation is that some market participants are actually irrationalin the sense that they believe that the value of an asset dependson extraneous information in addition to the economic fundamentals.The individuals who take actions based on these pseudo signals arecalled ‘noise’ traders.The notational convention followed in this chapter is to let uppercase letters denote variables in levels and lower case letters denote theirlogarithms, with the exception of interest rates, which are always denotedin lower case. As usual, stars are used to denote foreign countryvariables.6.1 Deviations From UIPLet s be the log spot exchange rate, f be the log one-period forwardrate, i be the one-period nominal interest rate on a domestic currency(dollar) asset and i ∗ is the nominal interest rate on the foreign currency(euro) asset. If uncovered interest parity holds, i t − i ∗ t =E t (s t+1 ) − s t ,but by covered interest parity, i t −i ∗ t = f t −s t . Therefore, unbiasednessof the forward exchange rate as a predictor of the future spot ratef t =E t (s t+1 ) is equivalent to uncovered interest parity.We begin by covering the basic econometric analyses used to detectthese deviations.

6.1. DEVIATIONS FROM UIP 163Hansen and Hodrick’s Tests of UIPHansen and Hodrick [71] use generalized method of moments (GMM)to test uncovered interest parity. The GMM method is covered inchapter 2.2. The Hansen—Hodrick problem is that a moving-average serialcorrelation is induced into the regression error when the predictionhorizon exceeds the sampling interval of the data.The Hansen—Hodrick ProblemToseehowtheproblemarises,letf t,3 be the log 3-month forward exchangerate at time t, s t be the log spot rate, I t be the time t information ⇐(102)setavailabletomarketparticipants,andJ t be the time t informationset available to you, the econometrician. Even though you are workingwith 3-month forward rates, you will sample the data monthly. Youwant to test the hypothesisH 0 :E(s t+3 |I t )=f t,3 .In setting up the test, you note that I t is not observable but since J t isasubsetofI t and since f t,3 is contained in J t , you can use the law ofiterated expectations to testH 0 0 :E(s t+3|J t )=f t,3 ,which is implied by H 0 . You do this by taking a vector of economicvariables z t−3 in J t−3 , running the regressions t − f t−3,3 = z 0 t−3 β + ² t,3,and doing a joint test that the slope coefficients are zero.Under the null hypothesis, the forward rate is the market’s forecastof the spot rate 3 months ahead f t−3,3 =E(s t |J t−3 ). The observations,however, are collected every month. Let J t =(² t ,² t−1 ,...,z t ,z t−1 ,...).The regression error formed at time t − 3is² t = s t − E(s|J t−3 ). Att − 3, E(² t |J t−3 )=E(s t − E(s t |J t−3 )) = 0 so the error term is un- ⇐(103)predictable at time t − 3 when it is formed. But at time t − 2andt − 1 you get new information and you cannot say that E(² t |J t−1 )=E(s t |J t−1 )−E[E(s t |J t−3 )|J t−1 ] is zero. Using the law of iterated expectations,the Þrst autocovariance of the error E(² t ² t−1 )=E(² t−1 E(² t |J t−1 ))

164CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYneed not be zero. You can’t say that E(² t ² t−2 )iszeroeither.Youcan,however, say that E(² t ² t−k ) = 0 for k ≥ 3. When the forecast horizonof the forward exchange rate is 3 sampling periods, the error term ispotentially correlated with 2 lags of itself and follows an MA(2) process.If you work with a k − period forward rate, you must be preparedfor the error term to follow an MA(k-1) process.Generalized least squares procedures, such as Cochrane-Orcutt orHildreth-Lu, covered in elementary econometrics texts cannot be usedto handle these serially correlated errors because these estimators areinconsistent if the regressors are not econometrically exogenous. Researchersusually follow Hansen and Hodrick by estimating the coefficientvector by least squares and then calculating the asymptotic covariancematrix assuming that the regression error follows a movingaverage process. Least squares is consistent because the regression error² t , being a rational expectations forecast error under the null, isuncorrelated with the regressors, z t−3 . 1Hansen-Hodrick Regression Tests of UIPHansen and Hodrick ran two sets of regressions. In the Þrst set, theindependent variables were the lagged forward exchange rate forecasterrors (s t−3 −f t−6,3 ) of the own currency plus those of cross rates. In thesecond set, the independent variables were the own forward premiumand those of cross rates (s t−3 −f t−3,3 ). They rejected the null hypothesisat very small signiÞcance levels.Let’s run their second set of regressions using the dollar, pound,1 To compute the asymptotic covariance matrix of the least-squares vector,follow the GMM interpretation of least squares developed in chapter 2.2. Assumethat ² t is conditionally homoskedastic, and let w t = z t−3 ² t . We haveE(w t wt)=E(² 0 2 t z t−3 z 0 t−3 )=E(E[²2 t z t−3 z 0 t−3 |z t−3 ]) = γ 0E(z t−3 z 0 t−3 )=γ 0Q 0 ,whereγ 0 = E(² 2 t ) and Q = E(z t−3 z 0 t−3). Now, E(w t wt−1) 0 = E(² t ² t−1 z t−3 z 0 t−4) =E(E[² t ² t−1 z t−3 z 0 t−4 |z t−3 ,z t−4 ]) = E(z t−3 z0 t−4 E[² t² t−1 |z t−3 ,z t−4 ]) = γ 1 Q 1 , whereγ 1 =E(² t ² t−1 ), and Q 1 =E(z t−3 z t−4 ). By an analogous argument, E(w t wt−2) 0 =γ 2 Q 2 ,andE(w t wt−k 0 )=0, fork ≥ 3. Now, D =E(∂(z t ² t)/∂β 0 )=Q 0 so theasymptotic covariance matrix for the least squares estimator is, (Q 0 0 W−1 Q 0 ) −1where W = γ 0 Q 0 + P 2j=1 γ j(Q j + Q 0 j ). Actually, Hansen and Hodrick used weeklyobservations with the 3-month forward rate which leads the regression error tofollow an MA(11).

6.1. DEVIATIONS FROM UIP 165yen, and deutschemark. The dependent variable is the realized forwardcontract proÞt, which is regressed on the own and cross forward premia.The 350 monthly observations are formed by taking observations fromevery fourth Friday. From March 1973 to December 1991, the dataare from the Harris Bank Foreign Exchange Weekly Review extendingfrom March 1973 to December 1991. From 1992 to 1999, the data ⇐(107)are from Datastream. The Wald test that the slope coefficients arejointly zero with p-values are given in Table 6.1. The Wald statisticsare asymptotically χ 2 3 under the null hypothesis. Two versions of theasymptotic covariance matrix are estimated. Newey and West with 6lags (denoted Wald(NW[6])), and Hansen-Hodrick with 2 lags (denotedWald(HH[2])). In these data, UIP is rejected at reasonable levels ofsigniÞcance for every currency except for the dollar-deutschemark rate.Table 6.1: Hansen-Hodrick tests of UIPUS-BP US-JY US-DM DM-BP DM-JY BP-JYWald(NW[6]) 16.23 400.47 5.701 66.77 46.35 294.31p-value 0.001 0.000 0.127 0.000 0.000 0.000Wald(HH[2]) 16.44 324.85 4.299 57.81 32.73 300.24p-value 0.001 0.000 0.231 0.000 0.000 0.000Notes: Regression s t −f t−3,3 = z 0 t−3 β +² t,3 estimated on monthly observations from1973,3 to 1999,12. Wald is the Wald statistic for the test that β = 0. Asymptoticcovariance matrix estimated by Newey-West with 6 lags (NW[6]) and by Hansen—Hodrick with 2 lags (HH[2]).The Advantage of Using Overlapping ObservationsThe Hansen—Hodrick correction involves some extra work. Are the beneÞtsobtained by using the extra observations worth the extra costs?Afterall, you can avoid inducing the serial correlation into the regressionerror by using nonoverlapping quarterly observations but then youwould only have 111 data points. Using the overlapping monthly observationsincreases the nominal sample size by a factor of 3 but theeffective increase in sample size may be less than this if the additionalobservations are highly dependent.

166CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYTable 6.2: Monte Carlo Distribution of OLS Slope Coefficients andT-ratios using Overlapping and Nonoverlapping Observations.Overlapping percentiles RelativeT Observations 2.5 50 97.5 Range50 yes slope 0.778 0.999 1.207 0.471t NW (-2.738) (-0.010) (2.716) 1.207t HH [-2.998] [-0.010] [3.248] 1.38316 no slope 0.543 0.998 1.453t OLS ((-2.228)) ((-0.008)) ((2.290))100 yes slope 0.866 0.998 1.126 0.474t NW (-2.286) (-0.025) (2.251) 1.098t HH [-2.486] [-0.020] [2.403] 1.18333 no slope 0.726 0.996 1.274t OLS ((-2.105)) ((-0.024)) ((2.026))300 yes slope 0.929 1.001 1.074 0.509t NW (-2.071) (0.021) (2.177) 1.041t HH [-2.075] [-0.016] [2.065] 1.014100 no slope 0.858 1.003 1.143t OLS ((-2.030)) ((0.032)) ((2.052))Notes: True slope = 1. t NW : Newey—West t-ratio. t HH : Hansen—Hodrick t-ratio.t OLS : OLS t-ratio. Relative range is ratio of the distance between the 97.5 and2.5 percentiles in the Monte Carlo distribution for the statistic constructed usingoverlapping observations to that constructed using nonoverlapping observations.(108)⇒The advantage that one gains by going to monthly data are illustratedin table 6.2 which shows the results of a small Monte Carlo experimentthat compares the two (overlapping versus nonoverlapping)strategies. The data generating process isy t+3 = x t + ² t+3 ,x t =0.8x t−1 + u t ,² tiid∼ N(0, 1),u tiid∼ N(0, 1),where T is the number of overlapping (monthly) observations. y t+3 isregressed on x t and Newey-West t-ratios t NW are reported in parentheses.5 lags were used for T =50, 100 and 6 lags used for T = 300.

6.1. DEVIATIONS FROM UIP 167Hansen-Hodrick t-ratios t HH are given in square brackets and OLS t-ratios t OLS are given in double parentheses. The relative range is the2.5 to 97.5 percentile of the distribution with overlapping observationsdivided by the 2.5 to 97.5 percentile of the distribution with nonoverlappingobservations. 2 The empirical distribution of each statistic isbased on 2000 replications.You can see that there deÞnitely is an efficiency gain to using overlappingobservations. The range encompassing the 2.5 to 97.5 percentilesof the Monte Carlo distribution of the OLS estimator shrinksapproximately by half when going from nonoverlapping (quarterly) tooverlapping (monthly) observations. The tradeoff is that for very smallsamples, the distribution of the t-ratios under overlapping observationsare more fat-tailed and look less like the standard normal distributionthan the OLS t-ratios.Fama Decomposition RegressionsAlthough the preceding Monte Carlo experiment suggested that youcan achieve efficiency gains by using overlapping observations, in theinterests of simplicity, we will go back to working with the log oneperiodforward rate, f t = f t,1 to avoid inducing the moving averageerrors.DeÞne the expected excess nominal forward foreign exchange payoffto bep t ≡ f t − E t [s t+1 ], (6.1)where E t [s t+1 ]=E[s t+1 |I t ]. You already know from the Hansen—Hodrickregressions that p t is non zero and that it evolves overtime as a randomprocess. Adding and subtracting s t from both sides of (6.1) givesf t − s t =E t (s t+1 − s t )+p t . (6.2)Fama [48] shows how to deduce some properties of p t using the analysisof omitted variables bias in regression problems. First, considerthe regression of the ex post forward proÞt f t − s t+1 on the currentperiod forward premium f t − s t . Second, consider the regression of the2 Forexample,wegettherow1 relative range value 0.471 for the slope coefficientfrom (1.207-0.778)/(1.453-0.543).

168CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYone-period ahead depreciation s t+1 − s t on the current period forwardpremium. The regressions aref t − s t+1 = α 1 + β 1 (f t − s t )+ε 1t+1 , (6.3)s t+1 − s t = α 2 + β 2 (f t − s t )+ε 2t+1 . (6.4)(6.3) and (6.4) are not independent because when you add them togetheryou getα 1 + α 2 = 0,β 1 + β 2 = 1, (6.5)ε 1t+1 + ε 2t+1 = 0. (6.6)In addition, these regressions have no structural interpretation. Sowhy was Fama interested in running them? Because it allowed him toestimate moments and functions of moments that characterize the jointdistribution of p t and E t (s t+1 − s t ).The population value of the slope coefficient in the Þrst regression(6.3) is β 1 =Cov[(f t − s t+1 ), (f t − s t )]/Var[f t − s t ]. Using the deÞnitionof p t , it follows that the forward premium can be expressed as,f t − s t = p t +E(∆s t+1 |I t ) whose variance is Var(f t − s t )=Var(p t )+Var[E(∆s t+1 |I t )]+2Cov[p t , E(∆s t+1 |I t )]. Now add and subtract E(s t+1 |I t )to the realized proÞt togetf t − s t+1 = p t − u t+1 where u t+1 = s t+1 −E(s t+1 |I t )=∆s t+1 − E(∆s t+1 |I t ) is the unexpected depreciation. Nowyou have, Cov[(f t −s t+1 ), (f t −s t )] = Cov[(p t −u t+1 ), (p t +E(∆s t+1 |I t ))]=Var(p t )+Cov[p t , E(∆s t+1 |I t ))]. With the aid of these calculations,the slope coefficient from the Þrst regression can be expressed asβ 1 =Var(p t )+Cov[p t , E t (∆s t+1 )]Var(p t )+Var[E t (∆s t+1 )] + 2Cov[p t , E t (∆s t+1 )] . (6.7)In the second regression (6.4), the population value of the slope coefficientis β 2 =Cov[(∆s t+1 ), (f t −s t )]/Var(f t −s t ). Making the analogoussubstitutions yieldsβ 2 =Var[E t (∆s t+1 )] + Cov[p t ,E t (∆s t+1 )]Var(p t )+Var[E t (∆s t+1 )] + 2Cov[p t , E t (∆s t+1 )] . (6.8)

6.1. DEVIATIONS FROM UIP 169Table 6.3: Estimates of Regression Equations (6.3) and (6.4)US-BP US-JY US-DM DM-BP DM-JY BP-JYˆβ 2 -3.481 -4.246 -0.796 -1.645 -2.731 -4.295t(β 2 =0) (-2.413) (-3.635) (-0.542) (-1.326) (-1.797) (-2.626)t(β 2 =1) (-3.107) (-4.491) (-1.222) (-2.132) (-2.455) (-3.237)ˆβ 1 4.481 5.246 1.796 2.645 3.731 5.295Notes: Nonoverlapping quarterly observations from 1976.1 to 1999.4. t(β 2 =0)(t(β 2 = 1) isthet-statistictotestβ 2 =0(β 2 = 1).Let’s run the Fama regressions using non-overlapping quarterly observationsfrom 1976.1 to 1999.4 for the British pound (BP), yen (JY),deutschemark (DM) and dollar (US). We get the following results.There is ample evidence that the forward premium contains usefulinformation for predicting the future depreciation in the (generally) signiÞcantestimates of β 2 .Sinceˆβ 2 is signiÞcantly less than 1, uncoveredinterest parity is rejected. The anomalous result is not that β 2 6=1,but that it is negative. The forward premium evidently predicts thefuture depreciation but with the “wrong” sign from the UIP perspective.Recall that the calibrated Lucas model in chapter 4 also predictsanegativeβ 2 for the dollar-deutschemark rate.The anomaly is driven by the dynamics in p t . Wehaveevidencethat it is statistically signiÞcant. The next question that Fama asks iswhether p t is economically signiÞcant. Is it big enough to be economicallyinteresting? To answer this question, we use the estimates andthe slope-coefficient decompositions (6.7) and (6.8) to get informationabout the relative volatility of p t .First note that ˆβ 2 < 0. From (6.8) it follow that p t must be negativelycorrelated with the expected depreciation,Cov[p t , E(∆s t+1 |I t )] < 0. By (6.5), the negative estimate of β 2 impliesthat ˆβ 1 > 0. By (6.7), it must be the case that Var(p t ) is large enoughto offset the negative Cov(p t , E t (∆s t+1 )). Since ˆβ 1 − ˆβ 2 > 0, it followsthat Var(p t ) > Var(E(∆s t+1 |I t )), which at least places a lower boundon the size of p t .

170CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY(109)⇒Estimating p tWe have evidence that p t = f t − E(s t+1 |I t ) evolves as a random process,but what does it look like? You can get a quick estimate of p tby projecting the realized proÞt f t − s t+1 = p t − u t+1 on a vector ofobservations z t where u t+1 = s t+1 − E(s t+1 |I t ) is the rational predictionerror. Using the law of iterated expectations and the property thatE(u t+1 |z t )=0,youhaveE(f t − s t+1 |z t )=E(p t |z t ). If you run theregressionf t − s t+1 = z 0 tβ + u t+1 ,you can use the Þtted value of the regression as an estimate of the exante payoff, ˆp t = z 0 ˆβ. tA slightly more sophisticated estimate can be obtained from a vectorerror correction representation that incorporates the joint dynamics ofthe spot and forward rates. Here, the log spot and forward rates are assumedto be unit root processes and the forward premium is assumed tobe stationary. The spot and forward rates are cointegrated with cointegrationvector (1, −1). As shown in chapter 2.6, s t and f t have a vectorerror correction representation which can be represented equivalentlyas a bivariate vector autoregression in the forward premium (f t − s t )and the depreciation ∆s t .Let’s pursue the VAR option. Let y t=(f t − s t , ∆s t ) 0 follow thek−th order VARy t=kXj=1A j y t−j+ v t .Let e 2 =(0, 1) be a selection vector such that e 2 y t= ∆s t picks off thedepreciation, and H t =(y t,y t−1,...) be current and lagged values ofh Pkj=1 iy t.ThenE(∆s t+1 |H t )=e 2 E(y t+1|H t )=e 2 A j y t+1−j ,andyou

6.1. DEVIATIONS FROM UIP 171A. US-UK50403020100-10-20-30-40-501976 1978 1980 1982 1984 1986 1988 1990 1992504030B. US-Germany20100-10-20-301976 1978 1980 1982 1984 1986 1988 1990 1992C. US-Japan50403020100-10-20-30-401976 1978 1980 1982 1984 1986 1988 1990 1992Figure 6.1: Time series point estimates of p t (boxes) with 2-standarderror bands and point estimates of E t (∆s t+1 )(circles).

172CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYcan estimate p t with⎡ˆp t =(f t − s t ) − e ⎣ 2kXj=1⎤Â j y ⎦ t+1−j. (6.9)(110)⇒Mark and Wu [102] used the VAR method to get quarterly estimatesof p t for the US dollar relative to the deutschemark, pound, and yen.Their estimates, shown in Figure 6.1, show that of E(∆s t+1 |H t )arepersistent for the pound and yen. Both ˆp t and Ê(∆s t+1|H t )alternatebetween positive and negative values but they change sign infrequently.The cross-sectional correlation across the three exchange rates is alsoevident. Each of the series spikes in early 1980 and 1981, the ˆp t saregenerally positive during the period of dollar strength from mid-1980 to1985 and are generally negative from 1990 to late 1993. You can alsosee in the Þgures the negative covariance between ˆp t and Êt(∆s t+1 )deduced by Fama’s regressions.Deviations from uncovered interest parity are a stylized fact of theforeign exchange market landscape. But whether the stochastic p t termßoating around is the byproduct of an inefficient market is an unresolvedissue. As per Fama’s deÞnition, we say that the foreign exchangemarket is efficient if the relevant prices are determined in accordancewith a model of market equilibrium. One possibility is that p t is a riskpremium. At this point, we revisit the Lucas model and use it to placesome structure on p t .6.2 Rational Risk PremiaHodrick [75] and Engel [44] show how to use the Lucas model to priceforward foreign exchange. We follow their use of Lucas model to understanddeviations from uncovered interest parity.Recall that forward foreign exchange contracts are like nominalbonds in the Lucas model in that they are not actually traded. Weare calculating shadow prices that keep them off the market. Let S t isthe nominal spot exchange rate expressed as the home currency priceofaunitofforeigncurrencyandF t be the price the foreign currencyfor one-period ahead delivery.

6.2. RATIONAL RISK PREMIA 173The intertemporal marginal rate of substitution will play a key role.In aggregate asset-pricing applications, it is common to work with percapita consumption data. One way to justify using such data in theutility function in Lucas’s two-country model is to assume that the periodutility function is homothetic and that the relative price betweenthe home good and the foreign good (the real exchange rate) is constant.This allows you to write the representative agent’s intertemporalmarginal rate of substitution between t and t +1asµ t+1 = βu0 (C t+1 ), (6.10)u 0 (C t )where u 0 (C t ) is the representative agent’s marginal utility evaluated atequilibrium consumption. 3Let P t be the domestic price level and let β isthesubjectivediscountfactor. A speculative position in a forward contract requires noinvestment at time t. If the agent is behaving optimally, the expectedmarginal utility from the real payoff from buying the foreign currencyforward is E t [u 0 (c t+1 )(F t − S t+1 )/P t+1 ] = 0. To express the Euler equationin terms of stationary random variables so that their unconditionalvariances and unconditional covariances between random variables exist,multiply both sides by β and divide by u 0 (c t )toget"#F t − S t+1E t µ t+1 . =0, (6.11)P t+1(6.11) is key to understanding the demand for forward foreign exchangerisk-premia in the intertemporal asset pricing framework. Keep in mindthat the intertemporal marginal rate of substitution varies inverselywith consumption growth so that when the agent experiences the goodstate, consumption growth is high and the intertemporal marginal rateof substitution is low.3 If the period utility function in Lucas’s two-good model isu(C t )= C1−γ twith C1−γ t = Cxt θ C1−θ yt the intertemporal marginal rate of substitutionis β(C t+1 /C t ) 1−γ (C xt /C xt+1 ). But if the relative price between X and Y isconstant, the growth rate of consumption of X is the same as the growth rate ofthe consumption index and the intertemporal marginal rate of substitution becomesthat in (6.10)

174CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYCovariance decomposition and Euler equations. We will use the propertythat the covariance between any two random variables X t+1 andY t+1 can be decomposed asCov t (X t+1 ,Y t+1 )=E t (X t+1 Y t+1 ) − E t (X t+1 )E t (Y t+1 ).For a particular deÞnition of X and Y , the theory, embodied in (6.11)restricts E t (X t+1 Y t+1 ) = 0. Using this restriction in the covariancedecomposition and rearranging givesThe Real Risk PremiumE t (Y t+1 )= −Cov t(X t+1 ,Y t+1 ). (6.12)E t (X t+1 )(111)⇒Set Y t+1 =(F t − S t+1 )/P t+1 and X t+1 = µ t+1 in (6.11) and use (6.12)to get" #Ft − S t+1E t = −Cov h³ ´ iFt−S t+1t ,µt+1P t. (6.13)P t+1 E t µ t+1The forward rate is in general not the rationally expected future spot inthe Lucas model. The expected forward contract payoff is proportionalto the conditional covariance between the payoff and the intertemporalmarginal rate of substitution. The factor of proportionality is−1/E t (µ t+1 ) which is the ex ante gross real interest rate multiplied by−1.h iHow do we make sense of (6.13)? Suppose that E Ft−S t+1t P t+1< 0.Then the covariance on the right side is positive. You expect to generateaproÞt by buying the foreign currency (euros) forward and resellingthem in the spot market at E t (S t+1 ). A corresponding strategy thatexploits the deviation from uncovered interest parity is to borrow thehome currency (dollars) and lend uncovered in the foreign currency(euros). The market pays a premium to those investors who are willingto hold euro-denominated assets. It follows that the euro must be therisky currency. If you are holding the euro forward, the high payoffstates occur when h iF t −S t+1P t+1is negative. By the covariance term in(6.13), these states are associated with low realizations of µ t+1 . But

6.2. RATIONAL RISK PREMIA 175µ t+1 is low when consumption growth is high. What it boils down to isthis. Holding the euro forward pays off well in good states of the worldbut you don’t need an asset to pay off well in the good state. You wantassets to pay off well in the bad state—when you really need it. But theforward euro will pay off poorly in the bad state and in that sense it isrisky.h iIf the euro is risky the dollar is safe. If EFt −S t+1t P t+1< 0andyoubuy the dollar forward, you expect to realize a loss. It might seemlike a strange idea to buy an asset with expected negative payoff, butthis is something that risk-averse individuals are willing to do if theasset provides consumption insurance by providing high payoffs inbad(low growth) consumption states. The expected negative payoff can beviewed as an insurance premium.To summarize, in Lucas’s intertemporal asset pricing model, therisk of an asset lies in the covariance of its payoff with something thatindividuals care about—namely consumption. Assets that generate highpayoffs in the bad state offer insurance against these bad states and areconsidered safe. A high payoff during the good state is less valuable tothe individual than it is during the bad state due to the concavity ofthe utility function. Risk-averse individuals require compensation byway of a risk premium to hold the risky assets.Risk-neutral forward exchange. If individuals are risk neutral, the intertemporalmarginal rate of substitution µ t+1 is constant. Since thecovariance of any random variable with a constant is 0, (6.13) becomesà ! à !Ft St+1E t =E t . (6.14)P t+1 P t+1So even under risk-neutrality the forward rate is not the rationallyexpected future spot rate because you need to divide by the futureand stochastic price level. To see more clearly how covariance risk isrelated to the fundamentals, it is useful to take a look at expectednominal speculative proÞts.

176CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYThe Nominal Risk Premium.Multiply (6.11) by P t and divide through by S t to get"Ã ! µFt P t − S # t+1E t µ t+1 =0.P t+1 S tLetµ m t+1 = µ P tt+1 . (6.15)P t+1Since 1 P tis the purchasing power of one domestic currency unit andu 0 (C t)P tis the marginal utility of money, we will call µ m t+1 the intertemporalmarginal rate of substitution of money. In chapter 4, (equation(4.62)) we found that the price of a one-period riskless domestic currencynominal bond is (1 + i t ) −1 =E t (µ m t+1).Using (6.12), set Y t+1 = (Ft−S t+1)P t+1and X t+1 = µ m t+1. Because F tS tisknown at date t, it can be treated as a constant and you get¸··Ft − S t+1E t =(1+i t )Cov tS tµ m t+1 , S t+1S t¸. (6.16)Perhaps now you can see more clearly why the foreign currency (euro)is risky h when i the forward euro contract offersanexpectedproÞt. IfE Ft−S t+1t P t+1< 0, the covariance in (6.16) must be negative. In the badstate, µ m t+1 is high because consumption growth is low. This is associatedwith a weakening of the euro (low values of S t+1S t). The euro is riskybecause its value is positively correlated with consumption. Agents consumeboth the domestic and the foreign goods but the foreign currencybuysfewerforeigngoodsinthebadstateofnatureandisthereforeabad hedge against low consumption states.Pitfalls in pricing nominal contracts. Suppose that individuals are riskneutral. Then µ m t+1 = βP tP t+1and the covariance in (6.16) need not be0 and again you can see that the forward rate is not necessarily therationally expected future spot rate. Agents care about real proÞts,not nominal proÞts. Under risk neutrality, equilibrium expected realproÞts are 0, but in order to achieve zero expected real proÞts, theforward rate may have to be a biased predictor of the future spot.

6.3. TESTING EULER EQUATIONS 177This is why market efficiency does not mean that the exchange ratemust follow a random walk or that uncovered interest parity must hold.The Lucas model predicts that in equilibrium, it is the marginal utilityof the forward contract payoff that is unpredictable and that deviationsfrom UIP can emerge as compensation for risk bearing.6.3 Testing Euler EquationsUsing the methods of Hansen and Singleton [73], Mark [100] estimatedand tested the Euler equation restrictions using 1-month forward exchangerates. Modjtahedi [106] goes a step further and tests impliedEuler equation restrictions across the entire a term structure availablefor forward rates (1, 3, 6, and 12 months). The strategy is to estimatethe coefficient of relative risk aversion γ and test the orthogonalityconditions implied by the Euler equation (6.11) using GMM.Here, we use non-overlapping quarterly observations on dollar ratesof the pound, deutschemark, and yen from 1973.1 to 1997.1 and revisitMark’s analysis. To write the problem compactly, let r t+1 be the 3 × 1forward foreign exchange payoff vectorr t+1 =and let the 3 × 1vectorw t+1 be⎡⎢⎣(F 1t −S 1t+1 )(S 1t )(F 2t −S 2t+1 )(S 2t )(F 3t −S 3t+1 )(S 3t )⎤⎥⎦ ,w t+1 = µ m t+1r t+1 , (6.17)where µ m t+1 is the US representative investor’s intertemporal marginalrate of substitution of money under CRRA utility, u(C) =C 1−γ /(1−γ).Using the notation developed here to rewrite the Euler equations(6.11) you getE[w t+1 |I t ]=0. (6.18)Divide both sides by β sothatyouonlyneedtoestimateγ. (6.18)says that w t+1 is uncorrelated with any time-t information. Let z tbe a k−dimensional vector of time-t ‘instrumental variables,’ available

178CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYto you, the econometrician. Then (6.18) implies the following 3 × kestimable and testable equations 4E[w t+1 ⊗ z t ]=E[(µ m t+1 r t+1) ⊗ z t ]=0. (6.19)Now the question is what to choose for z t ? It is not a good idea touse too many variables because the estimation problem will becomeintractable and the small sample properties of the GMM estimator willsuffer. A good candidate is the forward premium since we know thatit is directly relevant to the problem at hand. Furthermore, it is notnecessary to use all the possible orthogonality conditions. To reducethe dimensionality of the estimation problem further, for each currencyi, letz it ="#1(F it −S it )S itbeavectorofinstrumentalvariablesconsistingoftheconstant1,andthe normalized forward premium. Estimating γ from the system of sixequations⎡⎤w 1t+1 z 1t⎢⎥E ⎣ w 2t+1 z 2t ⎦ , (6.20)w 3t+1 z 3tgives ˆγ =48.66 with asymptotic standard error of 79.36. The coef-Þcient of relative risk aversion is uncomfortably large and impreciselyestimated. However, the test of the Þve overidentifying restrictionsgives a chi-square statistic of 7.20 (p-value=0.206) does not reject atstandard levels of signiÞcance.Why does the data force ˆγ to be so big? We can get some intuitionby recasting the problem as a regression. Suppose you look at justone currency. If ( C t PC t+1, tP t+1, F t−S t+1S t) are jointly lognormally distributedthen w t+1 is also lognormal. 5 Taking logs, of both sides of (6.17), youµ 4 a11 a⊗ denotes the Kronecker product. Let A = 12and B be any n × ka 21 aµ 22a11 B amatrix. Then A ⊗ B =12 B.a 21 B a 22 B5 A random variable X is said to be lognormally distributed if ln(X) is normallydistributed.

6.3. TESTING EULER EQUATIONS 179getµ Ã ! Ã !Ft − S t+1 PtCtln+ln = −γ ln +lnw t+1 .S t P t+1 C t+1ln(C t /C t+1 ) is correlated with ln w t+1 so you don’t get consistent estimateswith OLS—but you do get consistency with instrumental variablesand this is what GMM does. However, the regression analogytells us that the large estimate of γ and its large standard error canbe attributed to high variability in the excess return combined withlow variability in consumption growth. The difficulty that the Lucasmodel under CRRA utility to explain the data with small values of γ isnot conÞned to the foreign exchange market. The corresponding difficultyfor the model to simultaneously explain historical stock and bondreturns is what Mehra and Prescott [105] call the ‘equity premiumpuzzle’.Volatility BoundsHansen and Jagannathan [72] propose a framework to evaluate the extentto which the Euler equations from representative agent asset pricingmodels satisfy volatility restrictions on the intertemporal marginalrate of substitution.We will Þrst derive a lower bound on the volatility of the intertemporalmarginal rate of substitution predicted by the Euler equations ofthe intertemporal asset pricing model. Let r t+1 be an N-dimensionalvector of holding period returns from t to t+1 available to the agent,and µ t+1 = βu 0 (C t+1 )/u 0 (C t ) be the intertemporal marginal rate of substitution.We need to write the Euler equations in returns form. For equities,they take the form 1 = E t (µ t+1 rt+1) e wherert+1 e is the gross return. 6 Itreads—an asset with expected payoff E t (µ t+1 rt+1) e costsoneunitoftheconsumption good. An analogous returns form of the Euler equationholds for bonds. In the case of forward foreign exchange contracts, thereis no investment required in the current period so the returns form for(Fthe Euler equation is 0 = E t µ t−S t+1 )t+1 P t. Thus, the returns form of6 Take the equity Euler equation (4.12)) and divide both sides by e t u 1,t+1 . Letr e t+1 =(e t+1 + x t+1 )/e t to get the expression in the text.

180CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYthe Euler equations for asset pricing can generically be represented asv =E t (µ t+1 r t+1 ), (6.21)where v isavectorofconstantswhosei − th element v i =1ifassetiis a stock or bond, and v i =0ifasseti is a forward foreign exchangecontract.Taking the unconditional expectation on both sides of (6.21) andusing the law of iterated expectations givesv = E(µ t+1 r t+1 ). (6.22)(112)⇒Let θ µ ≡ E(µ t ), σµ 2 ≡ E(µ t − θ µ ) 2 , θ r ≡ E(r t ), andΣ r ≡ E(r t − θ r )(r t − θ r ) 0 .Project(µ t − θ µ )onto(r t − θ r )toobtain(µ t − θ µ )=(r t − θ r ) 0 β µ+ u t , (6.23)where β µis a vector of least squares projection coefficients, u t is theleast squares projection error andFurthermore, you know thatβ µ= Σ −1r E(r t − θ r )(µ t − θ µ ). (6.24)E(r t − θ r )(µ t − θ µ )=E(r t µ t ) −θ r θ µ , (6.25)| {z }vwhere E(r t µ t )=v comes from the returns form of the Euler equations.Upon substituting (6.25) into (6.24), we get, β µ= Σ −1r (v − θ r θ µ ).Computing the variance of the intertemporal marginal rate of substitutiongivesσ 2 µ = E(µ t − θ µ ) 2= E(µ t − θ µ ) 0 (µ t − θ µ )= E[(r t − θ r ) 0 β µ+ u t ] 0 [(r t − θ r ) 0 β µ+ u t ]= E[β 0 (r µ t − θ r )(r t − θ r ) 0 β µ]+σu 2 + β0 E(r µ t − θ r )u t + Eu t (r t − θ r ) 0 β µ| {z }(a)= E(µ t r t − θ µ θ r ) 0 Σ −1r Σ r Σ −1r E(µ t r t − θ µ θ r )+σu2= (v − θ µ θ r ) 0 Σ −1r (v − θ µθ r )+σu 2 . (6.26)

6.3. TESTING EULER EQUATIONS 181The term labeled (a) above is zero because u t is the least-squares projectionerror and is by construction orthogonal to r t .Sinceσ 2 u ≥ 0, thevolatility or standard deviation of the intertemporal marginal rate ofsubstitution must lie above σ r whereσ µ ≥ σ r ≡q(v − θ µ θ r ) 0 Σ −1r (v − θ µθ r ). (6.27)The right side of (6.27) is the lower bound on the volatility of theintertemporal marginal rate of substitution. If the assets are all equitiesor bonds v is a vector of ones and the volatility bound is a parabola in(θ µ , σ µ ) space. If the assets are all forward foreign exchange contracts,v is a vector of zeros and the lower volatility bound is a ray from theoriginh iσ r = θ µ θ0rΣ −1 1/2r θ r . (6.28)How does one construct and use the volatility bound in practice?First determine v and calculate θ r and Σ r from asset price data. Thenusing (6.28) you trace out σ r as a function of θ µ . Next, for a given functionalform of the utility function, use consumption data to calculatethe volatility of the intertemporal marginal rate of substitution, σ µ .Compare this estimate to the volatility bound and determine whetherthe bound is satisÞed.When we do this using quarterly US consumption and CPI dataand dollar exchange ratesqfor the pound, deutschemark, and yen from1973.1 to 1997.1, we get θ 0 rΣ −1 θ r =0.309. Now let the utility function ⇐(113)be CRRA with relative risk aversion coefficient γ. As we vary γ, wegenerate the entries in the following table.γ θ µ σ µ σ r2 0.982 0.015 0.3034 0.974 0.031 0.30110 0.953 0.078 0.29420 0.923 0.159 0.28530 0.901 0.248 0.27840 0.886 0.349 0.27350 0.879 0.469 0.27260 0.881 0.615 0.272

182CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYYou can see that σ µ < σ r for values of γ below 30. This means ⇐(115)that exchange rate payoffs are too volatile relative to the fundamentals(the intertemporal marginal rate of substitution) over this range of γ.Note how the GMM estimate of γ = 48 obtained earlier in this chapteris consistent with this result. In order to explain the data, the Lucasmodel with CRRA utility requires people to be very risk averse. Manypeople feel that the degree of risk aversion associated with γ =48isunrealistically high and would rule out many observed risky gamblesundertaken by economic agents.0.70.60.5C =60C =5 0IMRSV o la tility0.40.3C =4 0Lower Volatility Bound0.2C =3 0C =2 00.1C =1 0C =4C =200.86 0.88 0.9 0.92 0.94 0.96 0.98 1MeanFigure 6.2: Mean and volatility estimates of the intertemporal marginalrate of substitution (IMRS) with β =0.99 and alternative values of γunder constant relative risk aversion utility and lower bound implied byforward exchange payoffs of the pound, deutschemark, and yen, 1973.1to 1997.1.(116)⇒The mean and volatility of the intertemporal marginal rate of substitution(θ µ , σ µ ) for alternative values of γ and the lower volatility bound(σ r =0.309θ µ ) implied by the data are illustrated in Figure 6.2. 7

6.4. APPARENT VIOLATIONS OF RATIONALITY 1836.4 Apparent Violations of RationalityWe’ve seen that there are important dimensions of the data that the Lucasmodel with CRRA utility cannot explain. 8 What other approacheshave been taken to explain deviations from uncovered interest parity?This section covers the peso problem approach and the noise traderparadigm. Both approaches predict that market participants make systematicforecast errors. In the peso problem approach, agents have rationalexpectations but don’t know the true economic environment withcertainty. In the noise trading approach, some agents are irrational.Before tackling these issues, we want to have some evidence thatmarket participants actually do make systematic forecast errors. So weÞrst look at a line of research that studies the properties of exchangerate forecasts compiled by surveys of actual foreign exchange marketparticipants. The subjective expectations of market participants arekey to any theory in international Þnance. The rational expectationsassumption conveniently allows the economic analyst to model thesesubjective expectations without having to collect data on people’s expectationsper se. If the rational expectations assumption is wrong, itsviolation may be the reason that underlies asset-pricing anomalies suchas the deviation from uncovered interest parity.7 Backus, Gregory, and Telmer [4] investigate the lower volatility bound (6.28)implied by data on the U.S. dollar prices of the Canadian-dollar, the deutschemark,the French-franc, the pound, and the yen. They compute the bound for aninvestor who chases positive expected proÞts by deÞning forward exchange payoffson currency i as I it (F i,t − S i,t+1 )/S i,t where I it = 1 if E t (f i,t − s i,t+1 ) > 0andI it = 0 otherwise. The bound computed in the text does not make this adjustmentbecause it is not a prediction of the Lucas model where investors may be willingto take a position that earns expected negative proÞt ifitprovidesconsumptioninsurance. Using the indicator adjustment on returns lowers the volatility boundmaking it more difficult for the asset pricing model to match this quarterly dataset.8 The failure of the model to generate sufficiently variable risk premiums to explainthe data cannot be blamed on the CRRA utility function. Bekaert [9] obtainssimilar results with utility speciÞcations where consumption exhibits durability andwhen utility displays ‘habit persistence’.

184CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYProperties of Survey Expectations(117)⇒Instead of modeling the subjective expectations of market participantsas mathematical conditional expectations, why not just ask people whatthey think? One line of research has used surveys of exchange rate forecastsby market participants to investigate the forward premium bias(deviation from UIP). Froot and Frankel [65], study surveys conductedby the Economist’s Financial Report from 6/81—12/85, Money MarketServices from 1/83—10/84, and American Express Banking Corporationfrom 1/76—7/85, Frankel and Chinn [58] employ a survey compiledmonthly by Currency Forecasters’ Digest from 2/88 through 2/91, andCavaglia et. al. [23] analyze forecasts on 10 USD bilateral rates and 8deutschemark bilateral rates surveyed by Business International Corporationfrom 1/86 to 12/90. The survey respondents were asked toprovide forecasts at horizons of 3, 6, and 12 months into the future.The salient properties of the survey expectations are captured intwo regressions. Let ŝ e t+1 be the median of the survey forecast of thelog spot exchange rate s t+1 reported at date t. TheÞrst equation is theregression of the survey forecast error on the forward premium∆ŝ e t+1 − ∆s t+1 = α 1 + β 1 (f t − s t )+² 1t+1 . (6.29)If survey respondents have rational expectations, the survey forecast errorrealized at date t+1 will be uncorrelated with any publicly availableat time t and the slope coefficient β 1 in (6.29) will be zero.The second regression is the counterpart to Fama’s decompositionand measures the weight that market participants attach to the forwardpremium in their forecasts of the future depreciation∆ŝ e t+1 = α 2 + β 2 (f t − s t )+² 2,t+1 . (6.30)Survey respondents perceive there to be a risk premium to the extentthat β 2 deviates from one. That is because if a risk premium exists,it will be impounded in the regression error and through the omittedvariables bias will cause β 2 to deviate from 1.Table 6.4 reports selected estimation results drawn from the literature.Two main points can be drawn from the table.1. The survey forecast regressions generally yield estimates of β 1that are signiÞcantly different from zero which provides evidence

6.4. APPARENT VIOLATIONS OF RATIONALITY 185Table 6.4: Empirical Estimates from Studies of Survey ForecastsData SetEconomist MMS AMEX CFD BIC—USD BIC—DEMHorizon: 3-monthsβ 1 2.513 6.073 – – 5.971 1.930t(β 1 =1) 1.945 2.596 – – 1.921 -0.452t(β 2 =1) 1.304 -0.182 – 0.423 1.930 0.959t-test 1.188 -2.753 – -2.842 5.226 -1.452Horizon: 6-monthsβ 1 2.986 – 3.635 – 5.347 1.841t(β 1 =1) 1.870 – 2.705 – 2.327 -0.422β 2 1.033 – 1.216 – 1.222 0.812t(β 2 =1) 0.192 – 1.038 – 1.461 -4.325Horizon: 12-monthsβ 1 0.517 – 3.108 – 5.601 1.706t(β 1 =1) 0.421 – 2.400 – 3.416 0.832β 2 0.929 – 0.877 1.055 1.046 0.502t(β 2 =1) -0.476 – -0.446 0.297 0.532 -6.594Notes: Estimates from the Economist, Money Market Services, and American Expresssurveys are from Froot and Frankel [65]. Estimates from the CurrencyForecasters’ Digest survey are from Frankel and Chinn [58], and estimates from theBusiness International Corporation (BIC) survey from Cavaglia et. al. [23]. BIC—USD is the average of individual estimates for 10 dollar exchange rates. BIC—DEMis the average over 8 deutschemark exchange rates.

186CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYagainst the rationality of the survey expectations. In addition,the slope estimates typically exceed 1 indicating that survey respondentsevidently place too much weight on the forward ratewhen predicting the future spot. That is, an increase in the forwardpremium predicts that the survey forecast will exceed thefuture spot rate.2. Estimates of β 2 are generally insigniÞcantly different from 1. Thissuggests that survey respondents do not believe that there is arisk premium in the forward foreign exchange rate. Respondentsuse the forward rate as a predictor of the future spot. They areputting too much weight on the forward rate and are formingtheir expectations irrationally in light of the empirically observedforward rate bias.We should point out that some economists are skeptical about theaccuracy of survey data and therefore about the robustness of resultsobtained from the analyses of these data. They question whether thereare sufficient incentives for survey respondents to truthfully report theirpredictions and believe that you should study what market participantsdo, not what they say.6.5 The ‘Peso Problem’On the surface, systematic forecast errors suggests that market participantsare repeatedly making the same mistake. It would seem thatpeople cannot be rational if they do not learn from their past mistakes.The ‘peso problem’ is a rational expectations explanation forpersistent and serially correlated forecast errors as typiÞed in the surveydata. Until this point, we have assumed that economic agents knowwith complete certainty, the model that describes the economic environment.That is, they know the processes including the parametervalues governing the exogenous state variables, the forms of the utilityfunctions and production functions and so forth. In short, they knowand understand everything that we write down about the economicenvironment.

6.5. THE ‘PESO PROBLEM’ 187In ‘peso problem’ analyses, agents may have imperfect knowledgeabout some aspects of the underlying economic environment. Likeapplied econometricians, rational agents have observed an insufficientnumber of data points from which to exactly determine the true structureof the economic environment. Systematic forecast errors can ariseas a small sample problem.A Simple ‘Peso-Problem’ Example.The ‘peso problem’ was originally studied by Krasker [87] who observeda persistent interest differential in favor of Mexico even thoughthe nominal exchange rate was Þxed by the central bank. By coveredinterest arbitrage, there would also be a persistent forward premium,since if i is the US interest rate and i ∗ is the Mexican interest rate,i t − i ∗ t = f t − s t < 0. If the Þx ismaintainedatt +1,wehavearealizationof f t s 0 and a probability 1 − p that the s 0 pegwill be maintained. The process governing the exchange rate iss t+1 =(s1 with probability ps 0 with probability 1 − p . (6.31)The 1-period ahead rationally expected future spot rate isE t (s t+1 ) = ps 1 +(1− p)s 0 . As long as the peg is maintained andp>0, we will observe the sequence of systematic, serially correlated,but rational forecast errorss 0 − E t (s t+1 )=p(s 0 − s 1 ) < 0. (6.32)If the forward exchange rate is the market’s expected future spot rate,we have a rational explanation for the forward premium bias. Although ⇐(119)the forecast errors are serially correlated, they are not useful in predictingthe future depreciation.

188CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY(120)⇒Lewis’s ‘Peso-Problem’ with Bayesian LearningLewis [93] studies an exchange rate pricing model in the presence ofthe peso-problem. The stochastic process governing the fundamentalsundergo a shift, but economic agents are initially unsure as whethera shift has actually occurred. Such a regime shift may be associatedwith changes in the economic, policy, or political environment. Oneexample of such a phenomenon occurred in 1979 when the Federal Reserveswitched its policy from targeting interest rates to one of targetingmonetary aggregates. In hindsight, we now know that the Fed actuallydid change its operating procedures, but at the time, one may not havebeen completely sure. Even when policy makers announce a change,there is always the possibility that they are not being truthful.Lewis works with the monetary model of exchange rate determination.The switch in the stochastic process that governs the fundamentalsoccurs unexpectedly. Agents update their prior probabilities aboutthe underlying process as Bayesians and learn about the regime shiftbut this learning takes time. The resulting rational forecast errors aresystematic and serially correlated during the learning period.As in chapter 3, we let the fundamentals be f t = m t −m ∗ t −φ(y t −yt ∗ ),where m is money and y is real income and φ is the income elasticity ofmoney demand. 9 For convenience, the basic difference equation (3.9)that characterizes the model is reproduced heres t = γf t + ψE t (s t+1 ), (6.33)where γ =1/(1 + λ), and ψ = λγ, andλ is the income elasticity ofmoney demand. The process that governs the fundamentals are knownby foreign exchange market participants and evolves according to arandom walk with drift term δ 0f t = δ 0 + f t−1 + v t , (6.34)where v tiid∼ N(0, σ 2 v).We will obtain the no-bubbles solution using the method of undeterminedcoefficients (MUC). To MUC around this problem, begin with(6.33). From the Þrst term we see that s t depends on f t . s t also depends9 Note: f denotes the fundamentals here, not the forward exchange rate.

6.5. THE ‘PESO PROBLEM’ 189on E t (s t+1 ) which is a function of the currently available informationset, I t .Sincef t is the only exogenous variable and the model is linear,it is reasonable to conjecture that the solution has forms t = π 0 + π 1 f t . (6.35)Now you need to determine the coefficients π 0 and π 1 that make (6.35)the solution. From (6.34), the one-period ahead forecast of the fundamentalsis, E t f t+1 = δ 0 + f t . If (6.35) is the solution, you can advancetime by one period and take the conditional expectation as of date t togetE t (s t+1 )=π 0 + π 1 (δ 0 + f t ). (6.36)Substitute (6.35) and (6.36) into (6.33) to obtainπ 0 + π 1 f t = γf t + ψ(π 0 + π 1 δ 0 + π 1 f t ). (6.37)In order for (6.37) to be a solution, the coefficients on the constant andon f t on both sides must be equal. Upon equating coefficients, you seethat the equation holds only if π 0 = λδ 0 and π 1 = 1. The no bubblessolution for the exchange rate when the fundamentals follow a randomwalk with drift δ 0 is therefores t = λδ 0 + f t . (6.38)A possible regime shift. Now suppose that market participants are toldat date t 0 that the drift of the process governing the fundamentals mayhave increased to δ 1 > δ 0 . Agents attach a probabilityp 0t =Prob(δ = δ 0 |I t ) that there has been no regime change and aprobability p 1t =Prob(δ = δ 1 |I t ) that there has been a regime changewhere I t is the information set available to agents at date t. Agents usenew information as it becomes available to update their beliefs aboutthe true drift. At time t, they form expectations of the future values ofthe fundamental according toE t (f t+1 ) = p 0t E(δ 0 + v t + f t )+p 1t E(δ 1 + v t + f t )= p 0t δ 0 + p 1t δ 1 + f t . (6.39)

190CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYUse the method of undetermined coefficients again to solve for theexchange rate under the new assumption about the fundamentals byconjecturing the solution to depend on f t and on the two possible driftparameters δ 0 and δ 1s t = π 1 f t + π 2 p 0t δ 0 + π 3 p 1t δ 1 . (6.40)The new information available to agents is the current period realizationof the fundamentals which evolves according to a random walk.Since the new information is not predictable, the conditional expectationof the next period probability at date t is the current probability,E t (p 0t+1 )=p 0t . 10 Using this information, advance time by one periodin (6.40) and take date-t expectations to getE t s t+1 = π 1 (f t + p 0t δ 0 + p 1t δ 1 )+π 2 p 0t δ 0 + π 3 p 1t δ 1= π 1 f t +(π 1 + π 2 )p 0t δ 0 +(π 1 + π 3 )p 1t δ 1 . (6.41)Substitute (6.40) and (6.41) into (6.33) to getπ 1 f t +π 2 p 0t δ 0 +π 3 p 1t δ 1 = γf t +ψπ 1 (p 0t δ 0 +p 1t δ 1 +f t )+ψπ 2 p 0t δ 0 +ψπ 3 p 1t δ 1 ,(6.42)and equate coefficients to obtain π 1 =1,π 2 = π 3 = λ. This gives thesolutions t = f t + λ(p 0t δ 0 + p 1t δ 1 ). (6.43)Now we want to calculate the forecast errors so that we can see howthey behave during the learning period. To do this, advance the timesubscript in (6.43) by one period to gets t+1 = f t+1 + λ(p 0t+1 δ 0 + p 1t+1 δ 1 ).and take time t expectations to getE t s t+1 = f t + p 0t δ 0 + p 1t δ 1 + λp 0t δ 0 + λp 1t δ 1= f t +(1+λ)(p 0t δ 0 + p 1t δ 1 ). (6.44)10 This claim is veriÞed in problem 6 at the end of the chapter.

6.5. THE ‘PESO PROBLEM’ 191The time t+1 rational forecast error iss t+1 − E t (s t+1 ) = λ[δ 0 (p 0t+1 − p 0t )+δ 1 (p 1t+1 − p 1t )]+∆f t+1 − (p 0t δ 0 + p 1t δ 1 )] | {z }E t∆f t+1= λ(δ 1 − δ 0 )[p 1t+1 − p 1t ]+δ 1 + v t+1 − [δ 0 +(δ 1 − δ 0 )p 1t ].The regime probabilities p 1t and the updated probabilities p 1t+1 − p 1tare serially correlated during the learning period. The rational forecasterror therefore contains systematic components and is serially correlated,but the forecast errors are not useful for predicting the futuredepreciation. To determine explicitly the sequence of the agent’s beliefprobabilities, we use,Bayes’ Rule: for events A i ,i =1,...,N that partition the samplespace S, andanyeventB with Prob(B) > 0P(A i |B) =P(A i )P(B|A i )P Nj=1P(A j )P(B|A j ) .To apply Bayes rule to the problem at hand, let news of the possibleregime shift be released at t = 0. Agents begin with the unconditional ⇐(121)probability, p 0 =P(δ = δ 0 ), and p 1 =P(δ = δ 1 ). In the period after theannouncement t = 1, apply Bayes’ Rule by setting B =(∆f 1 ), A 1 = δ 1 ,A 2 = δ 0 to get the updated probabilitiesp 0 P(∆f 1 |δ 0 )p 0,1 =P(δ = δ 0 |∆f 1 )=p 0 P(∆f 1 |δ 0 )+p 1 P(∆f 1 |δ 1 ) . (6.46)As time evolves and observations on ∆f t are acquired, agents updatetheir beliefs according top 0,2 =p 0 P(∆f 2 , ∆f 1 |δ 0 )P(δ 0 |∆f 2 , ∆f 1 )=p 0 P(∆f 2 , ∆f 1 |δ 0 )+p 1 P(∆f 2 , ∆f 1 |δ 1 ) ,p 0,3 =p 0 P(∆f 3 , ∆f 2 , ∆f 1 |δ 0 )P(δ 0 |∆f 3 , ∆f 2 , ∆f 1 )=p 0 P(∆f 3 , ∆f 2 , ∆f 1 |δ 0 )+p 1 P(∆f 3 , ∆f 2 , ∆f 1 |δ 1 ) ,...p 0,T = P(δ 0 |∆f T ,...,∆f 1 )=p 0 P(∆f T ,...,∆f 1 |δ 0 )p 0 P(∆f T ,...,∆f 1 |δ 0 )+p 1 P(∆f T ,...,∆f 1 |δ 1 ) .

192CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYThe updated probabilities p 0t =P(δ 0 |∆f t ,...,∆f 1 ) are called the posteriorprobabilities. An equivalent way to obtain the posterior probabilitiesisp 0,1 =p 0,2 =.p 0t =p 0 P(∆f 1 |δ 0 )p 0 P(∆f 1 |δ 0 )+p 1 P(∆f 1 |δ 1 ) ,p 0,1 P(∆f 2 |δ 0 )p 0,1 P(∆f 2 |δ 0 )+p 1,1 P(∆f 2 |δ 1 ) ,p 0,t−1 P(∆f t |δ 0 )p 0,t−1 P(∆f t |δ 0 )+p 1,t−1 P(∆f t |δ 1 ) .How long is the learning period? To start things off, you need to specifyan initial prior probability, p 0 =P(δ = δ 0 ). 11 Let δ 0 =0, δ 1 =1,andlet v have a discrete probability distribution with the probabilities,P(v = −5) =66 3 P(v = −1) =11 2 P(v =3)=663P(v = −4) =66 3 P(v =0)=11 2 P(v =4)=663P(v = −3) =66 3 P(v = 1) =11 2 P(v =5)=663P(v = −2) =11 1 P(v =2)=111We generate the distribution of posterior probabilities, learningtimes, and forecast error autocorrelations by simulating the economy2000 times. Figure 6.3 shows the median of the posterior probabilitydistribution when the initial prior is 0.95. The distribution of learningtimes and autocorrelations is not sensitive to the initial prior. Thelearning time distribution is quite skewed with the 5, 50, and 95 percentilesof the distribution of learning times being 1, 14, and 66 periodsrespectively. Judging from the median of the distribution, Bayesianupdaters quickly learn about the true economy. Since the forecast errorsare serially correlated only during the learning period, we calculatethe autocorrelation of the forecast errors only during the learning period.The median autocorrelations at lags 1 through 4 of the forecast11 Lewis’s approach is to assume that learning is complete by some date T>t 0 inthe future at which time p 0,T = 0. Having pinned down the endpoint, she can workbackwards to Þnd the implied value of p 0 that is consistent with learning havingbeen completed by T .

6.6. NOISE-TRADERS 19310.90.80.70.60.50.40.30.20.101 2 3 4 5 6 7 8 9 10 11 12 13 14 15PeriodFigure 6.3: Median posterior probabilities of δ = δ 0 when truth is δ = δ 1with initial prior of 0.95.errors computed from the Þrst 14 periods are -0.130, -0.114, -0.098, and-0.078.This simple example serves as an introduction to rational learningin peso-problems. However, the rapid rate at which learning takes placesuggests that a single regime switch is insufficient to explain systematicforecast errors observed over long periods of time as might be the case inforeign exchange rates. If the peso problem is to provide a satisfactoryexplanation of the data a model with richer dynamics with recurrentregime shifts, as outlined in Evans [47], is needed.⇐(124)6.6 Noise-TradersWe now consider the possibility that some market participants are notfully rational. Mark and Wu [102] present a model in which a mixture

194CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYof rational and irrational agents produce spot and forward exchange dynamicsthat is consistent with the Þndings from survey data. The modeladapts the overlapping-generations noise trader model of De Long et.al. [38] to study the pricing of foreign currencies in an environmentwhere heterogeneous beliefs across agents generate trading volume andexcess currency returns.The irrational ‘noise’ traders are motivated by Black’s [14] suggestionthat the real world is so complex that some (noise) traders areunable to distinguish between pseudo-signals and news. These individualsthink that the pseudo-signals contain information about assetreturns. Their beliefs regarding prospective investment returns seemdistorted by waves of excessive optimism and pessimism. The resultingtrading dynamics produce transitory deviations of the exchangerate from its fundamental value. Short-horizon rational investors bearthe risk that they may be required to liquidate their positions at a timewhen noise-traders have pushed asset prices even farther away from thefundamental value than they were when the investments were initiated.The ModelWe consider a two-country constant population partial equilibrium model.It is an overlapping generations model where people live for two periods.When people are born, they have no assets but they do have a fullstomach and do not consume in the Þrst period of life. People makeportfolio decisions to maximize expected utility of second period wealthwhichisusedtoÞnance consumption when old.The home country currency unit is called the ‘dollar’ and the foreigncountry currency unit is called the ‘euro.’ In each country, there is aone-period nominally safe asset in terms of the local currency. Bothassets are available in perfectly elastic supply so that in period t, peoplecan borrow or lend any amount they desire at the gross dollar rate ofinterest R t =(1+i t ), or at the gross euro rate of interest, R ∗ t =(1+i ∗ t ).The nominal interest rate differential—and hence by covered interestparity, the forward premium—is exogenous.In order for Þnancial wealth to have value, it must be denominatedin the currency of the country in which the individual resides. Thus inthe second period, the domestic agent must convert wealth to dollars

6.6. NOISE-TRADERS 195and the foreign agent must convert wealth to euros. We also assumethat the price level in each country is Þxed at unity. Individuals thereforeevaluate wealth in national currency units. The portfolio problemis to decide whether to borrow the local currency and to lend uncoveredin the foreign currency or vice-versa in an attempt to exploit deviationsfrom uncovered interest parity, as described in chapter 1.1.The domestic young decide whether to borrow dollars and lend eurosor vice versa. Let λ t be the dollar value of the portfolio position taken.If the home agent borrows dollars and lends euros the individual hastaken a long euro positions which we represent with positive values ofλ t . To take a long euro position, the young trader borrows λ t dollars atthe gross interest rate R t and invests λ t /S t euros at the gross rate R ∗ t .When old, the euro payoff Rt ∗ (λ t /S t ) is converted into (S t+1 /S t )Rt ∗ λ tdollars. If the agent borrows euros and lends dollars, the individualhas taken a long dollar position which we represent with negative λ t .A long position in dollars is achieved by borrowing −λ t /S t euros andinvesting the proceeds in the dollar asset at R t . In the second period,the domestic agent sells −(S t+1 /S t )Rt ∗ λ t dollars in order to repay theeuro debt −Rt ∗ (λ t /S t ). In either case, the net payoff is the numberof dollars at stake multiplied by the deviation from uncovered interestparity, [(S t+1 /S t )R ∗ t − R t ]λ t . We use the approximations (S t+1 /S t ) '(1 + ∆s t+1 )and(R t /Rt ∗ )=(F t /S t ) ' 1+x t to express the net payoffas 12 [∆s t+1 − x t ]Rt ∗ λ t. (6.47)The foreign agent’s portfolio position is denoted by λ ∗t with positivevalues indicating long euro positions. To take a long euro position, theforeign young borrows λ ∗t dollars and invests (λ ∗t /S t ) euros at the grossinterest rate R ∗ t . Next period’s net euro payoff is (Rt ∗ /S t − R t /S t+1 )λ ∗t .A long dollar position is achieved by borrowing −(λ ∗t /S t ) euros andinvesting −λ ∗t dollars. The net euro payoff in the second period is−(R t /S t+1 − R ∗ t /S t )λ ∗t . Using the approximation (F t S t )/(S t S t+1 ) '12 These approximations are necessary in order to avoid dealing with Jenseninequality terms when evaluating the foreign wealth position which render themodel intractable. Jensen’s inequality is E(1/X) > 1/(EX). So we have[(S t+1 /S t )R ∗ t − R t]λ t =[(1 + ∆s t+1 )R ∗ t − (1 + x t)R ∗ t ]λ t, which is (6.47).

196CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY1+x t − ∆s t+1 , the net euro payoff is 13[∆s t+1 − x t ]R ∗ λ ∗tt . (6.48)S tThe foreign exchange market clears when net dollar sales of thecurrent young equals net dollar purchases of the current oldλ t + λ ∗t =Fundamental and Noise TradersS tS t−1R ∗ t−1λ t−1 + R t−1 λ ∗t−1 . (6.49)Afractionµ of domestic and foreign traders are fundamentalists whohave rational expectations. The remaining fraction 1 − µ are noisetraders whose beliefs concerning future returns from their portfolio investmentsare distorted. Let the speculative positions of home fundamentalistand home noise traders be given by λ f t and λ n t respectively.Similarly, let foreign fundamentalist and foreign noise trader positionsbe λ f ∗t and λ n ∗t. The total portfolio position of domestic residents isλ t = µλ f t +(1− µ)λ n t and of foreign residents is λ ∗t = µλ f ∗t +(1− µ)λ n ∗t.We denote subjective date—t conditional expectations genericallyas E t (·). When it is necessary to make a distinction we will denotethe expectations of fundamentalists denoted by E t (·). Similarly, theconditional variance is generically denoted by V t (·) with the conditionalvariance of fundamentalists denoted by V t (·).Utility displays constant absolute risk aversion with coefficient γ.The young construct a portfolio to maximize the expected utility ofnext period wealth³E t −e−γW t+1´. (6.50)Both fundamental and noise traders believe that conditional on time-tinformation, W t+1 is normally distributed. As shown in chapter 1.1.1,maximizing (6.50) with (perceived) normally distributed W t+1 is equivalentto maximizingE t (W t+1 ) − γ 2 V t(W t+1 ). (6.51)13 To get (6.48), −(R t /S t+1 − R ∗ t /S t)λ ∗t = −λ ∗t [(R ∗ t F t)/(S t S t+1 ) − (R ∗ t /S t)]= −λ ∗t (R ∗ t /S t)[(S t F t )/(S t S t+1 ) − 1] =−λ ∗t R ∗ t /S t[1 + x t − ∆s t+1 ]

6.6. NOISE-TRADERS 197The relevant uncertainty in the model shows up in the forward premiumwhich in turn inherits its uncertainty from the interest rates R t andR ∗ t , through the covered interest parity condition. The randomness ofone of the interest rates is redundant. Therefore, the algebra can besimpliÞed without loss of generality by letting the uncertainty be drivenby R t alone and Þx R ∗ =1.A Fundamentals (µ =1)EconomySuppose everyone is rational (µ = 1) so that E t (·) = E t (·) andV t (·) = V t (·). Second period wealth of the fundamentalist domesticagent is the portfolio payoff plus c dollars of exogenous ‘labor’ incomewhichispaidinthesecondperiod.14 The forward premium,(F t /S t )=(R t /R ∗ )=R t ' 1+x t inherits its stochastic properties fromR t , which evolves according to the AR(1) processx t = ρx t−1 + v t , (6.52)iidwith 0 < ρ < 1, and v t ∼ (0, σv). 2 Second period wealth can now bewritten asW ft+1 =[∆s t+1 − x t ]λ f t + c. (6.53)People evaluate the conditional mean and variance of next periodwealth asE t (Wt+1) f =[E t (∆s t+1 ) − x t ]λ f t + c, (6.54)V t (Wt+1) f =σs 2 (λf t ) 2 , (6.55)where σs 2 = V t (∆s t+1 ). The domestic fundamental trader’s problem isto choose λ f t to maximizewhich is attained by setting[E t (∆s t+1 ) − x t ]λ f t + c − γ 2 (λf t ) 2 σ 2 s , (6.56)λ f t = [E t(∆s t+1 ) − x t ]. (6.57)γσs214 The exogenous income is introduced to lessen the likelihood of negative secondperiod wealth realizations, but as in De Long et. al., we cannot rule out such apossibility.

198CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY(6.57) displays the familiar property of constant absolute risk aversionutility in which portfolio positions are proportional to the expectedasset payoff. The factor of proportionality is inversely related to theindividual’s absolute risk aversion coefficient. Recall that individualsundertake zero-net investment strategies. The portfolio position in oursetup does not depend on wealth because traders are endowed with zeroinitial wealth.The foreign fundamental trader faces an analogous problem. Thesecond period euro-wealth of fundamentalist foreign agents is the payofffrom portfolio investments plus an exogenous euro payment of ‘labor’income c ∗ , W f ∗t+1 = [∆s t+1 − x t ] λf ∗tS t+ c ∗ . The solution is to chooseλ f ∗t = S t λ f t . Because individuals at home and abroad have identicaltastes but evaluate wealth in national currency units, they will pursueidentical investment strategies by taking positions of the same size asmeasured in monetary units of the country of residence.These portfolios combined with the market clearing condition (6.49)imply the difference equation 15E t ∆s t+1 − x t = Γ t (E t−1 ∆s t − x t−1 ), (6.58)where Γ t ≡ [(S t /S t−1 )+S t−1 R t−1 ]/(1 + S t ). The level of the exchangerate is indeterminate but it is easily seen that a solution for the rate ofdepreciation is∆s t = 1 ρ x t = x t−1 + 1 ρ v t. (6.59)The independence of v t and x t−1 implies E t (∆s t+1 )=x t and the fundamentalssolution therefore does not generate a forward premium biasbecause uncovered interest parity holds in the fundamentals equilibriumeven when agents are risk averse. The reason is that under homogeneousexpectations and common knowledge, you demand the same riskpremium as I do, and we want to do the same transaction. Since wecannot Þnd a counterparty to take the opposite side of the transaction,no trades take place. The only way that no trades will occur inequilibrium is for uncovered interest parity to hold.15 The left side of the market clearing condition (6.49) is λ t + λ ∗t =(1 + S t )λ t =(1 + S t )/(γσ s )[E t ∆s t+1 − x t ]. The right side is, (S t /S t−1 )R ∗ λ t−1 + R t−1 S t−1 λ t−1=[(S t /S t−1 )+(1 + x t−1 )S t−1 ]λ t−1 . Finally, using λ t−1 =[E t−1 ∆s t − x t−1 ]/(γσs 2),we get (6.58).

6.6. NOISE-TRADERS 199ANoiseTrader(µ 0 for they believe the dollarwill be weaker in the future than what is justiÞed by the fundamentals.We specify the noise distortion to conform with the evidence fromsurvey expectations in which respondents appear to place excessiveweight on the forward premium when predicting future changes in theexchange raten t = kx t + u t , (6.62)iidwhere k>0,u t ∼ N(0, σu). 2 The domestic noise trader’s problem isto maximize λ n t (E t ∆s t+1 − x t + n t ) − γ(λ n t ) 2 σs/2. 2 The solution is tochooseλ n t = λf t + n t. (6.63)γσs2The noise trader’s position deviates from that of the fundamentalist bya term that depends on the distortion in their beliefs, n t .The foreign noise trader holds similar beliefs, solves an analogousproblem and choosesλ n ∗t = S tλ n t . (6.64)Substituting these optimal portfolio positions into the market clearingcondition (6.49) yields the stochastic difference equation[E t ∆s t+1 − x t ]+(1− µ)n t = Γ t ([E t−1 ∆s t − x t−1 ]+(1− µ)n t−1 ). (6.65)

200CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYUsing the method of undetermined coefficients, you can verify thatis a solution.∆s t = 1 ρ x t −(1 − µ)n t − (1 − µ)u t−1 , (6.66)ρProperties of the SolutionFirst, fundamentalists and noise traders both believe, ex ante, thatthey will earn positive proÞts from their portfolio investments. It is thedifferences in their beliefs that lead them to take opposite sides of thetransaction. When noise traders are excessively pessimistic and takeshort positions in the dollar, fundamentalists take the offsetting longposition. In equilibrium, the expected payoff of fundamentalists andnoise-traders are respectivelyE t ∆s t+1 − x t = −(1 − µ)n t , (6.67)E t ∆s t+1 − x t = µn t . (6.68)On average, the forward premium is the subjective predictor of thefuture depreciation: µE t ∆s t+1 +(1−µ)E t ∆s t+1 = x t . As the measure ofnoise traders approaches 0 (µ → 1), the fundamentals solution with notrading is restored. Foreign exchange risk, excess currency movements,and trading volume are induced entirely by noise traders. Neither typeof trader is guaranteed to earn proÞts or losses, however. The ex postproÞt depends on the sign of∆s t+1 − x t = −(1 − µ)n t + 1 ρ [1 − k(1 − µ)]v t+1 − 1 − µρ u t+1, (6.69)which can be positive or negative.Matching Fama’s regressions. To generate a negative forward premiumbias, substitute (6.62) and (6.52) into (6.66) to get∆s t+1 =[1− k(1 − µ)]x t + ξ t+1 , (6.70)where ξ t+1 ≡ (1/ρ)[1 − k(1 − µ)]v t+1 − (1 − µ)/ρu t+1 − (1 − µ)u t isan error term which is orthogonal to x t . If [1 − k(1 − µ)] < 0, the

6.6. NOISE-TRADERS 201implied slope coefficient in a regression of the future depreciation onthe forward premium is negative.Next, if we compute the implied second moments of the deviationfrom uncovered interest parity and the expected depreciationCov([x t − E t (∆s t+1 )],E t (∆s t+1 )) =k(1 − µ)(1 − k(1 − µ))σx 2 − (1 − µ) 2 σu, 2 (6.71)Var(x t − E t (∆s t+1 )) = (1 − µ) 2 [k 2 σx 2 + σ2 u ], (6.72)Var(E t (∆s t+1 )) = Var(x t − E t (∆s t+1 )) + [1 − 2k(1 − µ)]σx 2 . (6.73)We see that 1 − k(1 − µ) < 0alsoimplesthatFama’sp t covariesnegatively with and is more volatile than the rationally expected de- ⇐(125)preciation. The noise-trader model is capable of matching the stylizedfacts of the data as summarized by Fama’s regressions.Matching the Survey Expectations. The survey research on expectationspresents results on the behavior of the mean forecast from a survey ofindividuals. Let ˆµ be the fraction of the survey respondents comprisedof fundamentalists and 1 − ˆµ be the fraction of the survey respondentsmade up of noise traders.Suppose the survey samples the proportion of fundamentalists andnoise traders in population without error (ˆµ = µ). Then the meansurvey forecast of depreciation is ∆ŝ e t+1 = µE t (∆s t+1 )+(1−µ)E t (∆s t+1 )= µ[1−k(1−µ)]x t +µ(µ−1)u t +(1−µ)(1+µk)x t +(1−µ)µu t = x t ,whichpredicts that β 2 = 1. There is no risk premium if ˆµ = µ. In additionto β 2 =1,wehaveβ =1− k(1 − µ) =1− β 1 ,andβ 1 = k(1 − µ), which ⇐(126)amounts to one equation in two unknowns k and µ, sothecoefficientof over-reaction k cannot be identiÞed here.We can ‘back out’ the implied value of over-reaction k if we arewilling to make an assumption about survey measurement error. Ifˆµ 6= µ, then∆ŝ e t+1 = ˆµE t (∆s t+1 )+(1− ˆµ)E t (∆s t+1 )=[1+k(µ −ˆµ)]x t +(µ − ˆµ)u t , which implies, β 2 =1+k(µ − ˆµ), β 1 = k(1 − ˆµ),and β = 1 − k(1 − µ). For given values of ˆµ, β 1 , and β, wehave,k = β 1 /(1 − ˆµ), and µ =(β − 1+k)/k. For example, if we assume thatˆµ =0.5, the 3-month horizon BIC-US results in Table 6.4 imply thatk =11.94 and µ =0.579.

202CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYForeign Exchange Market Efficiency Summary1. The Þnancial market is said to be efficient if there are no unexploitedexcess proÞt opportunities available. What is excessivedepends on a model of market equilibrium. Violations of uncoveredinterest parity in and of themselves does not mean that theforeign exchange market is inefficient.2. The Lucas model–perhaps the most celebrated asset pricingmodel of the last 20 years–provides a qualitative and elegantexplanation for why uncovered interest parity doesn’t hold. Thereason is that risk-averse agents must be compensated with arisk premium in order for them to hold forward contracts in arisky currency. The forward rate becomes a biased predictor ofthe future spot rate because this risk premium is impoundedinto the price of a forward contract. But the Lucas model requireswhat many people regard as an implausibly coefficient ofrelative risk aversion to generate sufficiently large and variablerisk premia to be consistent with the volatility of exchange ratereturns data.3. Analyses of survey data from professional foreign exchange marketparticipants predictions of future exchange rates Þnd thatthe survey forecast error is systematic. If you believe the surveydata, these systematic prediction errors may be the reason thatuncovered interest parity doesn’t hold.4. Market participant’s systematic forecast errors can be consistentwith rationality. A class of models called ‘peso-problem’models show how rational agents make systematic predictionerrors when there is a positive probability that the underlyingstructure may undergo a regime shift.5. On the other hand, it may be the case that some market participantsare indeed irrational in the sense that they believe thatpseudo signals are important determinants of asset returns. Thepresence of such noise traders generate equilibrium asset pricesthat deviate from their fundamental values.

6.6. NOISE-TRADERS 203Problems1. (Siegel’s [128] Paradox) Let S t be the spot dollar price of the euroand F t be the 1-period forward rate in dollars per euro. The claim isif investors are risk-neutral and the forward foreign exchange market ⇐(128)is efficient, the forward rate is the rational expectation of the futurespot rate. From the US perspective we write this asE t (S t+1 )=F t .The risk-neutral, rational-expectations, efficient market statement froman European perspective is(1/F t )=E t (1/S t+1 )since from the euro-price of the dollar is the reciprocal of the dollareurorate. Both statements cannot possibly be true. Why not? (Hint:Use Jensen’s inequality).2. Let the Euler equation for a domestic investor that speculates in forwardforeign exchange beF t = E t[u 0 (c t+1 )(S t+1 /P t+1 )]E t [u 0 ,(c t )/P t+1 ]where u 0 (c) is marginal utility of real consumption c and P is thedomestic price level. From the foreign perspective, the Euler equationis1= E t[u 0 (c ∗ t+1 /(S t+1Pt+1 ∗ )]F t E t [u 0 (c ∗ t )/P t+1 ∗ ]where c ∗ is foreign consumption and P ∗ is the foreign price level.Suppose further that both domestic and foreign agents are risk neutral.Show that Siegel’s paradox does not pose a problem now that payoffsarestatedinrealterms. 163. We saw that the slope coefficient in a regression of s t −s t−1 on f t −s t−1is negative. McCallum [103] shows regressing s t − s t−2 on f t − s t−2yields a slope coefficient near 1. How can you explain McCallum’sresult?16 Engel’s [43] empirical work showed that regression test results on forward exchangerate unbiasedness done with nominal exchange rates were robust to speciÞcationsin real terms so evidently Siegel’s paradox is not economically important.

204CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY4. (Kaminsky and Peruga [82]). Suppose that the data generating processfor observations on consumption growth, inßation, and exchange ratesis given by the lognormal distribution, and that the utility functionis u(c) =c 1−γ . Let lower case letters denote variables in logarithms.We have ∆c t+1 =ln(C t+1 /C t ) be the rate of consumption growth,∆s t+1 =ln(S t+1 /S t ) be the depreciation rate, ∆p t+1 =ln(P t+1 /P t )be the inßation rate, and f t =ln(F t ) be the log one-period forwardrate.If ln(Y ) ∼ N(µ, σ 2 ), then Y is said to be log-normally distributed andhhE e ln(y)i =E(Y )=eµ+ σ22i. (6.74)Let J t consist of lagged values of c t ,s t ,p t and f t be the date t informationset available to the econometrician. Conditional on J t , lety t+1 = (∆s t+1 , ∆c t+1 , ∆p t+1 ) 0 be normally distributed with conditionalmean E(y⎡ t+1 |J t ) = (µ st ,µ⎤ ct ,µ pt ) 0 and conditional covarianceσ sst σ sct σ spt⎢⎥matrix Σ t = ⎣ σ cst σ cct σ cpt ⎦ . Let a t+1 = ∆s t+1 − ∆p t+1 andσ pst σ pct σ pptb t+1 = f t − s t − ∆p t+1 . Show thatµ st − f t = γσ cst + σ spt − σ sst2 . (6.75)5. Testing the volatility restrictions (Cecchetti et. al. [25]). This exercisedevelops the volatility bounds analysis so that we can do classicalstatistical hypothesis tests to compare the implied volatility of theintertemporal marginal rate of substitution and the lower volatilitybound. Begin deÞning φ as a vector of parameters that characterizethe utility function, and ψ as a vector of parameters associated withthe stochastic process governing consumption growth.Stack the parameters that must be estimated from the data into thevector θθ =⎛⎜⎝µ rvec(Σ r )ψ⎞⎟⎠ ,where vec(Σ r ) is the vector obtained by stacking all of the uniqueelements of the symmetric matrix, Σ r . Let θ 0 be the true value of θ,

6.6. NOISE-TRADERS 205and let ˆθ be a consistent estimator of θ 0 such that√T (ˆθ − θ0 ) D → N(0, Σ θ ).Assume that consistent estimators of both θ 0 and Σ θ are available.Now make explicit the fact that the moments of the intertemporalmarginal rate of substitution and the volatility bound depend on sampleinformation. The estimated mean and standard deviation of predictedby the model are, ˆµ µ = µ µ (φ; ˆψ) andˆσ µ = σ µ (φ; ˆψ), while theestimated volatility bound isrˆσ r = σ r (φ; ˆθ) =³ˆµq− µ µ (φ; ˆψ)ˆµ r´0 ˆΣ−1 r³ˆµ q− µ µ (φ; r´ ˆψ)ˆµ .Let∆(φ; ˆθ) =σ M (φ; ˆψ) − σ r (φ; ˆθ),be the difference between the estimated volatility bound and the estimatedvolatility of the intertemporal marginal rate of substitution.Using the ‘delta method,’ (a Þrst-order Taylor expansion about thetrue parameter vector), show that√T (∆(φ; ˆθ) − ∆(φ; θ0 )) → D N(0, σ∆),2whereµ µ ∂∆∂∆σ∆ 2 =∂θ 0 (ˆθ − θ 0 )(ˆθ − θ 0 ) 0 .θ 0∂θ θ 0How can this result be used to conduct a statistical test of whether aparticular model attains the volatility restrictions?6. (Peso problem). Let the fundamentals, f t = m t −m ∗ t −λ(y t −yt ∗ ) followthe random walk with drift, f t+1 = δ 0 + f t + v t+1 ,wherev t ∼ iid withE(v t )=0andE(vt 2 )=σv. 2 Agents know the fundamentals processwith certainty. Forward iteration on (6.33) yields the present valueformula∞Xs t = γ E t (f t+j ).j=1Verify the solution (6.38) by direct substitution of E t (f t+j ).Now let agents believe that the drift may have increased to δ = δ 1 .Show that E t (f t+j )=f t + j(δ 0 − δ 1 )p 0t + jδ 1 . Use direct substitutionof this forecasting formula in the present value formula to verify thesolution (6.43) in the text.

206CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

Chapter 7The Real Exchange RateIn this chapter, we examine the behavior of the nominal exchange ratein relation to domestic and foreign goods prices in the short run andin the long run. A basic theoretical framework that underlies the empiricalexamination of these prices is the PPP doctrine encountered inchapter 3. The ßexible price models of chapters 3 through 5 assumethat the the law-of-one price holds internationally, and by implication,that purchasing-power parity holds. In empirical work, we deÞne the(log) real exchange rate between two countries as the relative pricebetween a domestic and foreign commodity basketq = s + p ∗ − p. (7.1)Under purchasing-power parity, the log real exchange rate is constant(speciÞcally, q =0).The prediction that q t is constant is clearly false–a fact we discoveredafter examining Figures 3.1 in chapter 3.1. This result is not new.So given the obvious short-run violations of PPP, the interesting thingsto study are whether these international pricing relationships hold inthe long run, and if so, to see how much time it takes to get to thelong-run.Why would we want to know this? Because real exchange rateßuctuations can have important allocative effects. A prolonged realappreciation may have an adverse effect on a country’s competitivenessas the appreciation raises the relative price of home goods and induces207

208 CHAPTER 7. THE REAL EXCHANGE RATEexpenditures to switch from home goods toward foreign goods. Domesticoutput might then be expected to fall in response. Although thedomestic traded-goods sector is hurt, consumers evidently beneÞt. Onthe other hand, a real depreciation may be beneÞcial to the tradedgoodssector and harmful to consumers. The foreign debt of manydeveloping countries, is denominated in US dollars, however, so a realdepreciation reßects a real increase in debt servicing costs. These expenditureswitching effects are absent in the ßexible price theories thatwe have covered thus far.So what leads you to conclude that PPP does not hold in the longrun. Would this make any sense? What theory predicts that PPPdoes not hold? The Balassa [6]—Samuelson [124] model, which is developedin this chapter provides one such theory. The Balassa—Samuelsonmodel predicts that the long-run real exchange rate depends on relativeproductivity trends between the home and foreign countries. If relativeproductivity is governed by a stochastic trend, the real exchangerate will similarly be driven and will not exhibit any mean-revertingbehavior.Theresearchonrealexchangeratebehaviorraisesmanyquestions,but as we will see, offers few concrete answers.7.1 Some Preliminary IssuesThe Þrst issue that you confront in real exchange rate research is thatdata on price levels are generally not available. Instead, you typicallyhave access to a price index Pt I , which is the ratio of the price level P tin the measurement year to the price level in a base year P 0 . Lettingstars denote foreign country variables and lower case letters to denotevariables in logarithms, the empirical log real exchange rate uses priceindices and amounts toq t =(p 0 − p ∗ 0 )+s t + p ∗ t − p t. (7.2)s t +p ∗ t −p t is the relative price of the foreign commodity basket in termsof the domestic basket. This term is 0 if PPP holds instantaneously,and is mean-reverting about 0 if PPP is violated in the short run butholds in the long run. Tests of whether PPP holds in the long run

7.2. DEVIATIONS FROM THE LAW-OF-ONE PRICE 209typically ask whether q t is stationary about a Þxedmeanbecauseevenif PPP holds, measured q t will be (p 0 − p ∗ 0) which need not be 0 due tothe base year normalization of the price indices.An older literature made the distinction between absolute PPP (s t +p ∗ t − p t = 0) and relative PPP (∆s t + ∆p ∗ t − ∆p t = 0). By taking Þrstdifferences of the observations, the arbitrary base-year price levels dropout under relative PPP. In this chapter, when we talk about PPP, wemean absolute PPP.A second issue that you confront in this line of research is that thereare as many empirical real exchange rates as there are price indices. Asdiscussed in chapter 3.1, you might use the CPI if your main interestis to investigate the Casellian view of PPP because the CPI includesprices of a broad range of both traded and nontraded Þnal goods. ThePPI has a higher traded-goods component than the CPI and is viewedby some as a crude measure of traded-goods prices. If a story aboutaggregate production forms the basis of your investigation, the grossdomestic product deßator may make better sense.7.2 Deviations from the Law-Of-One PriceThe root cause of deviations from PPP must be violations of the law-ofoneprice. Such violations are easy to Þnd. Just check out the price ofunleaded regular gasoline at two gas stations located at different cornersof the same intersection. More puzzling, however, is that internationalviolations of the law-of-one price are several orders of magnitude largerthan intranational violations. There is a large empirical literature thatstudies international violations of the law-of-one price. We will considertwo of the many contributions that have attracted attention ofinternational macroeconomists.Isard’s Study of the Law-Of-One PriceIsard [79] collected unit export and unit import transactions pricesfor the US, Germany, and Japan from 1970 to 1975 at 4 and 5 digitstandard international trade classiÞcation (SITC) levels for machineditems. Isard deÞnes the relative export price to be the ratio of the US

210 CHAPTER 7. THE REAL EXCHANGE RATEdollar price of German exports of these items to the dollar price of USexports of the same items. Between 1970 and 1975, the dollar fell by55.2 percent while at the same time the relative export price of internalcombustion engines, office calculating machinery, and forklift trucksincreased by 48.1 percent, 47.7 percent, and 39.1 percent, respectivelyin spite of the fact that German and US prices are both measured indollars. Evidently, nominal exchange rate changes over this Þve-yearperiod had a big effect on the real exchange rate.In a separate regression analysis, he obtains 7-digit export commoditieswhich he matches to 7-digit import unit values in which theimports are distinguished by country of origin. The dependent variableis the US import unit value from Canada, Japan, and Germany, respectively,divided by the unit values of US exports to the rest of the world,both measured in dollars. If the law-of-one price held, this ratio wouldbe 1. Instead, when the ratio is regressed on the DM price of the dollar,the slope coefficient is positive but is signiÞcantly different from 1 forGermany and Japan. The slope coefficients and implied standard errorsfor Germany and Japan are reproduced in Table 7.1. 1 The estimatesfor Germany indicate that import and export prices exhibit insufficientdependence on the exchange rate to be consistent with the law-of-oneprice, whereas the estimates for Japan suggest that there is too muchdependence.While Isard’s study provides evidence of striking violations of thelaw-of-one price, it is important to bear in mind that these results weredrawn from a very short time-series sample taken from the 1970s. Thiswas a time period of substantial international macroeconomic uncertaintyand one in which people may have been relatively unfamiliarwith the workings of the ßexible exchange rate system.1 A potential econometric problem in Isard’s analysis is that he runs the regressionR t = a 0 + a 1 S t + a 2 D t + e t + ρe t−1 where R t is the ratio of import to export prices,S t is the DM price of the dollar, and D t is a dummy variable that splits up thesample. The problem is that the regression is run by Cochrane—Orcutt to controlfor serial correlation in the error term, e t , which is inconsistent if the regressors arenot strictly (econometrically) exogenous.

7.2. DEVIATIONS FROM THE LAW-OF-ONE PRICE 211Table 7.1: Slope coefficients in Isard’s regression of the US import toexport price ratio on nominal exchange rateImports from Germany Imports from JapanSoap Tires Wallpaper Soap Tires Wallpaper0.094 0.04 0.03 15.49 6.28 6.79(0.04) (0.02) (0.01) (13.8) (1.04) (1.28)Engel and Rogers on the BorderEngel and Rogers [46] ask what determines the volatility of the percentagechange in the price of 14 categories of consumer prices sampledin various US and Canadian cities from Sept. 1978 through Dec. 1994. 2Let p ijt be the price of good i in city j at time t, measuredinUSdollars.Let σ ijk be the volatility of the percentage change in the relativeprice of good i in cities j and k. Thatis,σ ijk is the time-series samplestandard deviation of ∆ ln(p ijt /p ikt ). In addition, deÞne D jk as the logarithmof the distance between cities j and k. The idea of the distancevariable is to capture potential effects of transportation costs that maycause violations of the law-of-one price between two locations. Let B jkbe a dummy variable that is 1 if cities j and k are separated by theUS-Canadian border and 0 otherwise, and let X 0 i be a vector of controlvariables, such as a separate dummy variable for each good i and/or foreach city in the sample. Engel and Rogers run restricted cross-sectionregressionsσ ijk = αD jk + βB jk + X 0 i γ i + u ijk,and obtain ˆβ = 10.6 × 10 −4 (s.e.=3.25 × 10 −4 ), ˆα = 11.9 × 10 −3(s.e.=0.42 × 10 −3 ), ¯R2 =0.77. The regression estimates imply thatthe border adds 11.9 × 10 −3 to the average volatility (standard deviation)of prices between two pairs of cities. Based on the estimate of2 The cities are Baltimore, Boston, Chicago, Dallas, Detroit, Houston, Los Angeles,Miami, New York, Philadelphia, Pittsburgh, San Francisco, St. Louis, WashingtonD.C., Calgary, Edmonton, Montreal, Ottawa, Quebec, Regina, Toronto, Vancouver,and Winnipeg.

212 CHAPTER 7. THE REAL EXCHANGE RATEα, this is equivalent to an additional 75,000 miles of distance betweentwo cities in the same country. In addition, the border was found toaccount for 32.4 percent of the variation in the σ ijk , while log distancewas found to explain 20.3 percent.The striking differences between within country violations of thelaw-of-one price and across country violations raise but do not answerthe question, “Why is the border is so important?” This is still an openquestion but possible explanations include,1. Barriers to international trade, such as tariffs, quotas, and nontariffbarriers such as bureaucratic red tape imposed on foreignbusinesses. The Engel-Rogers sample spans periods of pre- andpost-trade liberalization between the US and Canada. In subsampleanalysis, they reject the trade barrier hypothesis.2. Labor markets are more integrated and homogeneous within countriesthan they are across countries. This might explain why therewould be less volatility in per unit costs of production across citieswithin the same country and more per unit cost volatility acrosscountries.3. Nominal price stickiness. Goods prices seem to respond to macroeconomicshocks and news with a lag and behave more sluggishlythan asset prices and nominal exchange rates. Engel and RogersÞnd that this hypothesis does not explain all of the relative pricevolatility. 34. Pricing to market. This is a term used to describe how Þrmswith monopoly power engage in price discrimination between segmenteddomestic and foreign markets characterized by differentelasticities of demand.3 The experiment they run here is as follows. Instead of measuring the relativeintercity price as p ijt /(S t p ∗ ikt )whereS is the nominal exchange rate, p is the USdollar price and p ∗ is the Canadian dollar price, replace it with (p ijt /P t )/(Pt ∗/p∗ ikt )where P and P ∗ are the overall price levels in the US and Canada respectively. If theborder effect is entirely due to sticky prices, the border should be insigniÞcant whenthe alternative price measure is used. But in fact, the border remains signiÞcant sosticky nominal prices can provide only a partial explanation.

7.3. LONG-RUN DETERMINANTS OF THE REAL EXCHANGE RATE213What About the Long-Run?Since the international law-of-one price and purchasing-power parityhas Þrmly been shown to break down in the short run, the next stepmight be to ask whether purchasing-power parity holds in the long run.Recent work on this issue proceeds by testing for a unit root in the logreal exchange rate. The null hypothesis in popular unit-root tests isthat the series being examined contains a unit root. But before we jumpin we should ask whether these tests are interesting from an economicperspective. In order for unit-root tests on the real exchange rate tobe interesting, the null hypothesis (that the real exchange rate has aunit root) should have a Þrm theoretical foundation. Otherwise, if wedo not reject the unit root, we learn only that the test has insufficientpower to reject a null hypothesis that we know to be false, and if wedo reject the unit root, we have only conÞrmed what we believed to betrue in the Þrst place.The next section covers the Balassa-Samuelson model which providesa theoretical justiÞcationforPPPtobeviolatedeveninthelongrun.7.3 Long-Run Determinants of the RealExchange RateWe study a two-sector small open economy. The sectors are a tradablegoodssector and a nontradable-goods sector. The terms of trade (therelative price of exports in terms of imports) are given by world conditionsand are assumed to be Þxed. Before formally developing themodel, it will be useful to consider the following sectoral decompositionof the real exchange rate.Sectoral Real Exchange Rate DecompositionLet P T be the price of the tradable-good and P N be the price of the ⇐(129)nontradable-good, and let the general price level be given by the Cobb-Douglas formP =(P T ) θ (P N ) 1−θ , (7.3)

214 CHAPTER 7. THE REAL EXCHANGE RATEP ∗ =(PT ∗ )θ (PN ∗ )1−θ , (7.4)where the shares of the traded and nontraded-goods are identical athomeandabroad(θ ∗ = θ). The log real exchange rate can be decomposedasq =(s + p ∗ T − p T )+(1− θ)(p ∗ N − p∗ T ) − (1 − θ)(p N − p T ), (7.5)where lower case letters denote variables in logarithms. We adopt thecommodity arbitrage view of PPP (chapter 3.1) and assume that thelaw-of-one price holds for traded goods. It follows that the Þrst termon the right hand side of (7.5), which is the deviation from PPP forthe traded good, is 0. The dynamics of the real exchange rate is thencompletely driven by the relative price of the tradable good in terms ofthe nontraded good.The Balassa—Samuelson ModelNow, we need a theory to understand the behavior of the relative priceof tradables in terms of nontradables. It turns out if, i) factor marketsand Þnal goods markets are competitive, ii) production takes placeunder constant returns to scale, iii) capital is perfectly mobile internationally,iv) labor is internationally immobile but mobile between thetradable and nontradable sectors, then the relative price of nontradablegoods in terms of tradable goods is determined entirely by theproduction technology. Demand (preferences) does not matter at all.The theory is viewed as holding in the long run and therefore omittime subscripts. To Þx ideas, let there be only one traded good and onenontraded good. Capital and labor are supplied elastically. Let L T (L N )and K T (K N ) be labor and capital employed in the production of thetraded Y T (nontraded Y N )good. A T (A N ) is the technology level in thetraded (nontraded) sector. The two goods are produced according toCobb-Douglas production functionsY T = A T L (1−α T )T K (α T )T , (7.6)Y N = A N L (1−α N )N K (α N )N . (7.7)The balance of trade is assumed to be zero which must be true inthe long run. Let the traded good be the numeraire. The small open

7.3. LONG-RUN DETERMINANTS OF THE REAL EXCHANGE RATE215economy takes the price of traded goods as given. We’ll set P T =1. Ris the rental rate on capital, W is the wage rate, and P N is the price ofnontraded goods, all stated in terms of the traded good.Competitive Þrms take factor and output prices as given and chooseK and L to maximize proÞts. The intersectoral mobility of labor andcapital equalizes factor prices paid in the tradable and nontradablesectors. The tradable-good Þrm chooses K T and L T to maximize proÞtsA T L (1−α T )T K α TT − (WL T + RK T ). (7.8)The nontradable-good Þrm’s problem is to choose K N and L N to maximizeP N A N L (1−α N )N K α NN − (WL N + RK N ). (7.9)Let k ≡ (K/L) denote the capital—labor ratio. It follows from theÞrst order conditionsR = A T α T (k T ) α T −1 , (7.10)R = P N A N α N (k N ) α N −1 , (7.11)W = A T (1 − α T )(k T ) α T, (7.12)W = P N A N (1 − α N )(k N ) α N. (7.13)The international mobility of capital combined with the small countryassumption implies that R is exogeneously given by the world rentalrate on capital. (7.10)-(7.13) form four equations in the four unknowns(P N ,W,k T ,k N ).To solve the model, Þrst obtain the traded-goods sector capital-laborratio from (7.10)¸ 1·αT A T(1−α T )k T =. (7.14)RNext, substitute (7.14) into (7.12) to get the wage rateW =(1− α T )(A T )1(1−α T )Substituting (7.15) into (7.13), you getk N =⎛⎜⎝(1 − α T ) A(1 − α N )1(1−α T )T¸ ·αTαT1−α T. (7.15)R³ αTR´ αT1−α TP N A N⎞⎟⎠1α N. (7.16)

216 CHAPTER 7. THE REAL EXCHANGE RATE(130)⇒Finally, plug (7.16) into (7.11) to get the solution for relative price ofthe nontraded good in terms of the traded good(1−αP N = A N )(1−α T )TCR (α N −α T )(1−α T )(7.17)A Nwhere C is a positive constant. Now let a =ln(A),r =ln(R), andc =ln(C) and take logs of (7.17) to get the solution for the log relativeprice of nontraded goods in terms of traded goodsp N =µ 1 − αNa T − a N +1 − α TÃ(αN − α T )(1 − α T )!r + c. (7.18)Over time, the evolution of the log relative price of nontradables dependsonly on the technology and the exogenous rental rate on capital.We see that there are at least two reasons why the relative price ofnon-tradables in terms of tradables should increase with a country’sincome.First, suppose that the economy experiences unbiased technologicalgrowth where a N and a T increaseatthesamerate. p N will rise over timeif traded-goods production is relatively capital intensive (α N < α T ). Astandard argument is that tradables are manufactured goods whoseproduction is relatively capital intensive whereas nontraded goods aremainly services which are relatively labor intensive. Second, p N willincrease over time if technological growth is biased towards the capitalintensive sector. In this case, a T actually grows at a faster rate thana N . If either of these scenarios are correct, it follows that fast growingeconomies will experience a rising relative price of nontradables and by(7.5), a real appreciation over time.The implications for the behavior of the real exchange rate are asfollows. If the productivity factors grow deterministically, the deviationof the real exchange rate from a deterministic trend should be astationary process. But if the productivity factors contain a stochastictrend (chapter 2.6) the log real exchange rate will inherit the randomwalk behavior and will be unit-root nonstationary. In either case, PPPwill not hold in the long run.When we take the Balassa—Samuelson model to the data, it is temptingto think of services as being nontraded. It is also tempting to think

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 217that services are relatively labor intensive. While this may be true ofsome services, such as haircuts, it is not true that all services are nontradedor that they are labor intensive. Financial services are sold athome and abroad by international banks which make them traded, andtransportation and housing services are evidently capital intensive.7.4 Long-Run Analyses of Real ExchangeRatesEmpirical research into the long-run behavior of real exchange rateshas employed econometric analyses of nonstationary time series andis aimed at testing the hypothesis that the real exchange rate has aunit root. This research can potentially provide evidence to distinguishbetween the Casselian and the Balassa—Samuelson views of the world.Univariate Tests of PPP Over the FloatTo test whether PPP holds in the long run, you can use the augmentedDickey-Fullertest(chapter2.4)totestthehypothesisthattherealexchange rate contains a unit root. Using quarterly observations of theCPI-deÞned real exchange rate from 1973.1 to 1997.4 for 19 high-incomecountries, Table 7.2 shows the results of univariate unit-root tests forUS and German real exchange rates. Four lags of ∆q t and a constantwere included in the test equation. The p-values are the proportion ofthe Dickey—Fuller distribution that lies to the left (below) τ c . Includinga trend in the test regressions yields qualitatively similar results andare not reported.Statistical versus Economic SigniÞcance. Classical hypothesis testingis designed to establish statistical signiÞcance. Given a sufficiently longtime series, it may be possible to establish statistical signiÞcance of thestudentized coefficients to reject the unit root, but if the true value ofthe dominant root is 0.98, the half-life of a shock is still over 34 yearsand this stationary process may not be signiÞcantly different from atrue unit-root process in the economic sense.

218 CHAPTER 7. THE REAL EXCHANGE RATEIf that is indeed the case, then in light of the statistical difficultiessurrounding unit-root tests, it can be argued that we should not evencare whether the real exchange rate has a unit root but we should insteadfocus on measuring the economic implications of the real exchangerate’s behavior. What market participants care about is the degree ofpersistence in the real exchange rate and one measure of persistence isthe half life.The annualized half-lives reported in Table 7.2 are based on estimatesadjusted for bias by Kendall’s formula [equation (2.81)]. 4 Theaverage half-life is 3.7 years when the US is the numeraire country. Thatis, on average, it takes 3.7 years–quite a long time since the businesscycle frequency ranges from 1.25 to 8 years–for half of a shock to thelog real exchange rate to disappear. The average half-life is 2.6 yearswhen Germany is the numeraire county.Univariate tests using data from the post Bretton-Woods ßoat typicallycannot reject the hypothesis that the real exchange rate is drivenby a unit-root process. Using the US as the home country, only two ofthetestscanrejecttheunitrootatthe10percentlevelofsigniÞcance.The results are somewhat sensitive to the choice of the home (numeraire)country. 5 Part of the persistence exhibited in the real valueof the dollar comes from the very large swings during the 1980s. Thereal appreciation in the early 1980s and the subsequent depreciationwas largely a dollar phenomenon not shared by cross-rates. To illustrate,the evidence for purchasing-power parity is a little stronger whenGermany is used as the home country since here, the unit root can berejected at the 10 percent level of signiÞcance for German real exchangerates with several European countries.Univariate Tests for PPP Over Long Time SpansOne reason that the evidence against a unit root in q t is weak may bethat the power of the test is low with only 100 quarterly observations. 64 Christiano and Eichenbaum [27] put forth this argument in the context of theunitrootinGNP.5 A point made by Papell and Theodoridis [119].6 The power of a test is the probability that the test correctly rejects the nullhypothesis when it is false.

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 219Table 7.2: Augmented Dickey-Fuller Tests for a Unit Root in Post-1973Real Exchange RatesRelative to USRelative to GermanyCountry τ c (p-value) half-life τ c (p-value) half-lifeAustralia -1.895 (0.329) 4.582 -2.444 (0.124) 2.095Austria -2.434 (0.126) 3.208 -3.809 (0.004) 5.516Belgium -2.369 (0.138) 4.223 -2.580 (0.093) 2.914Canada -1.342 (0.621) – -2.423 (0.127) 2.914Denmark -2.319 (0.155) 3.733 -3.212 (0.017) 1.759Finland -2.919 (0.039) 2.421 -2.589 (0.089) 3.208France -2.526 (0.105) 2.761 -4.540 (0.001) 0.695Germany -2.470 (0.118) 3.025 – – –Greece -2.276 (0.169) 4.336 -2.360 (0.140) 1.278Italy -2.511 (0.107) 2.580 -1.855 (0.351) 5.709Japan -2.057 (0.252) 9.251 -1.930 (0.314) 11.919Korea -1.235 (0.677) 3.274 -2.125 (0.215) 1.165Netherlands -2.576 (0.094) 2.623 -2.676 (0.075) 2.969Norway -2.184 (0.193) 2.668 -2.573 (0.095) 2.539Spain -2.358 (0.140) 5.006 -2.488 (0.113) 2.861Sweden -2.042 (0.257) 5.516 -2.534 (0.103) 1.719Switzerland -2.670 (0.076) 2.215 -3.389 (0.011) 1.759UK -2.484 (0.113) 2.313 -2.272 (0.169) 3.274Notes: Half-lives are adjusted for bias and are measured in years. SigniÞcance atthe 10 percent level indicated in boldface.

220 CHAPTER 7. THE REAL EXCHANGE RATETable 7.3: ADF test and annual half-life estimates using over a centuryof real dollar—pound real exchange ratesLags τ c (p-value) half-life τ ct (p-value) half-life4 -3.074 (0.028) 6.911 -4.906 (0.001) 2.154PPIs 8 -2.122 (0.238) 10.842 -4.104 (0.007) 2.12612 -1.559 (0.510) 16.720 -2.754 (0.229) 2.7854 -3.148 (0.031) 3.659 -3.201 (0.096) 3.520CPIs 8 -3.087 (0.037) 3.033 -3.101 (0.124) 2.98212 -2.722 (0.073) 2.917 -2.720 (0.243) 2.885Bold face indicates signiÞcance at the 10 percentlevel.(131)⇒One way to get more observations is to go back in time and examine realexchange rates over long historical time spans. This was the strategyof Lothian and Taylor [94], who constructed annual real exchange ratesbetween the US and the UK from 1791 to 1990 and between the UKand France from 1803 to 1990 using wholesale price indices.Figure 7.1 displays the log nominal and log real exchange rate (multipledby 100) for the US-UK using CPIs. Using the “eyeball metric,”the real exchange rate appears to be mean reverting over this long historicalperiod. Table 7.3 presents ADF unit-root tests on annual datafor the US and UK. The real exchange rate deÞned over producer pricesextend from 1791 to 1990 and are Lothian and Taylor’s data. 7 The realexchange rate deÞned over consumer prices extend from 1871 to 1997.Half-lives are adjusted for bias with Kendall’s formula (eq. (2.81)).Using long time-span data, the augmented Dickey—Fuller test can rejectthe hypothesis that the real dollar-pound rate has a unit root. Thetest is sensitive to the number of lagged ∆q t values included in the testregression, however. The studentized coefficients are signiÞcant whena trend is included in the test equation which rejects the hypothesisthat the deviation from trend has a unit root. This result is consistentwith the Balassa—Samuelson model in which sectoral productivitydifferentials evolved deterministically.7 David Papell kindly provided me with Lothian and Taylor’s data.

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 2214020Real-20-400-60-80Nominal-1001871 1883 1895 1907 1919 1931 1943 1955 1967 1979 1991Figure 7.1: Real and nominal dollar-pound rate 1871-1997Variance Ratios of Real Exchange RatesWe can use the variance-ratio statistic (see chapter 2.4) to examinethe relative contribution to the overall variance of the real depreciationfrom a permanent component and a temporary component. Table 7.4shows variance ratios calculated on the Lothian—Taylor data along withasymptotic standard errors. 8The point estimates display a ‘hump’ shape. They initially riseabove 1 at short horizons then fall below 1 at the longer horizons. Thisis a pattern often found with Þnancial data. The variance ratio fallsbelow 1 because of a preponderance of negative autocorrelations at thelonger horizons. This means that a current jump in the real exchangerate tends to be offset by future changes in the opposite direction. Suchmovements are characteristic of mean—reverting processes.Even at the 20 year horizon, however, the point estimates indicatethat 23 percent of the variance of the dollar—pound real exchange rate8 Huizinga [77] calculated variance ratio statistics for the real exchange rate from1974 to 1986 while Grilli and Kaminisky [68] did so for the real dollar—pound ratefrom 1884 to 1986aswellasovervarioussubperiods.

222 CHAPTER 7. THE REAL EXCHANGE RATETable 7.4: Variance ratios and asymptotic standard errors of realdollar—sterling exchange rates. Lothian—Taylor data using PPIs.k 1 2 3 4 5 10 15 20VR k 1.00 1.07 0.951 0.906 0.841 0.457 0.323 0.232s.e. – 0.152 0.156 0.166 0.169 0.124 0.106 0.0872can be attributed to a permanent (random walk) component. Theasymptotic standard errors tend to overstate the precision of the varianceratios in small samples. That being said, even at the 20 yearhorizon VR 20 for the dollar—pound rate is (using the asymptotic standarderror) signiÞcantly greater than 0 which implies the presence of apermanent component in the real exchange rate. This conclusion contradictsthe results in Table 7.3 that rejected the unit-root hypothesis.Summary of univariate unit-root tests. We get conßicting evidenceabout PPP from univariate unit-root tests. From post Bretton—Woodsdata, there is not much evidence that PPP holds in the long run whenthe US serves as the numeraire country. The evidence for PPP withGermany as the numeraire currency is stronger. Using long-time spandata, the tests can reject the unit-root, but the results are dependenton the number of lags included in the test equation. On the other hand,the pattern of the variance ratio statistic is consistent with there beinga unit root in the real exchange rate.The time period covered by the historical data span across the Þxedexchange rate regimes of the gold standard and the Bretton Woodsadjustable peg system as well as over ßexible exchange rate periodsof the interwar years and after 1973. Thus, even if the results on thelong-span data uniformly rejected the unit root, we still do not havedirect evidence that PPP holds during a pure ßoating regime.Panel Tests for a Unit Root in the Real Exchange RateLet’s return speciÞcally to the question of whether long-run PPP holdsover the ßoat. Suppose we think that univariate tests have low power

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 223Table 7.5: Levin—Lin Test of PPPNumer- Time Half- Halfaireeffect τ c life τ ct life τc ∗ τct∗yes -8.593 2.953 -9.927 1.796 -1.878 -0.920(0.021) (0.070) (0.164) (0.093)[0.009] [0.074] [0.117] [0.095]US no -6.954 5.328 -7.415 3.943 – –(0.115) (0.651)[0.168] [0.658]yes -8.017 3.764 -9.701 1.816 -1.642 -0.628(0.018) (0.106) (0.154) (0.421)Ger- [0.022] [0.127] [0.158] [0.442]many no -10.252 3.449 -11.185 1.859 – –(0.000) (0.007)[0.001] [0.006]Notes: Bold face indicates signiÞcance at the 10 percent level. Half-lives are basedon bias-adjusted ˆρ by Nickell’s formula [eq.(2.82)] and are stated in years. Nonparametricbootstrap p-values in parentheses. Parametric bootstrap p-values in squarebrackets.because the available time-series are so short. We will revisit the questionby combining observations across the 19 countries that we examinedin the univariate tests into a panel data set. We thus have N =18real exchange rate observations over T = 100 quarterly periods.The results from the popular Levin—Lin test (chapter 2.5) are presentedin Table 7.5. 9 Nonparametric bootstrap p-values in parenthesesand parametric bootstrap p-values in square brackets. τ ct indicates alinear trend is included in the test equations. τ c indicates that only aconstant is included in the test equations. τc∗ and τct ∗ are the adjustedstudentized coefficients (see chapter 2.5). When we account for thecommon time effect, the unit root is rejected at the 10 percent levelboth when a time trend is and is not included in the test equationswhen the dollar is the numeraire currency. Using the deutschemark asthe numeraire currency, the unit root cannot be rejected when a trend9 Frankel and Rose [59], MacDonald [97], Wu [135], and Papell conduct Levin—Lintests on the real exchange rate.

224 CHAPTER 7. THE REAL EXCHANGE RATETable 7.6: Im—Pesaran—Shin and Maddala—Wu Tests of PPPNumer-Im—Pesaran—Shinaire ¯τ c (p-val) [p-val] ¯τ ct (p-val) [p-val]US -2.259 (0.047) [0.052] -2.385 (0.302) [0.307]Ger. -2.641 (0.000) [0.000] -3.119 (0.000) [0.001]Numer-Maddala—Wuaire ¯τ c (p-val) [p-val] ¯τ ct (p-val) [p-val]US 66.902 (0.083) [0.088] 40.162 (0.351) [0.346]Ger. 101.243 (0.000) [0.000] 102.017 (0.000) [0.000]Nonnparametric bootstrap p-values in parentheses. Parametric bootstrap p-valuesin square brackets. Bold face indicates signiÞcance at the 10 percent level.is included. The asymptotic evidence against the unit root is very weak.Next,wetesttheunitrootwhenthecommontimeeffect is omitted.Here, the evidence against the unit root is strong when thedeutschemark is the numeraire currency, but not for the dollar. Thebias-adjusted approximate half-life to convergence range from 1.7 to 5.3years, which many people still consider to be a surprisingly long time.Table 7.6 shows panel tests of PPP using the Im, Pesaran, andShin test and the Maddala—Wu test. Here, I did not remove the commontime effect. These tests are consistent with the Levin-Lin testresults. When the dollar is the numeraire, we cannot reject that thedeviation from trend is a unit root. When the deutschemark is thenumeraire currency, the unit root is rejected whether or not a trend isincluded. The evidence against a unit root is generally stronger whenthe deutschemark is used as the numeraire currency.Canzoneri, Cumby, and Diba’s test of Balassa-SamuelsonCanzoneri, Cumby, and Diba [21] employ IPS to test implications of theBalassa—Samuelson model. They examine sectoral OECD data for theUS, Canada, Japan, France, Italy, UK, Belgium, Denmark, Sweden,Finland, Austria, and Spain. They deÞne output by the “manufacturing”and “agricultural, hunting forestry and Þshing” sectors to betraded goods. Nontraded goods are produced by the “wholesale and

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 225Table 7.7: Canzoneri et. al.’s IPS tests of Balassa—SamuelsonAllEuropeanVariable countries G-7 Countries(p N − p T ) − (x T − x N ) -3.762 -2.422 –s t − (p T − p ∗ T )(dollar) -2.382 -5.319 –s t − (p T − p ∗ T )(DM) -1.775 – -1.565Notes: Bold face indicates asymptotically signiÞcant at the 10 percent level.retail trade,” “restaurants and hotels,” “transport, storage and communications,”“Þnance, insurance, real estate and business,” “communitysocial and personal services,” and the “non-market services” sectors.Their analysis begins with the Þrst-order conditions for proÞt maximizingÞrms. Equating (7.12) to (7.13), the relative price of nontrad- ⇐(133)ables in terms of tradables can be expressed asP NP T= 1 − α T A T k α TT1 − α N A N k α (7.19)NNwhere k = K/L is the capital labor ratio. By virtue of the Cobb-Douglas form of the production function, Ak α = Y/L is the averageproduct of labor. Let x T ≡ ln(Y T /L T )andx N ≡ ln(Y N /L N )denotethe log average product of labor. We rewrite (7.19) in logarithmic formasµ 1 − αTp N − p T =ln + x T − x N . (7.20)1 − α NTable 7.7 shows the standardized ¯t calculated by Canzoneri, Cumbyand Diba. All calculations control for common time effects. Theirresults support the Balassa—Samuelson model. They Þnd evidence thatthere is a unit root in p N − p T and in x T − x N , and that they arecointegrated, and there is reasonably strong evidence that PPP holdsfor traded goods.

226 CHAPTER 7. THE REAL EXCHANGE RATESize Distortion in Unit-Root TestsEmpirical researchers are typically worried that unit-root tests mayhave low statistical power in applications due to the relatively smallnumber of time series observations available. Low power means thatthe null hypothesis that the real exchange rate has a unit root will bedifficult to reject even if it is false. Low power is a fact of life becausefor any Þnite sample size, a stationary process can be arbitrarily wellapproximated by a unit-root process, and vice versa. 10 The conßictingevidence from post 1973 data and the long time-span data are consistentwith the hypothesis that the real exchange rate is stationary but thetests suffer from low statistical power.The ßip side to the power problem is that the tests suffer size distortionin small samples. Engel [45] suggests that the observational equivalenceproblem lies behind the inability to reject the unit root duringthe post Bretton Woods ßoat and the rejections of the unit root in theLothian—Taylor data and argues that these empirical results are plausiblygenerated by a permanent—transitory components process with aslow—moving permanent component. Engel’s point is that the unit-roottests have more power as T grows and are more likely to reject withthe historical data than over the ßoat. But if the truth is that the realexchange rate contains a small unit root process, the size of the testwhich is approximately equal to the power of the test, is also higherwhen T is large. That is, the probability of committing a type I erroralso increases with sample size and that the unit-root tests suffer fromsize distortion with the sample sizes available.10 Think of the permanent—transitory components decomposition. T < ∞ observationsfrom a stationary AR(1) process will be observationally equivalent to Tobservations of a permanent—transitory components model with judicious choice ofthe size of the innovation variance to the permanent and the transitory parts. Thisis the argument laid forth in papers by Blough [16], Cochrane [30], and Faust [50].

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 227Real Exchange Rate Summary1. Purchasing-power parity is a simple theory that links domesticand foreign prices. It is not valid as a short-run proposition butmost international economists believe that some variant of PPPholds in the long run.2. There are several explanations for why PPP does not hold. TheBalassa—Samuelson view focuses on the role of nontraded goods.Another view, that we will exploit in the next chapter, is thatthe persistence exhibited in the real exchange rate is due tonominal rigidities in the macroeconomy where Þrms are reluctantto change nominal prices immediately following shocks ofreasonably small magnitude.

228 CHAPTER 7. THE REAL EXCHANGE RATEProblems1. (Heterogeneous commodity baskets). Suppose there are two goods,both of which are internationally traded and for whom the law of oneprice holds,p 1t = s t + p ∗ 1t, p 2t = s t + p ∗ 2t,where p i is the home currency price of good i, p ∗ i is the foreign currencyprice, and s is the nominal exchange rate, all in logarithms. Assumefurther that the nominal exchange rate follows a unit-root process,s t = s t−1 + v t where v t is a stationary process, and that foreign pricesare driven by a common stochastic trend, zt∗p ∗ 1t = z ∗ t + ² ∗ 1t p ∗ 2t = z ∗ t + ² ∗ 2t.where z ∗ t = z ∗ t−1 + u t, ² ∗ it , (i = 1, 2) are stationary processes, and u t isiid with E(u t )=0,E(u 2 t )=σ 2 u. Show that even if the price levels areconstructed as,p t = φp 1t +(1 − φ)p 2t , p ∗ t = φ ∗ p ∗ 1t +(1 − φ ∗ )p ∗ 2t,with φ 6= φ ∗ ,thatp t − (s t + p ∗ t ) is a stationary process.

Chapter 8The Mundell-Fleming ModelMundell [108]—Fleming [54] is the IS-LM model adapted to the openeconomy. Although the framework is rather old and ad hoc the basicframework continues to be used in policy related research (Williamson [132],Hinkle and Montiel [107], MacDonald and Stein [98]). The hallmarkof the Mundell-Fleming framework is that goods prices exhibit stickinesswhereas asset markets–including the foreign exchange market–are continuously in equilibrium. The actions of policy makers play amajor role in these models because the presence of nominal rigiditiesopens the way for nominal shocks to have real effects. We begin with asimple static version of the model. Next, we present the dynamic butdeterministic Mundell-Fleming model due to Dornbusch [39]. Third, wepresent a stochastic Mundell-Fleming model based on Obstfeld [111].8.1 A Static Mundell-Fleming ModelThis is a Keynesian model where goods prices are Þxed for the durationof the analysis. The home country is small in sense that it takesforeign variables as Þxed. All variables except the interest rate are inlogarithms.Equilibrium in the goods market is given by an open economy versionof the IS curve. There are three determinants of the demand fordomestic goods. First, expenditures depend positively on own income ythrough the absorption channel. An increase in income leads to higher229

230 CHAPTER 8. THE MUNDELL-FLEMING MODELconsumption, most of which is spent on domestically produced goods.Second, domestic goods demand depends negatively on the interestrate i through the investment—saving channel. Since goods prices areÞxed, the nominal interest rate is identical to the real interest rate.Higher interest rates reduce investment spending and may encourage areduction of consumption and an increase in saving. Third, demand forhome goods depends positively on the real exchange rate s+p ∗ −p. Anincrease in the real exchange rate lowers the price of domestic goodsrelative to foreign goods leading expenditures by residents of the homecountry as well as residents of the rest of the world to switch towarddomestically produced goods. We call this the expenditure switchingeffect of exchange rate ßuctuations. In equilibrium, output equals expenditureswhich is given by the IS curvey = δ(s + p ∗ − p)+γy − σi + g, (8.1)where g is an exogenous shifter which we interpret as changes in Þscalpolicy. The parameters δ, γ, and σ are deÞned to be positive with0 < γ < 1.As in the monetary model, log real money demand m d − p dependspositively on log income y and negatively on the nominal interest rate iwhich measures the opportunity cost of holding money. Since the pricelevel is Þxed, the nominal interest rate is also the real interest rate, r.In logarithms, equilibrium in the money market is represented by theLM curvem − p = φy − λi. (8.2)The country is small and takes the world price level and world interestrate as given. For simplicity, we Þx p ∗ = 0. The domestic price level isalso Þxed so we might as well set p =0.Capital is perfectly mobile across countries. 1 International capitalmarket equilibrium is given by uncovered interest parity with static1 Given the rapid pace at which international Þnancial markets are becomingintegrated, analyses under conditions of imperfect capital mobility is becoming lessrelevant. However, one can easily allow for imperfect capital mobility by modelingboth the current account and the capital account and setting the balance of paymentsto zero (the external balance constraint) as an equilibrium condition. Seethe end-of-chapter problems.

8.1. A STATIC MUNDELL-FLEMING MODEL 231expectations 2 i = i ∗ . (8.3)Substitute (8.3) into (8.1) and (8.2). Totally differentiate the resultand rearrange to obtain the two-equation systemdm =dy ="φδ1 − γ ds − λ +φσ#di ∗ +1 − γφ dg, (8.4)1 − γδ1 − γ ds − σ1 − γ di∗ + dg1 − γ . (8.5)All of our comparative statics results come from these two equations.Adjustment under Fixed Exchange RatesDomestic credit expansion. Assume that the monetary authorities arecredibly committed to Þxing the exchange rate. In this environment,the exchange rate is a policy variable. As long as the Þx isineffect,we set ds = 0. Income y and the money supply m are endogenousvariables.Suppose the authorities expand the domestic credit component ofthe money supply. Recall from (1.22) that the monetary base is madeup of the sum of domestic credit and international reserves. In theabsence of any other shocks (di ∗ =0,dg = 0), we see from (8.4) thatthere is no long-run change in the money supply dm = 0 and from (8.5),there is no long-run change in output. The initial attempt to expandthe money supply by increasing domestic credit results in an offsettingloss of international reserves. Upon the initial expansion of domesticcredit, the money supply does increase. The interest rate must remainÞxed at the world rate, however, and domestic residents are unwillingto hold additional money at i ∗ . They eliminate the excess money by accumulatingforeign interest bearing assets and run a temporary balanceof payments deÞcit. The domestic monetary authorities evidently haveno control over the money supply in the long run and monetary policyis said to be ineffective as a stabilization tool under a Þxed exchangerate regime with perfect capital mobility.2 Agents expect no change in the exchange rate.

232 CHAPTER 8. THE MUNDELL-FLEMING MODELThe situation is depicted graphically in Figure 8.1. First, the expansionof domestic credit shifts the LM curve out. To maintain interestparity there is an incipient capital outßow. The central bank defendsthe exchange rate by selling reserves. This loss of reserves causes theLM curve to shift back to its original position.rr*LM(1)(2)aFFbISyy 0Figure 8.1: Domestic credit expansion shifts the LM curve out. Thecentral bank loses reserves to accommodate the resulting capital outßowwhich shifts the LM curve back in.(136)⇒Domestic currency devaluation. From (8.4)-(8.5), you havedy =[δ/(1−γ)]ds > 0anddm =[φδ/(1 −γ)]ds > 0. The expansionaryeffects of a devaluation are shown in Figure 8.2. The devaluation makesdomestic goods more competitive and expenditures switch towards domesticgoods. This has a direct effect on aggregate expenditures. In aclosed economy, the expansion would lead to an increase in the interestrate but in the open economy under perfect capital mobility, theexpansion generates a capital inßow. To maintain the new exchangerate, the central bank accommodates the capital ßows by accumulatingforeign exchange reserves with the result that the LM curve shifts out.One feature that the model misses is that in real world economies,the country’s foreign debt is typically denominated in the foreign currencyso the devaluation increases the country’s real foreign debt burden.

8.1. A STATIC MUNDELL-FLEMING MODEL 233rLMr*(1)ab(2)cFFISyy 0y 1Figure 8.2: Devaluation shifts the IS curve out. The central bankaccumulates reserves to accommodate the resulting capital inßow whichshifts the LM curve out.Fiscal policy shocks. The results of an increase in government spendingare dy =[1/(1 − γ)]dg and dm =[φ/(1 − γ)]dg which is expansionary. ⇐(137)Theincreaseing shifts the IS curve to the right and has a direct effecton expenditures. Fiscal policy works the same way as a devaluationand is said to be an effective stabilization tool under Þxed exchangerates and perfect capital mobility.Foreign interest rate shocks. An increase in the foreign interest ratehas a contractionary effect on domestic output and the money supply,dy = −(σ/(1 − γ))di ∗ ,anddm = −(λ + φσ/(1 − γ))di ∗ . The increasei ∗ creates an incipient capital outßow. To defend the exchange rate,the monetary authorities sell foreign reserves which causes the moneysupply to contract. The situation is depicted graphically in Figure 8.3.Implied International transmissions. Although we are working with thesmall-country version of the model, we can qualitatively deduce howpolicy shocks would be transmitted internationally in a two-countrymodel. If the increase in i ∗ was the result of monetary tightening inthe large foreign country, output also contracts abroad. We say that

234 CHAPTER 8. THE MUNDELL-FLEMING MODELrLMr*(2)y 1(1)FFISyy 0Figure 8.3: An increase in i ∗ generates a capital outßow, a loss of centralbank reserves, and a contraction of the domestic money supply.monetary shocks are positively transmitted internationally as they leadto positive output comovements at home and abroad. If the increase ini ∗ was the result of expansionary foreign government spending, foreignoutput expands whereas domestic output contracts. Aggregate expenditureshocks are said to be negatively transmitted internationally undera Þxed exchange rate regime.A currency devaluation has negative transmission effects. The devaluationof the home currency is equivalent to a revaluation of theforeign currency. Since the domestic currency devaluation has an expansionaryeffect on the home country, it must have a contractionaryeffect on the foreign country. A devaluation that expands the homecountry at the expense of the foreign country is referred to as a beggarthy-neighborpolicy.Flexible Exchange RatesWhen the authorities do not intervene in the foreign exchange market,s and y are endogenous in the system (8.4)-(8.5) and the authoritiesregain control over m, which is treated as exogenous.Domestic credit expansion. An expansionary monetary policy generatesan incipient capital outßow which leads to a depreciation of the

8.1. A STATIC MUNDELL-FLEMING MODEL 235rLMr*acFFy 0(1)by 1(2)ISyFigure 8.4: Expansion of domestic credit shifts LM curve out. Incipientcapital outßow is offset by depreciation of domestic currency whichshifts the IS curve out.home currency ds =[(1− γ)/φδ]dm > 0. The expenditure switchingeffect of the depreciation increases expenditures on the home good andhas an expansionary effect on output dy =(1/φ)dm > 0.The situation is represented graphically in Figure 8.4 where theexpansion of domestic credit shifts the LM curve to the right. In theclosed economy, the home interest rate would fall but in the small openeconomy with perfect capital mobility, the result is an incipient capitaloutßow which causes the home currency to depreciate (s increases) andthe IS curve to shift to the right. The effectiveness of monetary policyis restored under ßexible exchange rates.Fiscal policy. Fiscal policy becomes ineffective as a stabilization toolunder ßexible exchange rates and perfect capital mobility. The situationis depicted in Figure 8.5. An expansion of government spending isrepresentedbyaninitialoutwardshiftintheIScurvewhichleadstoanincipient capital inßow and an appreciation of the home currency ds =−(1/δ)dg < 0. The resulting expenditure switch forces a subsequentinward shift of the IS curve. The contractionary effects of the inducedappreciation offsets the expansionary effect of the government spendingleaving output unchanged dy = 0. The model predicts an international

236 CHAPTER 8. THE MUNDELL-FLEMING MODELr(1)LM(2)r*FFy 0ISyFigure 8.5: Expansionary Þscal policy shifts IS curve out. Incipientcapital inßow generates an appreciation which shifts the IS curve backto its original position.version of crowding out. Recipients of government spending expand atthe expense of the traded goods sector.Interest rate shocks. An increase in the foreign interest rate leads to anincipient capital outßow and a depreciation given byds =[(λ(1 − γ) +σφ)/φδ]di ∗ > 0. The expenditure-switching effectof the depreciation causes the IS curve in Figure 8.6 to shift out. Theexpansionary effect of the depreciation more than offsets the contractionaryeffect of the higher interest rate resulting in an expansion ofoutput dy =(λ/φ)di ∗ > 0.International transmission effects. If the interest rate shock was causedby a contraction in foreign money, the expansion of domestic outputwouldbeassociatedwithacontractionofforeignoutputandmonetarypolicy shocks are negatively transmitted from one country to anotherunder ßexible exchange rates. Government spending, on the other handis positively transmitted. If the increase in the foreign interest rate wasprecipitated by an expansion of foreign government spending, we wouldobserve expansion in output both abroad and at home.

8.2. DORNBUSCH’S DYNAMIC MUNDELL—FLEMING MODEL237rLM(1)r*FF(2)y 0y 1ISyFigure 8.6: An increase in the world interest rate generates an incipientcapital outßow, leading to a depreciation and an outward shift in theIS curve.8.2 Dornbusch’s Dynamic Mundell—FlemingModelAs we saw in Chapter 3, the exchange rate in a free ßoat behaves muchlike stock prices. In particular, it exhibits more volatility than macroeconomicfundamentals such as the money supply and real GDP. Dornbusch[39] presents a dynamic version of the Mundell—Fleming modelthat explains excess exchange rate volatility in a deterministic perfectforesight setting. The key feature of the model is that the asset marketadjusts to shocks instantaneously while goods market adjustment takestime.The money market is continuously in equilibrium which is representedby the LM curve, restated here asm − p = φy − λi. (8.6)To allow for possible disequilibrium in the goods market, let y denoteactual output which is assumed to be Þxed, andy d denote the demandfor home output. The demand for domestic goods depends on the real

238 CHAPTER 8. THE MUNDELL-FLEMING MODELexchange rate s + p ∗ − p, realincomey, andtheinterestratei 3y d = δ(s − p)+γy − σi + g, (8.7)wherewehavesetp ∗ =0.Denote the time derivative of a function x of time with a “dot”úx(t) =dx(t)/dt. Price level dynamics are governed by the ruleúp = π(y d − y), (8.8)where the parameter 0 < π < ∞ indexes the speed of goods marketadjustment. 4 (8.8) says that the rate of inßation is proportional toexcess demand for goods. Because excess demand is always Þnite, therate of change in goods prices is always Þnite so there are no jumps inprice level. If the price level cannot jump, then at any point in time itis instantaneously Þxed. The adjustment of the price-level towards itslong-run value must occur over time and it is in this sense that goodsprices are sticky in the Dornbusch model.International capital market equilibrium is given by the uncoveredinterest parity conditioni = i ∗ + ús e , (8.9)where ús e is the expected instantaneous depreciation rate. Let ¯s bethe steady-state nominal exchange rate. The model is completed byspecifying the forward—looking expectationsús e = θ(¯s − s). (8.10)Market participants believe that the instantaneous depreciation is proportionalto the gap between the current exchange rate and its long-runvalue but to be model consistent, agents must have perfect foresight.This means that the factor of proportionality θ must be chosen to beconsistent with values of the other parameters of the model. This perfectforesight value of θ can be solved for directly, (as in the chapter3 Making demand depend on the real interest rate results in the same qualitativeconclusions, but messier algebra.4 Low values of π indicate slow adjustment. Letting π → ∞ allows goodsprices to adjust instantaneously which allows the goods market to be in continuousequilibrium.

8.2. DORNBUSCH’S DYNAMIC MUNDELL—FLEMING MODEL239appendix) or by the method of undetermined coefficients. 5 Since wecan understand most of the interesting predictions of the model withoutexplicitly solving for the equilibrium, we will do so and simplyassume that we have available the model consistent value of θ suchthatús e = ús. (8.11)Steady-State EquilibriumLet an ‘overbar’ denote the steady-state value of a variable. The modelis characterized by a Þxed steady state with ús = úp =0andī = i ∗ , (8.12)¯p = m − φy + λī, (8.13)¯s = ¯p + 1 [(1 − γ)y + σī − g]. (8.14)δDifferentiating these long-run values with respect to m yieldsd¯p/dm =1,andd¯s/dm = 1. The model exhibits the sensible characteristicthat money is neutral in the long run. Differentiating thelong-run values with respect to g yields d¯s/dg = −1/δ = d(¯s − ¯p)/dg.Nominal exchange rate adjustments in response to aggregate expenditureshocks are entirely real in the long run and PPP does not hold ifthere are permanent shocks to the composition of aggregate expenditures,even in the long run.Exchange rate dynamicsThe hallmark of this model is the interesting exchange rate dynamicsthat follow an unanticipated monetary expansion. 6 Totally differentiating(8.6) but note that p is instantaneously Þxed and y is always Þxed,5 The perfect-foresight solution isθ = 1 2 [π(δ + σ/λ)+p π 2 (δ + σ/λ) 2 +4πδ/λ].6 Thisoftenusedexperimentbringsupanuncomfortablequestion. Ifagentshaveperfect foresight, how a shock be unanticipated?

240 CHAPTER 8. THE MUNDELL-FLEMING MODEL2.32.22.12.01.91 9 17 25 33 41 49 57Figure 8.7: Exchange Rate Overshooting in the Dornbusch model withπ =0.15, δ =0.15, σ =0.02, λ =5.the monetary expansion produces a liquidity effectdi = − 1 dm < 0. (8.15)λDifferentiate (8.9) while holding i ∗ constant and use d¯s = dm to getdi = θ(dm − ds). Use this expression to eliminate di in (8.15). Solvingfor the instantaneous depreciation yieldsµds = 1+ 1 dm > d¯s. (8.16)λθThis is the famous overshooting result. Upon impact, the instantaneousdepreciation exceeds the long-run depreciation so the exchangerate overshoots its long-run value. During the transition to the longrun, i

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL 241tals even when agents have perfect foresight. The implied dynamics areillustrated in Figure 8.7.If there were instantaneous adjustment (π = ∞), we would immediatelygo to the long run and would continuously be in equilibrium. Solong as π < ∞, the goods market spends some time in disequilibriumand the economy-wide adjustment to the long-run equilibrium occursgradually. The transition paths, which we did not solve for explicitlybut is treated in the chapter appendix, describe the disequilibrium dynamics.It is in comparison to the ßexible-price (long-run) equilibriumthat the transitional values are viewed to be in disequilibrium.There is no overshooting nor associated excess volatility in responseto Þscal policy shocks. You are invited to explore this further in theend-of-chapter problems.8.3 A Stochastic Mundell—Fleming ModelLet’s extend the Mundell-Fleming model to a stochastic environmentfollowing Obstfeld [111]. Let ytd be aggregate demand, s t be the nominalexchange rate, p t be the domestic price level, i t be the domesticnominal interest rate, m t be the nominal money stock, and E t (X t )bethe mathematical expectation of the random variable X t conditionedon date—t information. All variables except interest rates are in naturallogarithms. Foreign variables are taken as given so without loss ofgenerality we set p ∗ =0andi ∗ =0.The IS curve in the stochastic Mundell-Fleming model isy d t = η(s t − p t ) − σ[i t − E t (p t+1 − p t )] + d t , (8.17)where d t is an aggregate demand shock and i t − E t (p t+1 − p t )istheexante real interest rate. The LM curve ism t − p t = y d t − λi t, (8.18)where the income elasticity of money demand is assumed to be 1. Capitalmarket equilibrium is given by uncovered interest parityi t − i ∗ = E t (s t+1 − s t ). (8.19)

242 CHAPTER 8. THE MUNDELL-FLEMING MODELThe long-run or the steady-state is not conveniently characterized ina stochastic environment because the economy is constantly being hitby shocks to the non-stationary exogenous state variables. Instead of along-run equilibrium, we will work with an equilibrium concept given bythe solution formed under hypothetically fully ßexible prices. Then aslong as there is some degree of price-level stickiness that prevents completeinstantaneous adjustment, the disequilibium can be characterizedby the gap between sticky-price solution and the shadow ßexible-priceequilibrium.Let the shadow values associated with the ßexible-price equilibriumbe denoted with a ‘tilde.’ The predetermined part of the price level isE t−1˜p t which is a function of time t-1 information. Let θ(˜p t − E t−1˜p t )represent the extent to which the actual price level p t responds at datet to new information where θ is an adjustment coefficient. The stickypriceadjustment rule isp t =E t−1˜p t + θ(˜p t − E t−1˜p t ). (8.20)According to this rule, goods prices display rigidity for at most oneperiod. Prices are instantaneously perfectly ßexible if θ =1andtheyare completely Þxedone-periodinadvanceifθ = 0. Intermediatedegrees of price Þxity are characterized by 0 < θ < 1 which allowthe price level at t to partially adjust from its one-period-in-advancepredetermined value E t−1 (˜p t ) in response to period t news, ˜p t − E t−1˜p t .The exogenous state variables are output, money, and the aggregatedemand shock and they are governed by unit root processes. Outputand the money supply are driven by the driftless random walksy t = y t−1 + z t , (8.21)m t = m t−1 + v t , (8.22)iidwhere z t ∼ N(0, σz)andv 2 iidt ∼ N(0, σv). 2 The demand shock d t also is aunit-root processd t = d t−1 + δ t − γδ t−1 , (8.23)where δ tiid∼ N(0, σ 2 δ). Demand shocks are permanent, as represented byd t−1 but also display transitory dynamics where some portion 0 < γ < 1

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL 243of any shock δ t is reversed in the next period. 7 To solve the model, theÞrst thing you need is to get the shadow ßexible-price solution.Flexible Price SolutionUnder fully-ßexible prices, θ = 1 and the goods market is continuouslyin equilibrium y t = yt d . Let q t = s t − p t be the real exchange rate.Substitute (8.19) into the IS curve (8.17), and re-arrange to get˜q t = y à !t − d t ση + σ + E t˜q t+1 . (8.24)η + σThis is a stochastic difference equation in ˜q. It follows that the solutionfor the ßexible-price equilibrium real exchange rate is given bythe present value formula which you can get by iterating forward on(8.24). But we won’t do that here. Instead, we will use the method ofundetermined coefficients. We begin by conjecturing a guess solutionin which ˜q depends linearly on the available date t information˜q t = a 1 y t + a 2 m t + a 3 d t + a 4 δ t . (8.25)We then deduce conditions on the a−coefficients such that (8.25) solvesthe model. Since m t does not appear explicitly in (8.24), it probably isthecasethata 2 = 0. To see if this is correct, take time t conditionalexpectations on both sides of (8.25) to getE t˜q t+1 = a 1 y t + a 2 m t + a 3 (d t − γδ t ). (8.26)Substitute (8.25) and (8.26) into (8.24) to get⇐(139)a 1 y t + a 2 m t + a 3 d t + a 4 δ t= y t − d tη + σ + ση + σ [a 1y t + a 2 m t + a 3 (d t − γδ t )]7 Recursive backward substitution in (8.23) gives, d t = δ t +(1 − γ)δ t−1 +(1 −γ)δ t−2 + ···. Thus the demand shock is a quasi-random walk without drift in thatashockδ t has a permanent effect on d t , but the effect on future values (1 − γ) issmaller than the current effect.

244 CHAPTER 8. THE MUNDELL-FLEMING MODELNow equate the coefficients on the variables to geta 1 = 1 η = −a 3,a 2 = 0,a 4 = γ à !σ.η η + σThe ßexible-price solution for the real exchange rate is˜q t = y t − d tη+ γ ηÃση + σ!δ t , (8.27)where indeed nominal (monetary) shocks have no effect on ˜q t .Therealexchange rate is driven only by real factors—supply and demand shocks.Since both of these shocks were assumed to evolve according to unitroot process, there is a presumption that ˜q t alsoisaunitrootprocess.A permanent shock to supply y t leads to a real depreciation. Sinceγσ/(η(η + σ)) < (1/η), a permanent shock to demand δ t leadstoarealappreciation. 8To get the shadow price level, start from (8.18) and (8.19) to get˜p t = m t − y t + λE t (s t+1 − s t ). If you add λ˜p t to both sides, add andsubtract λE t˜p t+1 to the right side and rearrange, you get(1 + λ)˜p t = m t − y t + λE t (˜q t+1 − ˜q t )+λE t˜p t+1 . (8.28)By (8.27), E t (˜q t+1 − ˜q t )=[γ/(η + σ)]δ t , which you can substitute backinto (8.28) to obtain the stochastic difference equation˜p t = m t − y t1+λ + λγ(η + σ)(1 + λ) δ t + λ1+λ E t˜p t+1 . (8.29)Now solve (8.29) by the MUC. Let˜p t = b 1 m t + b 2 y t + b 3 d t + b 4 δ t , (8.30)be the guess solution. Taking expectations conditional on time-t informationgivesE t˜p t+1 = b 1 m t + b 2 y t + b 3 (d t − γδ t ). (8.31)8 Here is another way to motivate the null hypothesis that the real exchange ratefollows a unit root process in tests of long-run PPP covered in Chapter 7.

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL 245Substitute (8.31) and (8.30) into (8.29) to getb 1 m t + b 2 y t + b 3 d t + b 4 δ t= m t − y t1+λ + λγ(1 + λ)(η + σ) δ t (8.32)λ+1+λ [b 1m t + b 2 y t + b 3 (d t − γδ t )].Equate coefficients on the variables to getb 1 = 1 = −b 2 ,b 3 = 0,b 4 =λγ(1 + λ)(η + σ) . (8.33)Write the ßexible-price equilibrium solution for the price level as˜p t = m t − y t + αδ t , (8.34)whereλγα =(1 + λ)(η + σ) .A supply shock y t generates shadow deßationary pressure whereas demandshocks δ t and money shocks m t generate shadow inßationarypressure.The shadow nominal exchange rate can now be obtained by adding˜q t +˜p t˜s t = m t +Ã1 − ηη!y t − d Ã!t γση + η(η + σ) + α δ t . (8.35)Positive monetary shocks unambiguously lead to a nominal depreciationbut the effect of a supply shock on the shadow nominal exchange ratedepends on the magnitude of the expenditure switching elasticity, η.You are invited to verify that a positive demand shock δ t lowers thenominal exchange rate.

246 CHAPTER 8. THE MUNDELL-FLEMING MODELCollecting the equations that form the ßexible-price solution wehavey t = y t−1 + z t = y(z t ),˜q t = y t − d t γσ+η η(η + σ) δ t =˜q(z t , δ t ),˜p t = m t − y t + αδ t =˜p(z t , δ t ,v t ).The system displays a triangular structure in the exogenous shocks.Only supply shocks affect output, demand and supply shocks affectthe real exchange rate, while supply, demand, and monetary shocksaffect the price level. We will revisit the implications of this triangularstructure in Chapter 8.4.Disequilibrium DynamicsTo obtain the sticky-price solution with 0 < θ < 1, substitute thesolution (8.34) for ˜p t into the price adjustment rule (8.20), to getp t = m t−1 −y t−1 +θ[v t −z t +αδ t ]. Next, add and subtract (v t −z t +αδ t )to the right side and rearrange to getp t =˜p t − (1 − θ)[v t − z t + αδ t ]. (8.36)The gap between p t and ˜p t is proportional to current information(v t − z t + αδ t ), which we’ll call news. You will see below that thegap between all disequilibrium values and their shadow values are proportionalto this news variable. Monetary shocks v t and demand shocksδ t cause the price level to lie below its equilibrium value ˜p t while supplyshocks z t cause the current price level to lie above its equilibriumvalue. 9 Since the solution for p t does not depend on lagged values ofthe shocks, the deviation from full-price ßexibility values generated bycurrent period shocks last for only one period.Next, solve for the real exchange rate. Substitute (8.36) and aggregatedemand from the IS curve (8.17) into the LM curve (8.18) to9 The price-level responses to the various shocks conform precisely to the predictionsfrom the aggregate-demand, aggregate-supply model as taught in principlesof macroeconomics.

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL 247getm t −˜p t +(1−θ)[v t −z t +αδ t ]=d t +ηq t −(σ+λ)(E t q t+1 −q t )−λE t (p t+1 −p t ).(8.37)By (8.36) and (8.34) you know thatE t (p t+1 − p t )=−αδ t +(1− θ)[v t − z t + αδ t ]. (8.38)Substitute (8.38) and ˜p tequation in q tinto (8.37) to get the stochastic difference(η+σ+λ)q t = y t −d t +(1−θ)(1+λ)(v t −z t )−θ(1+λ)αδ t +(σ+λ)E t q t+1 .(8.39)Let the conjectured solution beq t = c 1 y t + c 2 d t + c 3 δ t + c 4 v t + c 5 z t . (8.40)It follows thatE t q t+1 = c 1 y t + c 2 (d t − γδ t ). (8.41)Substitute (8.40) and (8.41) into (8.39) to get⇐(140)(η + σ + λ)[c 1 y t + c 2 d t + c 3 δ t + c 4 v t + c 5 z t ]= y t − d t +(1− θ)(1 + λ)(v t − z t )−θ(1 + λ)αδ t +(σ + λ)[c 1 y t + c 2 (d t − γδ t )].Equating coefficients givesc 1 = 1 η = −c 2,c 3 =c 4 =γ(σ + λ) − ηαθ(1 + λ),η(η + σ + λ)(1 − θ)(1 + λ)= −c 5 ,η + σ + λand the solution isq t = y t − d tη+γ(σ + λ) − αηθ(1 + λ) (1 − θ)(1 + λ)d t +η(η + σ + λ)η + σ + λ (v t − z t ).⇐(141)

248 CHAPTER 8. THE MUNDELL-FLEMING MODEL(142)⇒Using the deÞnition of α and (8.27) to eliminate (y t −d t )/η, rewrite thesolution in terms of ˜q t and news(1 + λ)(1 − θ)q t =˜q t +η + σ + λ [v t − z t + αδ t ]. (8.42)Nominal shocks have an effect on the real exchange rate due to the rigidityin price adjustment. Disequilibrium adjustment in the real exchangerate runs in the opposite direction of price level adjustment. Monetaryshocks and demand shocks cause the real exchange rate to temporarilyrise above its equilibrium value whereas supply shocks cause the realexchange rate to temporarily fall below its equilibrium value.To get the nominal exchange rate s t = q t + p t , add the solutions forq t and p t(1 − θ)s t =˜s t +(1− η − σ)(η + σ + λ) [v t − z t + αδ t ]. (8.43)The solution displays a modiÞed form of exchange-rate overshootingunder the presumption that η + σ < 1 in that a monetary shock causesthe exchange rate to rise above its shadow value ˜s t . In contrast tothe Dornbusch model, both nominal and real shocks generate modiÞedexchange-rate overshooting. Positive demand shocks cause s t to riseabove ˜s t whereas supply shocks cause s t to fall below ˜s t .To determine excess goods demand, you know that aggregate demandisyt d = ηq t − σE t (∆q t+1 )+d t .Taking expectations of (8.42) yieldsE t (∆q t+1 )=γη + σ δ (1 + λ)(1 − θ)t −(η + σ + λ) [v t − z t + αδ t ].Substitute this and q t from (8.42) back into aggregate demand andrearrange to getyt d = y (1 + λ)(1 − θ)(η + σ)t + [v t − z t + αδ t ]. (8.44)(η + σ + λ)Goods market disequilibrium is proportional to the news v t − z t + αδ t .Monetary shocks have a short-run effect on aggregate demand, whichis the stochastic counterpart to the statement that monetary policy isan effective stabilization tool under ßexible exchange rates.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 2498.4 VAR analysis of Mundell—FlemingEven though it required tons of algebra to solve, the stochastic Mundell-Fleming with one-period nominal rigidity is still too stylized to takeseriously in formulating econometric speciÞcations. Modeling lag dynamicsin price adjustment is problematic because we don’t have a goodtheory for how prices adjust or for why they are sticky. Tests of overidentifyingrestrictions implied by dynamic versions of the Mundell—Fleming model are frequently rejected, but the investigator does notknow whether it is the Mundell-Fleming theory that is being rejected orone of the auxiliary assumptions associated with the parametric econometricrepresentation of the theory. 10Sims [129] views the restrictions imposed by explicitly formulatedmacroeconometric models to be incredible and proposed the unrestrictedVAR method to investigate macroeconomic theory without having toassume very much about the economy. In fact, just about the onlything that you need to assume are which variables to include in theanalysis. Unrestricted VAR estimation and accounting methods aredescribed in Chapter 2.1.The Eichenbaum and Evans VAREichenbaum and Evans [41] employ the Sims VAR method to the Þvedimensional vector-time-series consisting of i) US industrial production,ii) US CPI, iii) A US monetary policy variable iv) US—foreign nominalinterest rate differential, and v) US real exchange rate. They consid- ⇐(143)ered two measures of monetary policy. The Þrst was the ratio of thelogarithm of nonborrowed reserves to the logarithm of total reserves.The second was the federal funds rate. They estimated separate VARsusing exchange rates and interest rates for each of Þve countries: Japan,Germany, France, Italy, and the UK with monthly observations from1974.1 through 1990.5.Here, we will re-estimate the Eichenbaum—Evans VAR and do theassociated VAR accounting using monthly observations for the US, UK,Germany, and Japan from 1973.1 to 1998.1. All variables except inter-10 See Papell [117].

250 CHAPTER 8. THE MUNDELL-FLEMING MODELest rates are in logarithms. Let y t be US industrial production, p t be theUS consumer price index, nbr t be the log of non-borrowed bank reservesdivided by the log of total bank reserves, i t − i ∗ t be the 3 month USforeignnominal interest rate differential, q t be the real exchange rate,and s t be the nominal exchange rate. 11 For each US—foreign countrypair, two separate VARs were run–one using the real exchange rateand one with the nominal exchange rate. In the Þrst system, the VARis estimated for the 5-dimensional vector x t =(y t ,p t ,nbr t ,i t − i ∗ t ,q t ) 0 .In the second system, we used x t =(y t ,p t ,nbr t ,i t − i ∗ t ,s t ) 0 . 12The Þrst row of plots in Figure 8.8 shows the impulse response ofthe log real exchange rate for the US-UK, US-Germany, and US-Japan,following a one-standard deviation shock to nbr t . An increase in nbr tcorresponds to a positive monetary shock. The second row shows theresponses of the log nominal exchange rate with the same countries toa one-standard deviation shock to nbr t .Both the real and nominal exchange rates are found to depreciateupon impact but the maximal nominal depreciation occurs somemonths after the initial shock. The impulse response of both exchangerates is hump-shaped. There is evidently evidence of overshooting, butit is different from Dornbusch overshooting which is instantaneous. Thisunrestricted VAR response pattern has come to be known as delayedovershooting.Long-horizon (36 months ahead) forecast-error variance decompositionsof nominal exchange rates attributable to orthogonalized monetaryshocks are 16 percent for the UK, 24 percent for Germany, and10 percent for Japan. For real exchange rates, the percent of varianceattributable to monetary shocks is 23 percent for the UK and Germany,and 9 percent for Japan. Evidently, nominal shocks are prettyimportant in driving the dynamics of the real exchange rate.11 Interest rates for the US and UK are the secondary market 3-month TreasuryBill rate. For Germany, I used the interbank deposit rate. For Japan, the interestrate is the Japanese lending rate from the beginning of the sample to 1981.8, andis the private bill rate from 1981.9 to 1998.112 Using BIC (Chapter 2, equation 2.3) with the updated data indicated that theVARs required 3 lags. To conform with Eichenbaum and Evans, I included 6 lagsand a linear trend.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 2510.0160.0160.0140.0140.0140.0120.0120.010.010.0080.0080.0060.0060.0040.0040.0020.00200-0.0021 9 17 25 33 411 9 17 25 33 41 -0.0040.0120.010.0080.0060.0040.0020-0.002 1 9 17 25 33 41-0.004-0.0060.0120.010.0080.0060.0040.00202 10 18 26 34 420.0160.0140.0140.0120.0120.010.010.0080.0080.0060.0060.0040.0040.0020.00201 9 17 25 33 41-0.00201 9 17 25 33 41 -0.004Figure 8.8: Row 1: Impulse response of log real US-UK, US-German,US-Japan exchange rate to an orthogonalized one-standard deviationshock to nbr t . Row 2: Impulse responses of log nominal exchange rate.Clarida-Gali Structural VARIn Chapter 2.1, we discussed some potential pitfalls associated withthe unrestricted VAR methodology. The main problem is that theunrestricted VAR analyzes a reduced form of a structural model so wedo not necessarily learn anything about the effect of policy interventionson the economy. For example, when we examine impulse responsesfrom an innovation in y t , we do not know whether the underlying causewas due to a shock to aggregate demand or to aggregate supply or anexpansion of domestic credit.Blanchard and Quah [15] show how to use economic theory toplace identifying restrictions on the VAR, resulting in so-called struc-

252 CHAPTER 8. THE MUNDELL-FLEMING MODELtural VARs. 13 Clarida and Gali [28] employ Blanchard-Quah’ structuralVAR method using restrictions implied by the stochastic Mundell-Fleming model. To see how this works, consider the 3-dimensionalvector, x t =(∆(y t − yt ∗ ), ∆(p t − p ∗ t ), ∆q t ) 0 ,wherey is log industrialproduction, p is the log price level, and q is the log real exchange rateand starred variables are for the foreign country. Given the processesthat govern the exogenous variables (8.21) and (8.22), the stochasticMundell-Fleming model predicts that income and the real exchangerate are unit root processes, so the VAR should be speciÞed in termsof Þrst-differenced observations. The triangular structure also informsus that the variables are not cointegrated, since each of the variablesare driven by a different unit root process. 14As described in Chapter 2.1, Þrst Þt ap-th order VAR for x t andget the Wold moving average representationx t =∞Xj=0(C j L j )² t = C(L)² t , (8.45)where E(² t ² 0 t)=Σ, C 0 = I, andC(L) = P ∞j=0 C j L j is the one-sidedmatrix polynomial in the lag operator L. The theory predicts that inthe long run, x t is driven by the three dimensional vector of aggregatesupply, aggregate demand, and monetary shocks, v t =(z t , δ t ,v t ) 0 .The economic structure embodied in the stochastic Mundell-Flemingmodel is represented byx t =∞Xj=0(F j L j )v t = F(L)v t . (8.46)Because the underlying structural innovations are not observable, youare allowed to make one normalization. Take advantage of it by settingE(v t v 0 t = I). The orthogonality between the various structural shocksis an identifying assumption. To map the innovations ² t from the unrestrictedVAR into structural innovations v t , compare (8.45) and (8.46).It follows that² t = F 0 v t ⇒ ² t−j = F 0 v t−j ⇒ C j ² t−j = C j F 0 v t−j = F j v t−j .13 They are only identifying restrictions, however, and cannot be tested.14 Cointegration is discussed in Chapter2.6.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 253To summarizeF j = C j F 0 for all j ⇒ F(1) = C(1)F 0 . (8.47)Given the C j , which you get from unrestricted VAR accounting, (8.47)says you only need to determine F 0 after which the remaining F j follow.In our 3-dimensional system, F 0 is a 3 × 3 matrix with 9 uniqueelements. To identify F 0 , you need 9 pieces of information. Start with,Σ = G 0 G =E(² t ² 0 t)=F 0 E(v t v 0 t)F 0 0 = F 0 F 0 0 where G is the uniqueupper triangular Choleski decomposition of the error covariance matrixΣ. To summarizeΣ = G 0 G = F 0 F 0 0 . (8.48)Let g ij be the ijth element of G and f ij,0 be the ijth element of F 0 .Writing (8.48) out givesg11 2 = f 11,0 2 + f 12,0 2 + f 13,0 2 , (8.49)g 11 g 12 = f 11,0 f 21,0 + f 12,0 f 22,0 + f 13,0 f 23,0 , (8.50)g 11 g 13 = f 11,0 f 31,0 + f 12,0 f 32,0 + f 13,0 f 33,0 , (8.51)g12 2 +g2 22 = f 21,0 2 + f 22,0 2 + f 23,0 2 , (8.52)g 12 g 13 + g 22 g 23 = f 21,0 f 31,0 + f 22,0 f 32,0 + f 23,0 f 33,0 , (8.53)g13 2 +g23 2 + g33 2 = f31,0 2 + f32,0 2 + f33,0. 2 (8.54)G has 6 unique elements so this decomposition gives you 6 equationsin 9 unknowns. You still need three additional pieces of information.Get them from the long-run predictions of the theory.Stochastic Mundell-Fleming predicts that neither demand shocksnor monetary shocks have a long-run effect on output which we representby setting f 12 (1) = 0 and f 13 (1) = 0, where f ij (1) is the ijthelement of F(1) = P ∞j=0 F j . The model also predicts that moneyhas no long-run effect on the real exchange rate f 33 (1) = 0. SinceF(1) = C(1)F 0 , impose these three restrictions by settingf 13 (1) = 0 = c 11 (1)f 13,0 + c 12 (1)f 23,0 + c 13 (1)f 33,0 , (8.55)f 12 (1) = 0 = c 11 (1)f 12,0 + c 12 (1)f 22,0 + c 13 (1)f 32,0 , (8.56)f 33 (1) = 1 = c 31 (1)f 13,0 + c 32 (1)f 23,0 + c 33 (1)f 33,0 . (8.57)

254 CHAPTER 8. THE MUNDELL-FLEMING MODEL(8.49)—(8.57) form a system of 9 equations in 9 unknowns and implicitlydeÞne F 0 . Once the F j are obtained, you can do impulse response analysesand forecast error variance decompositions using the ‘structural’response matrices F j .Table 8.1: Structural VAR forecast error variance decompositions forreal exchange rate depreciation1month36 monthsSupply Demand Money Supply Demand MoneyBritain 0.378 0.240 0.382 0.331 0.211 0.458Germany 0.016 0.234 0.750 0.066 0.099 0.835Japan 0.872 0.011 0.117 0.810 0.071 0.119Clarida and Gali estimate a structural VAR using quarterly datafrom 1973.3 to 1992.4 for the US, Germany, Japan, and Canada Theirimpulse response analysis revealed that following a one-standard deviationnominal shock, the real exchange rate displayed a hump shape,initially depreciating then subsequently appreciating. Real exchangerate dynamics were found to display delayed overshooting.We’ll re-estimate the structural VAR using 4 lags and monthly datafor the US, UK, Germany, and Japan from 1976.1 through 1997.4. Thestructural impulse response dynamics of the levels of the variables aredisplayed in Figure 8.9. As predicted by the theory, supply shockslead to a permanent real deprecation and demand shocks lead to apermanent real appreciation. The US-UK real exchange rate does notexhibit delayed overshooting in response to monetary shocks. The realdollar-pound rate initially appreciates then subsequently depreciatesfollowing a positive monetary shock. The real dollar-deutschemark ratedisplays overshooting by Þrst depreciating and then subsequently appreciating.The real dollar-yen displays Dornbusch-style overshooting.Money shocks are found to contribute a large fraction of the forecasterror variance both the long run as well as at the short run for thereal exchange rate. The decompositions at the 1-month and 36-monthforecast horizons are reported in Table 8.1

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 2550.40.350.3Supply, US-UK0-0.05Demand, US-UK1 5 9 13 17 21 250.20.150.1Money, US-UK0.250.20.150.10.0501 5 9 13 17 21 25-0.1-0.15-0.2-0.250.050-0.05-0.1-0.15-0.21 5 9 13 17 21 250.20.150.10.050Supply, US-Germany1 5 9 13 17 21 250-0.05-0.1-0.15-0.2-0.25-0.3Demand, US-Germany1 5 9 13 17 21 250.30.20.10-0.1-0.2Money, US-Germany1 5 9 13 17 21 25-0.05-0.35-0.31.81.61.41.210.80.60.40.20Supply, US-Japan1 5 9 13 17 21 250-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9Demand, US-Japan1 5 9 13 17 21 250.40.350.30.250.20.150.10.050Money, US-Japan1 5 9 13 17 21 25Figure 8.9: Structural impulse response of log real exchange rate to supply,demand, and money shocks. Row 1: US-UK, row 2: US-Germany,row 3: US-Japan.

256 CHAPTER 8. THE MUNDELL-FLEMING MODELMundell-Fleming Models Summary1. The hallmark of Mundell-Fleming models is that they assumethat goods prices are sticky. Many people think of Mundell—Fleming models synonymously with sticky-price models. Becausethere exist nominal rigidities, these models invite an assessmentof monetary (and Þscal) policy interventions underboth Þxed and ßexible exchange rates. The models also providepredictions regarding the international transmission of domesticshocks and co-movements of macroeconomic variables at homeand abroad.2. The Dornbusch version of the model exploits the slow adjustmentin the goods market combined with the instantaneous adjustmentin the asset markets to explain why the exchange rate,which is the relative price of two monies (assets), may exhibitmore volatility than the fundamentals in a deterministic andperfect foresight environment. Explaining the excess volatilityof the exchange rate is a recurring theme in internationalmacroeconomics.3. The dynamic stochastic version of the model is amenable toempirical analysis. The model provides a useful guide for doingunrestricted and structural VAR analysis.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 257Appendix: Solving the Dornbusch ModelFrom (8.9) and (8.11), we see that the behavior of i(t) is completely determinedby that of s(t). This means that we need only determine the differentialequations governing the exchange rate and the price level to obtain acomplete characterization of the system’s dynamics.Substitute (8.9) and (8.11) into (8.6). Make use of (8.13) and rearrangeto obtainús(t) = 1 [p(t) − ¯p]. (8.58)λTo obtain the differential equation for the price level, begin by substituting(8.58) into (8.9), and then substituting the result into (8.8) to get ⇐(144)úp(t) =π[δ(s(t) − p(t)) + (γ − 1)y − σi ∗ − σ (p(t) − ¯p)+g]. (8.59)λHowever, in the long run0=π[δ(¯s − ¯p)+(γ − 1)y − σr ∗ + g], (8.60)the price dynamics are more conveniently characterized byúp(t) =π·δ(s(t) − ¯s) − (δ + σ )(p(t) − ¯p)¸, (8.61)λ⇐(145)which is obtained by subtracting (8.60) from (8.59).Now write (8.58) and (8.61) as the systemà ! à !ús(t) s(t) − ¯s= A, (8.62)úp(t) p(t) − ¯pwhereà !0 1/λA =.πδ −π(δ + σ/λ)(8.62) is a system of two linear homogeneous differential equations. We knowthat the solutions to these systems take the forms(t) = ¯s + αe θt , (8.63)p(t) = ¯p + βe θt . (8.64)We will next substitute (8.63) and (8.64) into (8.62) and solve for theunknown coefficients, α, β, andθ. First, taking time derivatives of (8.63)and (8.64) yieldsús = θαe θt , (8.65)úp = θβe θt . (8.66)

258 CHAPTER 8. THE MUNDELL-FLEMING MODELSubstitution of (8.65) and (8.66) into (8.62) yieldsà !α(A − θI 2 ) =0. (8.67)β(146)⇒In order for (8.67) to have a solution other than the trivial one (α, β) =(0, 0),requires that0 = |A − θI 2 | (8.68)= θ 2 − Tr(A)θ + |A|, (8.69)where Tr(A) =−π(δ + σ/λ) and|A| = −πδ/λ otherwise, (A − θI 2 ) −1 existswhich means that the unique solution is the trivial one, which isn’t veryinteresting. Imposing the restriction that (8.69) is true, we Þnd that itsroots areThe general solution isθ 1 = 1 2 [Tr(A) − qTr 2 (A) − 4|A|] < 0, (8.70)θ 2 = 1 2 [Tr(A)+ qTr 2 (A) − 4|A|] > 0. (8.71)s(t) = ¯s + α 1 e θ1t + α 2 e θ2t , (8.72)p(t) = ¯p + β 1 e θ1t + β 2 e θ2t . (8.73)This solution is explosive, however, because of the eventual dominance ofthe positive root. We can view an explosive solution as a bubble, in whichtheexchangerateandthepriceleveldivergesfromvaluesoftheeconomicfundamentals. While there are no restrictions within the model to rule outexplosive solutions, we will simply assume that the economy follows thestable solution by setting α 2 = β 2 = 0, and study the solution with thestable rootθ ≡ −θ 1 (8.74)= 1 2 [π(δ + σ/λ)+ qπ 2 (δ + σ/λ) 2 +4πδ/λ]. (8.75)Now, to Þndthestablesolution,wesolve(8.67)withthestablerootà !α0 = (A − θ 1 I 2 )βà !à !−θ1 1/λα=.πδ −θ 1 − π(δ + σ/λ) β(8.76)

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 259When this is multiplied out, you getIt follows that0 = −θ 1 α + β/λ, (8.77)µ0 = πδα − [θ 1 + π δ + σ ]β. (8.78)λα = β/θ 1 λ. (8.79)Because α is proportional to β, weneedtoimposeanormalization. Letthisnormalization be β = p o − ¯p where p o ≡ p(0). Then α =(p o − ¯p)/θ 1 λ =−[p o − ¯p]/θλ, whereθ ≡−θ 1 . Using these values of α and β in (8.63) and(8.64), yieldsp(t) = ¯p +[p o − ¯p]e −θt , (8.80)s(t) = ¯s +[s o − ¯s]e −θt , (8.81)where (s o − ¯s) =−[p o − ¯p]/θλ. This solution gives the time paths for theprice level and the exchange rate.To characterize the system and its response to monetary shocks, we willwant to phase diagram the system. Going back to (8.58) and (8.61), wesee that ús(t) = 0 if and only if p(t) = ¯p, while úp(t) = 0 if and only ifs(t) − ¯s =(1 + σ/λδ)(p(t) − ¯p). These points are plotted in Figure 8.10. Thesystem displays a saddle path solution.

260 CHAPTER 8. THE MUNDELL-FLEMING MODELs.p=0.s=0pFigure 8.10: Phase diagram for the Dornbusch model.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 261Problems1. (Static Mundell-Fleming with imperfect capital mobility). Let thetrade balance be given by α(s + p ∗ − p) − ψy. A real depreciationraises exports and raises the trade balance whereas an increase inincome leads to higher imports which lowers the trade balance. Letthe capital account be given by θ(i−i ∗ ), where 0 < θ < ∞ indexes thedegree of capital mobility. We replace (8.3) with the external balanceconditionα(s + p ∗ − p) − ψy + θ(i − i ∗ )=0,that the balance of payments is 0. (We are ignoring the service account.)When capital is completely immobile, θ = 0 and the balance ofpayments reduces to the trade balance. Under perfect capital mobility,θ = ∞ implies i = i ∗ which is (8.3).(a) Call the external balance condition the FF curve. Draw the FFcurve in r, y space along with the LM and IS curves.(b) Repeat the comparative statics experiments covered in this chapterusing the modiÞed external balance condition. Are any of theresults sensitive to the degree of capital mobility? In particular,how do the results depend on the slope of the FF curve in relationto the LM curve?2. How would the Mundell-Fleming model with perfect capital mobilityexplain the international co-movements of macroeconomic variables inChapter 5?3. Consider the Dornbusch model.(a) What is the instantaneous effect on the exchange rate of a shockto aggregate demand? Why does an aggregate demand shocknot produce overshooting?(b) Suppose output can change in the short run by replacing the IScurve (8.7) with y = δ(s − p) +γy − σi + g, replace the priceadjustment rule (8.8) with úp = π(y−ȳ), where long-run output isgiven by ȳ = δ(¯s − ¯p)+γȳ − σi ∗ + g. Under what circumstancesis the overshooting result (in response to a change in money)robust?

262 CHAPTER 8. THE MUNDELL-FLEMING MODEL

Chapter 9The New InternationalMacroeconomicsThe new international macroeconomics are a class of theories that embedimperfect competition and nominal rigidities in a dynamic generalequilibrium open economy setting. In these models, producers havemonopoly power and charge price above marginal cost. Since it is optimalin the short run for producers to respond to small ßuctuationsby changing output, these models explain why output is demand determinedin the short run when current prices are predetermined dueto some nominal rigidity. It follows from the imperfectly competitiveenvironment that equilibrium output lies below the socially optimallevel. We will see that this feature is instrumental in producing resultsthat are very different from Mundell—Fleming models. BecauseMundell—Fleming predictions can be overturned, it is perhaps inaccurateto characterize these models as providing the micro-foundationsfor Mundell-Fleming.These models also, and not surprisingly, are sharply distinguishedfrom the Arrow-Debreu style real business cycle models. Both classes oftheories are set in dynamic general equilibrium with optimizing agentsand well-speciÞed tastes and technology. Instead of being set in a perfectreal business cycle world, the presence of market imperfectionsand nominal rigidities permit international transfers of wealth in equilibriumand prevent equilibrium welfare from reaching the socially optimallevel of welfare. It therefore makes sense here to examine the263

264CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSwelfare effects of policy interventions whereas it does not make sensein real business cycle models since all real business cycle dynamics arePareto efficient.The genesis of this literature is the Obstfeld and Rogoff [113] Reduxmodel. This model makes several surprising predictions that arecontrary to Mundell—Fleming. The model is somewhat fragile, however,as we will see when we cover the pricing-to-market reÞnement by Bettsand Devereux [10].In this chapter, stars denote foreign country variables but lower caseletters do not automatically mean logarithms. Unless explicitly noted,variables are in levels. There is also a good deal of notation. For ease ofreference, Table 9.1 summarizes the notation for the Redux model andTable 9.2 lists the notation for the pricing-to-market model. The termshousehold, agent, consumer and individual are used interchangeably.The home currency unit is the ‘dollar’ and the foreign currency is the‘euro.’9.1 The Redux ModelWe are set in a deterministic environment and agents have perfectforesight. There are 2 countries, each populated by a continuum ofconsumer—producers. There is no physical capital. Each householdproduces a distinct and differentiated good using only its labor and theproduction of each household is completely specialized. Households arearranged on the unit interval, [0, 1] with a fraction n living in the homecountry and a fraction 1−n living in the foreign country. We will indexdomestic agents by z where 0

9.1. THE REDUX MODEL 265Home CountryForeign Countryz z*0 n1Figure 9.1: Home and foreign households lined up on the unit interval.provides individuals with indirect utility because higher levels of realcash balances help to lower shopping (transactions) costs.We assume that households have identical utility functions and wewill work with a representative household.Representative agent (household) in Redux model. Letc t (z) bethehome representative agent’s consumption of the domestic good z, andc t (z ∗ ) be the agent’s consumption of the foreign good z ∗ .Peoplehavetastes for all varieties of goods and the household’s consumption basketis a constant elasticity of substitution (CES) index that aggregatesacross the available varieties of goodsC t ==·Z 10·Z n0c t (u) θ−1θ du¸ θθ−1c t (z) θ−1θ dz +Z 1n¸ θc t (z ∗ ) θ−1 θ−1θ dz∗ , (9.1)where θ > 1 is the elasticity of substitution between the varieties. 1Let y t (z) bethetime-t output of individual z, M t be the domesticper capita money stock and P t be the domestic price level. Lifetimeutility of the representative domestic household is given byU t =∞Xj=0⎡β j ⎣ln C t+j +γà ! ⎤1−²Mt+j− ρ 1 − ² P t+j 2 y2 t+j (z) ⎦ , (9.2)⇐(147)1 In the discrete commodity formulation with N goods, the index can be written¸ θ·PNas C =z=1 c θ−1 θ−1θz ∆z where ∆z = 1. The representation under a continuumof goods takes the limit of the sums given by the integral formulation in (9.1).

266CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSwhere 0 < β < 1 is the subjective discount factor, C t+j is the CESindex given in (9.1) and M t /P t are real balances. The costs of forgoneleisure associated with work are represented by the term (−ρ/2)y 2 t (z).Let p t (z) be the domestic price of good z, S t be the nominal exchangerate, and p ∗ t (z) be the foreign currency price of good z. Akeyassumption is that prices are set in the producer’s currency. It followsthat the law of one price holds for every good 0

9.1. THE REDUX MODEL 267c t (z ∗ ) ="St p ∗ t (z∗ )P t# −θC t . (9.6)Analogously, foreign household lifetime utility is⎡Ã∞XM∗Ut ∗ = β j ⎣ln Ct+j ∗ t+j+j=0γ1 − ²with consumption and price indicesP ∗t+j! 1−²− ρ 2 y∗2t+j(z ∗ )⎤⎦ , (9.7)⇐(150)⇐(151)C ∗ t =P ∗t =·Z n0⎡Z n⎣0c ∗ θ−1t (z)θ dz +Z 1nà ! 1−θ Z pt (z)1dz +S t n¸ θc ∗ t (z∗ ) θ−1 θ−1θ dz∗ , (9.8)⎤[p ∗ t (z∗ )] 1−θ dz ∗ ⎦11−θ, (9.9)and individual demand for z and z ∗ goods" # −θc ∗ t (z) = pt (z)S t Pt∗ Ct ∗ ,"pc ∗ ∗t (z∗ ) = t (z ∗ # −θ)Ct ∗ .Every good is equally important in home and foreign householdsutility. It follows that the elasticity of demand 1/θ, in all goods marketswhether at home or abroad, is identical. Every producer has theidentical technology in production. In equilibrium, all domestic producersbehave identically to each other and all foreign producers behaveidentically to each other in the sense that they produce the same levelof output and charge the same price. Thus it will be the case that forany two domestic producers 0

268CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSIt follows that the home and foreign price levels, (9.4) and (9.9) simplifytoP t = [np t (z) 1−θ +(1− n)(S t p ∗ t (z∗ )) 1−θ ] 11−θ , (9.10)Pt ∗ = [n(p t (z)/S t ) 1−θ +(1− n)p ∗ t (z∗ ) 1−θ ] 11−θ , (9.11)and that PPP holds for the correct CES price indexP t = S t P ∗t . (9.12)Notice that PPP will hold for GDP deßators only if n =1/2.Asset Markets. The world capital market is fully integrated. There isan internationally traded one-period real discount bond which is denominatedintermsofthecompositeconsumption good C t . r t is thereal interest rate paid by the bond between t and t + 1. The bond isavailable in zero net supply so that bonds held by foreigners are issuedby home residents. The gross nominal interest rate is given by theFisher equation1+i t = P t+1(1 + r t ), (9.13)P tand is related to the foreign nominal interest rate by uncovered interestparity1+i t = S t+1(1 + i ∗ t ). (9.14)S tLet B t be the stock of bonds held by the domestic agent and Bt∗ bethe stock of bonds held by the foreign agent. By the zero-net supplyconstraint 0 = nB t +(1− n)Bt ∗ , it follows thatBt ∗ = −n1 − n B t. (9.15)(153)⇒The Government. For 0

9.1. THE REDUX MODEL 269Table 9.1: Notation for the Redux modeln Fraction of world population in home countryu Index across all individuals of the world 0

270CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSG t =G ∗ t =·Z 10·Z 10g t (u) θ−1θ du¸ θθ−1,gt ∗ (u) θ−1θ du¸ θθ−1.It follows that home government demand for individual goods are givenby replacing c t with g t and C t with G t in (9.5)—(9.6). The identicalreasoning holds for the foreign government demand function.Governments issue no debt. They Þnance consumption either throughmoney creation (seignorage) or by lump—sum taxes T t ,andTt ∗ .Negativevalues of T t and Tt∗ are lump—sum transfers from the government toresidents. The budget constraints of the home and foreign governmentsareG t = T t + M t − M t−1, (9.16)P tG ∗ t = T t ∗ + M t ∗ − Mt−1∗ . (9.17)Aggregate Demand. Let average world private and government consumptionbe the population weighted average of the domestic and foreigncounterpartsPt∗Ct w = nC t +(1− n)Ct ∗ , (9.18)G w t = nG t +(1− n)G ∗ t . (9.19)Then C w t + G w t is world aggregate demand. The total demand for anyhome or foreign good is given by" # −θyt d (z) = pt (z)(Ct w + G w t ), (9.20)P ty ∗dt (z ∗ )="p∗t (z ∗ # −θ)(Ct w + G w t ). (9.21)P ∗tBudget Constraints. Wealth that domestic agents take into the nextperiod (P t B t + M t ), is derived from wealth brought into the currentperiod ([1 + r t−1 ]P t B t−1 + M t−1 ) plus current income (p t (z)y t (z)) less

9.1. THE REDUX MODEL 271consumption and taxes (P t (C t +T t )). Wealth is accumulated in a similarfashion by the foreign agent. The budget constraint for home andforeign agents areP t B t +M t =(1+r t−1 )P t B t−1 +M t−1 +p t (z)y t (z)−P t C t −P t T t , (9.22)Pt ∗ B∗ t + M t∗ =(1+r t−1 )Pt ∗ B∗ t−1 + M t−1 ∗ + p∗ t (z∗ )yt ∗ (z∗ ) − Pt ∗ C∗ t − P t ∗ T t ∗ .(9.23)We can simplify the budget constraints by eliminating p(z) andp ∗ (z ∗ ).Because output is demand determined, re-arrange (9.20) to getp t (z)y t (z) =P t y t (z) θ−1θ [Ct w +G w t ] 1 θ , and substitute the result into (9.22).Do the same for the foreign household’s budget constraint using the zeronet supply constraint on bonds (9.15) to eliminate B ∗ to getC t = (1+r t−1 )B t−1 − B t − M t − M t−1− T tP t+y t (z) θ−1θ [Ct w + G w t ] 1 θ , (9.24)Ct ∗ = (1+r t−1 ) −nB t−11 − n + nB t1 − n − M t ∗ − Mt−1∗Pt∗ − Tt∗+yt ∗ (z∗ ) θ−1θ [Ct w + G w t ] 1 θ . (9.25)⇐(157)Euler Equations. C t ,M t , and B t are the choice variables for the domesticagent and Ct ∗ ,Mt ∗ , and Bt∗ are the choice variables for the foreignagent. For the domestic household, substitute the budget constraint(9.22) into the lifetime utility function (9.2) to transform the probleminto an unconstrained dynamic optimization problem. Do the samefor the foreign household. The Euler-equations associated with bondholding choice are the familiar intertemporal optimality conditionsC t+1 = β(1 + r t )C t , (9.26)Ct+1 ∗ = β(1 + r t )Ct ∗ . (9.27)The Euler-equations associated with optimal cash holdings are themoney demand functionsM tP t=" # 1γ(1 + it²)C t , (9.28)i t

272CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSM ∗ tP ∗t="γ(1 + i∗t )i ∗ tC ∗ t# 1², (9.29)where (1/²) is the consumption elasticity of money demand. 3Euler-equations for optimal “labor supply” are 4[y t (z)] θ+1θ=[y t (z ∗ ) ∗ ] θ+1θ="θ − 1ρθ"θ − 1ρθ##TheC −1t [C w t + G w t ] 1 θ , (9.30)C ∗−1t [C w t + G w t ] 1 θ . (9.31)It will be useful to consolidated the budget constraints of the individualand the government by combining (9.22) and (9.16) for the homecountry and (9.17) and (9.24) for the foreign countryC t =(1+r t−1 )B t−1 − B t + p t(z)y t (z)P t− G t , (9.32)Ct ∗ = −(1 + r nt−1)1 − n B t−1 + n1 − n B t + p∗ t (z ∗ )yt ∗ (z ∗ )Pt∗ − G ∗ t . (9.33)Because of the monopoly distortion, equilibrium output lies belowthe socially optimal level. Therefore, we cannot use the planner’s problemand must solve for the market equilibrium. The solution methodis to linearize the Euler equations around the steady state. To do so,we must Þrst study the steady state.The Steady StateConsider the state to which the economy converges following a shock.Let these steady state values be denoted without a time subscript. We3 The home-agent Þrst order condition is γ³M tP t´−² 1P t− 1P t C t+βP t+1 C t+1=0.Now using (9.26) to eliminate β and the Fisher equation (9.13) to eliminate (1 + r t )produces (9.28).4 “Supply” is placed in quotes since the monopolistically competitive Þrm doesn’thave a supply curve.

9.1. THE REDUX MODEL 273restrict the analysis to zero inßation steady states. Then the governmentbudget constraints (9.16) and (9.17) are G = T and G ∗ = T ∗ .By(9.26), the steady state real interest rate isr =(1 − β). (9.34)βFrom (9.32) and (9.33), and the steady state consolidated budget constraintsareC = rB + p(z)y(z) − G, (9.35)PC ∗ = −r nB1 − n + p∗ (z ∗ )y ∗ (z ∗ )− G ∗ . (9.36)P ∗The ‘0-steady state’. We have just described the forward-looking steadystate to which the economy eventually converges. We now specify thesteady-state from which we depart. This benchmark steady state hasno international debt and no government spending. We call it the ‘0-steady state’ and indicate it with a ‘0’ subscript, B 0 = G 0 = G ∗ 0 =0.From the domestic agent’s budget constraint (9.35), we have C 0 =(p 0 (z)/P 0 )y 0 (z). Since there is no international indebtedness, internationaltrade must be balanced, which means that consumption equalsincome C 0 = y 0 (z). It also follows from (9.35) that p 0 (z) =P 0 .Analogously,C0 ∗ = y0(z ∗ ∗ )andp ∗ 0(z ∗ )=P0∗ in the foreign country. By PPP,P 0 = S 0 P0 ∗ , and from the foregoing p 0 (z) =S 0 p ∗ 0(z ∗ ). That is, the dollarprice of good z is equal to the dollar price of the foreign good z ∗ inthe 0-equilibrium.It follows that in the 0-steady-state, world demand isC w 0 = nC 0 +(1− n)C ∗ 0 = ny 0 (z)+(1− n)y ∗ 0(z ∗ ).Substitute this expression into the labor supply decisions (9.30) and(9.31) to gety 0 (z) 2θ+1θ =y ∗ 0(z ∗ ) 2θ+1θ =Ãθ − 1ρθÃθ − 1ρθ!![ny 0 (z)+(1− n)y ∗ 0 (z∗ )] 1 θ[ny 0 (z)+(1− n)y ∗ 0(z ∗ )] 1 θ .

274CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSTogether, these relations tell us that 0-steady-state output at home andabroad are equal to consumption" # 1/2θ − 1y 0 (z) =y0(z ∗ ∗ )== C 0 = C0 ∗ = C0 w . (9.37)ρθNominal and real interest rates in the 0-steady state are equalizedwith (1+i 0 )/i 0 =1/(1−β). By (9.28) and (9.29), 0-steady state moneydemand isM 0P 0= M ∗ 0P ∗ 0=" # 1/²γy0 (z). (9.38)1 − βFinally by (9.38) and PPP, it follows that the 0-steady-state nominalexchange rate isS 0 = M 0M0∗ . (9.39)(9.39) looks pretty much like the Lucas-model solution (4.55).Log-Linear Approximation About the 0-Steady StateWe denote the approximate log deviation from the 0-steady state witha ‘hat’ so that for any variable ˆX t =(X t − X 0 )/X 0 ' ln(X t /X 0 ). Theconsolidated budget constraints (9.32) and (9.33) with B t−1 = B 0 =0becomeC t = p t(z)y t (z) − B t − G t , (9.40)P tCt ∗ = p∗ t (z ∗ )y ∗Pt∗ t (z∗ )+µ nBt1 − n− G ∗ t . (9.41)Multiply (9.40) by n and (9.41) by 1 − n and add together to get theconsolidated world budget constraintC w t= nà !pt (z)y t (z)+(1− n)P tÃp∗t (z ∗ )Log-linearizing (9.42) about the 0-steady state yieldsP ∗t!y ∗ t (z∗ ) − G w t . (9.42)Ĉ w t= n[ˆp t (z)+ŷ t (z) − ˆP t ]+(1− n)[ˆp ∗ t (z∗ )+ŷ ∗ t (z∗ ) − ˆP ∗t ] − ĝw t , (9.43)

9.1. THE REDUX MODEL 275⇒where ĝt w ≡ G w t /C0 w . 5 Do the same for PPP (9.12) and the domesticand foreign price levels (9.10)-(9.11) to getŜ t = ˆP t − ˆP t ∗ , (9.44)ˆP t = nˆp t (z)+(1− n)(Ŝt +ˆp ∗ t (z∗ )), (9.45)ˆP t ∗ = n(ˆp t (z) − Ŝt)+(1− n)ˆp ∗ t (z∗ ). (9.46)Log-linearizing the world demand functions (9.20) and (9.21) givesŷ t (z) = θ[ ˆP t − ˆp t (z)] + Ĉw t +ĝ w t , (9.47)ŷ ∗ t (z ∗ ) = θ[ ˆP ∗t − ˆp ∗ t (z ∗ )] + Ĉw t +ĝ w t . (9.48)Log-linearizing the ‘labor supply rules’ (9.30) and (9.31) gives(1 + θ)ŷ t (z) = −θĈt + Ĉw t +ĝt w , (9.49)(1 + θ)ŷt ∗ (z∗ ) = −θĈ∗ t + Ĉw t +ĝt w . (9.50)Log-linearizing the consumption Euler equations (9.26)—(9.27) givesĈ t+1 = Ĉt +(1− β)ˆr t , (9.51)Ĉ ∗ t+1 = Ĉ∗ t +(1− β)ˆr t , (9.52)and Þnally, log-linearizing the money demand functions (9.28) and(9.29) gives"ˆM t − ˆP t = 1 Ĉ t − β ˆr t + ˆP t+1 − ˆP t²1 − βˆM t ∗ − ˆP t ∗ = 1 Ã"Ĉ t ∗ ²− β ˆr t + ˆP t+1 ∗ − ˆP t∗1 − βÃ!#!#, (9.53). (9.54)5 The expansion of the Þrst term about 0-steady state values is,∆n(p t (z)/P t )y t (z) = n(y 0 (z)/P 0 )(p t (z) − p 0 (z)) + n(p 0 (z)/P 0 )(y t (z) − y 0 (z)) −n[(p 0 (z)y 0 (z))/P0 2 ](P t − P 0 ). When you divide by C0 w ,notethatC0w = y 0 (z) andP 0 = p 0 (z) togetn[ˆp t (z) − ˆP t +ŷ t (z)]. Expansion of the other terms follows in ananalogous manner.

276CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSLong-Run ResponseThe economy starts out in the 0-steady state. We will solve for the newsteady-state following a permanent monetary or government spendingshock. For any variable X, let ˆX ≡ ln(X/X 0 ), where X is the new(forward-looking) steady state value. Since log-linearized equations(9.43)—(9.50) hold for arbitrary t, they also hold across steady statesand from (9.43), (9.47), (9.48), (9.49) and (9.50) you getĈ w = n[ˆp(z)+ŷ(z) − ˆP ]+(1− n)[ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) − ˆP ∗ ] − ĝ w ,(9.55)ŷ(z) =θ[ ˆP − ˆp(z)] + Ĉw +ĝ w , (9.56)ŷ ∗ (z ∗ )=θ[ ˆP ∗ − ˆp ∗ (z ∗ )] + Ĉw +ĝ w , (9.57)(1 + θ)ŷ(z) =−θĈ + Ĉw +ĝ w , (9.58)(1 + θ)ŷ ∗ (z ∗ )=−θĈ∗ + Ĉw +ĝ w , (9.59)where ĝ = G/C0w and ĝ ∗ = G ∗ /C0 w . Log-linearizing the steady statebudget constraints (9.35) and (9.36) and letting ˆb = B/C0w yieldsĈ = rˆb +ˆp(z)+ŷ(z) − ˆP − ĝ, (9.60)µ nĈ ∗ = − rˆb +ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) −1 − nˆP ∗ − ĝ ∗ . (9.61)Together, (9.55)—(9.61) comprise 7 equations in 7 unknowns(ŷ, ŷ ∗ , (ˆp(z) − ˆP ), (ˆp ∗ (z ∗ ) − ˆP ∗ ), Ĉ,Ĉ∗ , Ĉw ). There is no easy way tosolve this system. You must bite the bullet and do the tedious algebrato solve this system of equations. 6 The solution for the steady statechanges isĈ = 1 2θ [(1 + θ)rˆb +(1− n)ĝ ∗ − (1 − n + θ)ĝ], (9.62)Ĉ ∗ = 1 "#n(1 + θ)r−2θ (1 − n) ˆb + nĝ − (n + θ)ĝ ∗ , (9.63)Ĉ w = −ĝw 2 , (9.64)ŷ(z) = 1"ĝw1+θ 2 − θĈ#, (9.65)6 Or you can use a symbolic mathematics software such as Mathematica or Maple.I confess that I used Maple.

9.1. THE REDUX MODEL 277ŷ ∗ (z ∗ )= 1"ĝw #1+θ 2 − θĈ∗ , (9.66)h(1 − n)(ĝ ∗ − ĝ)+rˆb i , (9.67)ˆp(z) − ˆP = 1 2θˆp ∗ (z ∗ ) − ˆP ∗ =n h(1 − n)(ĝ − ĝ ∗ ) − rˆb i . (9.68)(1 − n)2θFrom (9.62) and (9.63) you can see that a steady state transfer of wealthin the amount of B from the foreign country to the home country,raises home steady state consumption and lowers it abroad. The wealthtransfer reduces steady state home work effort (9.65) and raises foreignsteady state work effort (9.66). From (9.67), we see that this occursalong with ˆp(z) − ˆP >0 so that the relative price is high in the highwealth country. The underlying cause of the wealth redistribution hasnot yet been speciÞed. It could have been induced either by governmentspending shocks or monetary shocks.If the shock originates with an increase in home government consumption,∆G is spent on home and foreign goods which has a directeffect on home and foreign output. At home, however, higher governmentconsumption raises the domestic tax burden and this works toreduce domestic steady state consumption.The relative price of exports in terms of imports is called the termsof trade. To get the steady state change in the terms of trade, subtract(9.68) from (9.67), add S t to both sides and note that PPP impliesˆP − (Ŝ + ˆP ∗ )=0toget⇐(159)ˆp(z) − (Ŝ +ˆp∗ (z ∗ )) = 1 θ (ŷ∗ − ŷ) = 11+θ (Ĉ − Ĉ∗ ). (9.69)From (9.53) and (9.54), it follows that the steady state changes in ⇐(160)the price levels areˆP = ˆM − 1 Ĉ,²(9.70)ˆP ∗ = ˆM ∗ − 1 ² Ĉ∗ . (9.71)By PPP, (9.70), and (9.71) the long-run response of the exchange rateisŜ = ˆM − ˆM ∗ − 1 ² (Ĉ − Ĉ∗ ). (9.72)

278CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSShort-Run Adjustment under Sticky PricesWe assume that there is a one-period nominal rigidity in which nominalprices p t (z) andp ∗ t (z ∗ ) are set one period in advance in the producer’scurrency. 7 This assumption is ad hoc and not the result of a clearlyarticulated optimization problem. The prices cannot be changed withinthe period but are fully adjustable after 1 period. It follows that thedynamics of the model are fully described in 3 periods. At t − 1, theeconomy is in the 0-steady state. The economy is shocked at t, andthevariable X responds in the short run by ˆX t .Att + 1, we are in the newsteady state and the long-run adjustment is ˆX t+1 = ˆX ' ln(X/X 0 ).Date t + 1 variables in the linearized model are the new steady statevalues and date t hat values are the short-run deviations.From (9.45) and (9.46), the price-level adjustments areˆP t = (1− n)Ŝt, (9.73)ˆP t ∗ = −nŜt. (9.74)(161-163)⇒In the short run, output is demand determined by (9.47) and (9.48).Substituting (9.73) into (9.47) and (9.74) into (9.48) and noting thatindividual goods prices are sticky ˆp t (z) =ˆp ∗ t (z ∗ )=0,youhaveŷ t (z) = θ(1 − n)Ŝt + Ĉw t +ĝ w , (9.75)ŷ ∗ t (z ∗ ) = −θ(n)Ŝt + Ĉw t +ĝ w . (9.76)The remaining equations that characterize the short run are (9.51)-(9.54), which are rewritten asĈ = Ĉt +(1− β)ˆr t , (9.77)Ĉ ∗ = Ĉ∗ t +(1− β)ˆr t, (9.78)ˆM t − ˆP t = 1 ²"ˆM ∗ t − ˆP ∗t = 1 ²Ĉ t − β"ÃĈ ∗ t − βˆr t + ˆP − ˆP t1 − βÃ!#ˆr t + ˆP ∗ − ˆP t∗1 − β, (9.79)!#. (9.80)(164)⇒Using the consolidated budget constraints, (9.40)—(9.41) and the pricelevel response (9.73) and (9.74), the current account responds by

9.1. THE REDUX MODEL 27966)⇒ ˆbt = ŷ t (z) − (1 − n)Ŝt − Ĉt − ĝ t , (9.81)ˆb∗ t = ŷ ∗ t (z∗ )+nŜt − Ĉ∗ t − ĝ∗ t = −n1 − nˆb t . (9.82)We have not speciÞed the source of the underlying shocks, which mayoriginate from either monetary or government spending shocks. Sincethe role of nominal rigidities is most clearly illustrated with monetaryshocks, we will specialize the model to analyze an unanticipatedand permanent monetary shock. The analysis of governments spendingshocks is treated in the end-of-chapter problems.Monetary ShocksSet G t = 0 for all t in the preceding equations and subtract (9.78) from(9.77), (9.80) from (9.79), and use PPP to obtain the pair of equationsĈ − Ĉ∗ = Ĉt − Ĉ∗ t , (9.83)ˆM t − ˆM t ∗ − Ŝt = 1 β² (Ĉt − Ĉ∗ t ) −²(1 − β) (Ŝ − Ŝt). (9.84)Substitute Ŝ from (9.72) into (9.84) to getŜ t =(ˆM t − ˆM t ∗ ) − 1 ² (Ĉt − Ĉ∗ t ). (9.85)This looks like the solution that we got for the monetary approachexcept that consumption replaces output as the scale variable. Comparing(9.85) to (9.72) and using (9.83), you can see that the exchangerate jumps immediately to its long-run valueŜ = Ŝt. (9.86)Even though goods prices are sticky, there is no exchange rate overshootingin the Redux model.(9.85) isn’t a solution because it depends on Ĉt − Ĉ∗ t which is endogenous.To get the solution, Þrst note from (9.83) that you only need7 z-goods prices are set in dollars and z ∗ -goods prices are set in euros.

280CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSto solve for Ĉ − Ĉ∗ . Second, it must be the case that asset holdings immediatelyadjust to their new steady-state values, ˆb t = ˆb, becausewithone-period price stickiness, all variables must be at their new steadystate values at time t + 1. The extent of any current account imbalanceat t + 1 can only be due to steady-state debt service–not to changes inasset holdings. It follows that bond stocks determined at t which aretaken into t + 1 are already be at their steady state values. So, to getthe solution, start by subtracting (9.63) from (9.62) to getĈ − Ĉ∗ =(1 + θ)2θrˆb1 − n . (9.87)(167)⇒(168)⇒(169)⇒But ˆb/(1 − n) =ŷ t (z) − ŷt ∗ (z ∗ ) − Ŝt − (Ĉt − Ĉ∗ t ), which follows fromsubtracting (9.82) from (9.81) and noting that ˆb = ˆb t . In addition,ŷ t (z) − ŷt ∗ (z ∗ )=θŜt, which you get by subtracting (9.48) from (9.47),using PPP and noting that ˆp t (z) − ˆp ∗ t (z ∗ ) = 0. Now you can rewrite(9.87) asĈ − Ĉ∗ = (θ2 − 1)r(9.88)r(1 + θ)+2θŜt,and solve (9.85) and (9.88) to getŜ t =Ĉ t − Ĉ∗ t =²[r(1 + θ)+2θ]r(θ 2 − 1) + ²[r(1 + θ)+2θ] ( ˆM t − ˆM t ∗ ), (9.89)²[r(θ 2 − 1)]r(θ 2 − 1) + ²[r(1 + θ)+2θ] ( ˆM t − ˆM t ∗ ). (9.90)(170)⇒From (9.87) and (9.90), the solution for the current account isˆb =2θ²(1 − n)(θ − 1)r(θ 2 − 1) + ²[r(1 + θ)+2θ] ( ˆM t − ˆM ∗ t ). (9.91)(171)⇒(9.83), (9.90) and (9.69) together give the steady state terms of trade,ˆp(z) − ˆp ∗ (z ∗ ) − Ŝ = ²r(θ − 1)r(θ 2 − 1) + ²[r(1 + θ)+2θ] ( ˆM t − ˆM t ∗ ). (9.92)We can now see that money is not neutral since in (9.92) the monetaryshock generates a long-run change in the terms of trade. A domestic

9.1. THE REDUX MODEL 281money shock generates a home current account surplus (in (9.91)) andimproves the home wealth position and therefore the terms of trade.Home agents enjoy more leisure in the new steady state.From (9.89) it follows that the nominal exchange rate exhibits lessvolatility than the money supply. It also exhibits less volatility understicky prices than under ßexible prices since if prices were perfectlyßexible prices, money would be neutral and the effect of a monetaryexpansion on the exchange rate would be Ŝt = ˆM t − ˆM ∗ t .The short-run terms of trade decline by Ŝt since ˆp t (z) =ˆp ∗ t (z ∗ )=0. ⇐(172)Since there are no further changes in the exchange rate, it follows from(9.92) and (9.90) that the short-run increase in the terms of trade exceedsthe long-run increase. The partial reversal means there is overshootingin the terms of trade.To Þnd the effect of permanent monetary shocks on the real interestrate, use the consumption Euler equations (9.51) and (9.52) to getĈ w t = −(1 − β)ˆr t . (9.93)To solve for Ĉw t , use (9.73)—(9.74) to substitute out the short-run pricelevelchanges and (9.70)—(9.71) to substitute out the long-run price levelchanges from the log-linearized money demand functions (9.53)—(9.54) ⇐(173-174)Ĉ t +Ãβ²(1 − β)Ĉ − ² +β(1 − β)Ã! h iˆMt − (1 − n)Ŝt = βˆrt ,! hˆM∗t + nŜtĈt ∗ β+β)Ĉ∗ βi− ² += βˆrt .²(1 − (1 − β)Multiply the Þrst equation by n, the second by (1−n) then add togethernoting by (9.64) Ĉw =0. ThisgivesÃ!ββˆr t = Ĉw t − ² + ˆM t w .(1 − β)Now solve for the real interest rate gives the liquidity effectˆr t = −ò +β(1 − β)!ˆM w t . (9.94)A home monetary expansion lowers the real interest rate and raisesaverage world consumption. From the world demand functions (9.47)

282CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS(175)⇒and (9.48) it follows that domestic output unambiguously increases followinga the domestic monetary expansion. The monetary shock raiseshome consumption. Part of the new spending falls on home goods whichraises home output. The other part of the new consumption is spent onforeign goods but because ˆp ∗ t (z ∗ ) = 0, the increased demand for foreigngoods generates a real appreciation for the foreign country and leads toan expenditure switching effect away from foreign goods. As a result,it is possible (but unlikely for reasonable parameter values as shown inthe end-of-chapter problems) for foreign output to fall. Since the realinterest rate falls in the foreign country, foreign consumption followingthe shock behaves identically to home country consumption. Currentperiod foreign consumption must lie above foreign output. Foreignersgo into debt to Þnance the excess consumption and run a current accountdeÞcit. There is a steady-state transfer of wealth to the homecountry. To service the debt, foreign agents work harder and consumeless in the new steady state. To determine whether the monetary expansionis on balance, a good thing or a bad thing, we will perform awelfare analysis of the shock.Welfare Analysis(176)⇒(177)⇒We will drop the notational dependence on z and z ∗ . Beginning withthe domestic household, break lifetime utility into the three componentsarising from consumption, leisure, and real cash balances, U t = Ut c +U y t + Utm ,whereU c t =∞Xj=0U y t = − ρ 2U m t =γ1 − ²β j ln(C t+j ), (9.95)∞Xj=0∞Xβ j y 2 t+j , (9.96)j=0β j ÃMt+jP t+j! 1−². (9.97)It is easy to see that the surprise monetary expansion raises Utm so weneed only concentrate on Ut c and Ut y .Before the shock, Ut−1 c =ln(C 0 )+(β/(1 − β)) ln(C 0 ). After theshock, Utc =ln(C t )+(β/(1 − β)) ln(C). The change in utility due to

9.1. THE REDUX MODEL 283changes in consumption is∆Ut c = Ĉt +β Ĉ. (9.98)1 − βTo determine the effect on utility of leisure, in the 0-steady stateU y t−1 = −(ρ/2)[y0 2 +(β/(1 − β))y0]. 2 Directly after the shock,Ut y = −(ρ/2)[yt 2 +(β/(1 − β))y 2 ]. Using the Þrst-order approximation,yt 2 = y0 2 +2y 0 (y t − y 0 ), it follows that, ∆Uty = −(ρ/2)[(yt 2 − y0)+ 2 ⇐(178)(β/(1 − β))(y 2 − y0)]. 2 Dividing through by y 0 yields∆U y t= −ρ"y 2 0ŷt +Now use the fact that C 0 = y 0 = C0wà !(θ − 1)∆Ut c + ∆U ty = Ĉt − ŷ t +θAnalogously, in the foreign country∆U c∗t+ ∆U y∗t = Ĉ∗ t − Ã(θ − 1)θ!β(1 − β) y2 0ŷ#= ³ ´1/2,θ−1ρθ togetβ(1 − β)ŷ ∗ t +"Ĉ −"βĈ ∗ −(1 − β). (9.99)#(θ − 1)ŷ . (9.100)θ#(θ − 1)ŷ ∗ .θ(9.101)To evaluate (9.101), Þrst note that ŷ t = θ(1−n)Ŝt+Ĉw t which followsfrom (9.75). From (9.89) and (9.90) it follows that Ĉt = bŜt +Ĉ∗ t whereb =[r(θ 2 − 1)/(r(1 + θ) +2θ)]. Eliminate foreign consumption usingĈt ∗ =(Ĉw t − nĈt)/(1 − n) togetĈ t = (1 − n)r(θ2 − 1)Ŝ t +r(1 + θ)+2θĈw t . (9.102)Now plug (9.102) and (9.93) into (9.77) to get the long-run effect onconsumptionĈ = r(1 − n)(θ2 − 1)(9.103)[(r(1 + θ)+2θ)]Ŝt.Substitute Ĉ into (9.65) to get the long-run effect on home outputŷ =−rθ(1 − n)(θ − 1)Ŝ t . (9.104)r(1 + θ)+2θ

284CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSNow substituting these results back into (9.100) gives∆Ut c + ∆Ut y = (1 − à ! n)r(θ2 − 1)θ − 1 hθ(1 iŜ t +r(1 + θ)+2θĈw t −− n) Ŝ t +θĈw t+ β"r(1 − n)(θ 2 #− 1)Ŝ t(1 − β) r(1 + θ)+2θà !à !β θ − 1 rθ(1 − n)(θ − 1)+Ŝ t . (9.105)1 − β θ r(1 + θ)+2θ(180)⇒After collecting terms, the coefficient on Ŝt is seen to be 0. Substitutingr =(1− β)/β, youareleftwith∆U c t + ∆U y t= Ĉw tθ= −(1 − β)ˆr tθ=Ãβ + ²(1 − β)θ!ˆM w t > 0, (9.106)where the Þrst equality uses (9.93) and the second equality uses (9.94).Due to the extensive symmetry built into the model, the solutionsfor the foreign variables Ĉ∗ , Ĉ∗ t , ŷ ∗ , ŷt∗ are given by the same formulaederived for the home country except that (1 − n) is replaced with −n.It follows that the effect on ∆Utc∗ + ∆Ut y∗is identical to (9.106).One of the striking predictions of Redux is that the exchange rateeffects have no effect on welfare. All that is left of the monetary shockis the liquidity effect. The traditional terms of trade and current accounteffects that typically form the focus of international transmissionanalysis are of second order of importance in Redux. The reason is thatin the presence of sticky nominal prices, the monetary shock generatesa surprise depreciation and lowers the price level to foreigners. Homeproducers produce and sell more output but they also have to workharder which means less leisure. These two effects offset each other.The monetary expansion is positively transmitted abroad as it raisesthe leisure and consumption components of welfare by equal amountsin the two countries. Due to the monopoly distortion, Þrms set priceabove marginal cost, which leads to a level of output that is less than thesocially optimal level. The monetary expansion generates higher outputin the short run which moves both economies closer to the efficientfrontier. The expenditure switching effects of exchange rate ßuctuations

9.1. THE REDUX MODEL 285and associated beggar thy neighbor policies identiÞed in the Mundell-Fleming model are unimportant in the Redux model environment.It is possible, but unlikely for reasonable parameter values, that thedomestic monetary expansion can lower welfare abroad through its effectson foreign real cash balances. The analysis of this aspect of foreignwelfare is treated in the end-of-chapter problems.Summary of Redux Predictions. The law-of-one price holds for all goodsand as a consequence PPP holds as well. A permanent domestic monetaryshock raise domestic and foreign consumption. Domestic outputincreases and it is likely that foreign output increases but by a lesseramount. The presumption is that home and foreign consumption exhibita higher degree of co-movement than home and foreign output.Both home and foreign households experience the identical positivewelfare effect from changes in consumption and leisure. The monetaryexpansion moves production closer to the efficient level, which is distortedin equilibrium by imperfect competition. There is no exchangerate overshooting. The nominal exchange rate jumps immediately toits long-run value. The exchange rate also exhibits less volatility thanthe money supply.Many of these predictions are violated in the data. For example,Knetter [86] and Feenstra et. al. [52] Þnd that pass through of theexchange rate onto the domestic prices of imports is far from completewhereas there is complete pass-through in Redux. 8 Also, we saw inChapter 7 that deviations from PPP and deviations from the law-ofoneprice are persistent and can be quite large. Also, Redux does notexplain why international consumption displays lower degrees of comovementsthan output as we saw in Chapter 5.We now turn to a reÞnement of the Redux model in which theprice-setting rule is altered. The change in this one aspect of the modeloverturns many of the redux model predictions and brings us backtowards the Mundell—Fleming model.8 Pass-through is the extent to which the dollar price of US imports rise in responseto a 1-percent depreciation in the dollar currency.

286CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS9.2 Pricing to MarketThe integration of international commodity markets in the Redux modelrules out deviations from the law-of-one price in equilibrium. Were suchviolations to occur, they presumably would induce consumers to takeadvantage of international price differences by crossing the border tobuy the goods (or contracting with foreign consumers to do the shoppingfor them) in the lower price country resulting in the internationalprice differences being bid away.We will now modify the Redux model by assuming that domesticand foreign goods markets are segmented. Domestic (foreign) agents areunable to buy the domestically-produced good in the foreign (home)country. The monopolistically competitive Þrm has the ability to engagein price discrimination by setting a dollar price for domestic salesthat differs from the price it sets for exports. This is called pricing-tomarket.For concreteness, let the home country be the ‘US’ and the foreigncountry be ‘Europe.’ We assume that all domestic Þrms have the abilityto price-to-market as do all foreign Þrms. This is called ‘full’ pricingto-market.Betts and Devereux [10] allow the degree of pricing-tomarket–thefraction of Þrms that operate in internationally segmentedmarkets–to vary from 0 to 1. Both the Redux model and the nextmodel that we study are nested within their framework. The associatednotation is summarized in Table 9.2.Full Pricing-To-MarketWe modify Redux in two ways. The Þrst difference lies in the pricesettingopportunities for monopolistically competitive Þrms. The goodsmarket is integrated within the home country and within the foreigncountry, but not internationally. The second modiÞcation is in themenu of assets available to agents. Here, the internationally tradedasset is a nominal bond denominated in ‘dollars.’ The model is still setin a deterministic environment.Goods markets. AUSÞrm z, sells x t (z) units of output in the homemarket and exports v t (z) to the foreign country. Total output of the

9.2. PRICING TO MARKET 287US Þrm is y t (z) =x t (z)+v t (z). The per-unit dollar price of US salesis set at p t (z) and the per-unit euro price of exports is set at q ∗ t (z).AEuropeanÞrm z ∗ sells x ∗ t (z ∗ ) units of output in Europe at thepre-set euro price p ∗ t (z ∗ ) and exports v ∗ t (z ∗ )totheUSwhichitsellsatapre-setdollar price of q t (z ∗ ). Total output of the European Þrm isy ∗ t (z∗ )=x ∗ t (z∗ )+v ∗ t (z∗ ).Home Countryy=x+vx sells at home at dollarprice pForeign Countryv* sells at home atdollar price q0 nv sells abroad ateuro price q*y*=x*+v*x* sells in foreign countryat euro price p*1Figure 9.2: Pricing-to-market home and foreign households lined up onthe unit interval.Asset Markets. The internationally traded asset is a one-period nominalbond denominated in dollars. Restricting asset availability placespotential limits on the degree of international risk sharing that can beachieved. Since violations of the law of one price can now occur, so canviolations of purchasing power parity. It follows that that real interestrates can diverge across countries. Since intertemporal optimalityrequires that agents set the growth of marginal utility (consumptionin the log utility case) to be proportional to the real interest rate, theinternational inequality of real interest rates implies that home andforeign consumption will be not be perfectly correlated.The bond is sold at discount and has a face value of one dollar.Let B t be the dollar value of bonds held by domestic households, andBt ∗ be the dollar value of bonds held by foreign households. Bondsoutstanding are in zero net supply nB t +(1− n)Bt∗ = 0. The dollar

288CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSprice of the bond is1δ t ≡(1 + i t ) .The foreign nominal interest rate is given by uncovered interest parityà !(1 + i ∗ t )=(1+i Stt) .S t+1(181)⇒Households. We need to distinguish between hours worked, whichis chosen by the household, and output which is chosen by the Þrm.The utility function is similar to (9.2) in the Redux model except thathours of work h t (z) appears explicitly in place of output y t (z)⎡∞XU t = β j ⎣ln C t+j +γà ! ⎤1−²Mt+j− ρ 1 − ² P t+j 2 h2 ⎦t+j(z) . (9.107)j=0The associated price indices for the domestic and foreign householdsareP t =P ∗t =·Z n0·Z n0p t (z) 1−θ dz +q ∗ t (z) 1−θ dz +Z 1nZ 1nq t (z ∗ ) 1−θ dz ∗¸1/(1−θ) , (9.108)p ∗ t (z ∗ ) 1−θ dz ∗¸1/(1−θ) . (9.109)W t is the home country competitive nominal wage. The householdderives income from selling labor to Þrm z, W t h t (z). Household-z alsoowns Þrm-z from which it earns proÞts, π t (z). Nominal wealth takeninto the next period consists of cash balances and bonds (M t + δ t B t ).This wealth is the result of wealth brought into the current period(M t−1 + B t−1 ) plus current income (W t h t (z)+π t (z)) less consumptionand taxes (P t C t + P t T t ). The home and foreign budget constraints aregiven byM t + δ t B t = W t h t (z)+π t (z)+M t−1 + B t−1 − P t C t − P t T t , (9.110)M ∗ t +δ tB ∗ tS t= W ∗t h ∗ t (z)+π ∗ t (z)+M ∗ t−1 + B∗ t−1S t− P ∗t C ∗ t −P ∗t T ∗t . (9.111)Households take prices and Þrm proÞts as given and choose B t ,M t ,and h t . To derive the Euler-equations implied by domestic household

9.2. PRICING TO MARKET 289optimality, transform the household’s problem into an unconstraineddynamic choice problem by rewriting the budget constraint (9.110) interms of consumption and substituting this result into the utility function(9.107). Do the same for the foreign agent. The resulting Þrst-orderconditions can be re-arranged to yield, 9δ t P t+1 C t+1 = βP t C t , (9.112)µ St+1δ t Pt+1C ∗t+1∗· ¸M 1t γCt=P t 1 − δ tM ∗ tP ∗t=⎡⎣γC ∗ tS t1 − δ tS t+1S tW t= βP ∗t C ∗ t , (9.113)², (9.114)⎤⎦1², (9.115)h t (z) = 1 , (9.116)ρ P t C tW ∗th ∗ t (z) =1 ρ Pt ∗ Ct∗ . (9.117)Domestic household demand for domestic z−goods and for foreign z ∗ -goods are" # −θpt (z)c t (z) = C t . (9.118)P t⇐(183)9 Differentiating the utility function with respect to B t gives∂U t= −δ t β+ =0,∂B t P t C t P t+1 C t+1which is re-arranged as (9.112). Differentiating the utility function with respect toM t gives∂U t∂M t=−1P t C t+βP t+1 C t+1+ γ P tµMtP t −²=0.Re-arranging this equation and using (9.112) to substitute out P t+1 C t+1 = βP t C t /δ tresults in (9.114). The Þrst-order condition for hours is∂U t∂h t=W tP t C t− ρh t =0,from which (9.117) follows directly. Derivations of the Euler-equations for the foreigncountry follow analogously.

290CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS"c t (z ∗ qt (z ∗ # −θ))= C t , (9.119)P tForeign household demand for domestic z-goods and for and foreignz ∗ -goods are" #qc ∗ ∗ −θt (z) = t (z)Pt∗ Ct ∗ , (9.120)"p∗t (z ∗ # −θ)Ct ∗ . (9.121)c ∗ t (z ∗ )=Firms. Firms only employ labor. There is no capital in the model.The domestic and foreign production technologies are identical and arelinear in hours of worky t (z) =h t (z),P ∗ty ∗ t (z) =h∗ t (z).Domestic and foreign Þrm proÞts areπ t (z) = p t (z)x t (z)+S t qt ∗ (z)v t(z) − W t h t (z), (9.122)πt ∗ (z ∗ ) = p ∗ t (z ∗ )x ∗ t (z ∗ )+ q t(z ∗ )vt ∗ (z ∗ ) − Wt ∗ h ∗ t (z ∗ ). (9.123)S tThe domestic z-Þrm sets prices at the beginning of the period beforeperiod-t shocks are revealed. The monopolistically competitive Þrmmaximizes proÞts by choosing output to set marginal revenue equal tomarginal cost. Given the demand functions (9.118)—(9.121), the rulefor setting the price of home sales is the constant markup of price overcosts, 10 p t (z) =[θ/(θ − 1)]W t . The z-Þrm also sets the euro price of itsexports qt ∗ (z). Before period t monetary or Þscal shocks are revealed,the Þrm observes the exchange rate S t , and sets the euro price accordingto the law-of-one price S t qt ∗ (z) =p t (z). This is optimal, conditional onthe information available at the time prices are set because home and10 The domestic demand function is y = p −θ P θ C can be rewritten asp = PC 1/θ y −1/θ . Multiply by y to get total revenue. Differentiating with respectto y yields marginal revenue, [(θ − 1)/θ]PC 1/θ y −1/θ =[(θ − 1)/θ]p. Marginalcost is simply W . Equating marginal cost to marginal revenue gives the markuprule.

9.2. PRICING TO MARKET 291foreign market elasticity of demand is identical. Although the Þrm hasthe power to set different prices for the foreign and home markets itchooses not to do so. Once p t (z) andqt ∗ (z) are set, they are Þxed forthe remainder of the period. The foreign Þrm sets price according to asimilar technology.Since the elasticity of demand for all goods markets is identical andall Þrms have the identical technology, price-setting is identical amonghome Þrms and is identical among all foreign Þrmsp t (z) =S t qt ∗ (z) = θθ − 1 W t, (9.124)p ∗ t (z ∗ )= q t(z ∗ )= θS t θ − 1 W t ∗ . (9.125)Using (9.124) and (9.125), the formulae for the price indices (9.108)and (9.109) can be simpliÞed toP t = h np t (z) (1−θ) +(1− n)q t (z ∗ ) (1−θ)i 1(1−θ), (9.126)Pt ∗ = h nqt ∗ (z)(1−θ) +(1− n)p ∗ t (z∗ ) (1−θ)i 1(1−θ). (9.127)Output is demand determined in the short run and can either be soldto the domestic market or made available for export. The adding-upconstraint on output, sales to the home market and sales to the foreignmarket arey t (z) = x t (z)+v t (z), (9.128)" # −θpt (z)x t (z) =nC t , (9.129)P t" # −θpt (z)v t (z) =(1 − n)C ∗S t Pt∗t . (9.130)The analogous formulae for the foreign country areyt ∗ (z∗ ) = x ∗ t (z∗ )+vt ∗ (z∗ ), (9.131)"px ∗ ∗t (z∗ ) = t (z ∗ # −θ)(1 − n)Ct ∗ , (9.132)v ∗ t (z ∗ ) =Pt∗"St p ∗ t (z ∗ )P t# −θ(1 − n)C t . (9.133)⇐(186-187)

292CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSGovernment.seignorageGovernment spending is Þnanced by tax receipts andP t G t = P t T t + M t − M t−1 , (9.134)Pt ∗ t = P t ∗ t ∗ t ∗ − M t−1 ∗ . (9.135)In characterizing the equilibrium, it will help to consolidate the individual’sand government’s budget constraints. Substitute proÞts (9.122)-(9.123) and the government budget constraints (9.134)-(9.135) into thehousehold budget constraints (9.110)-(9.111) and use the zero-net supplyconstraint B ∗ t = −(n/(1 − n))B t from (9.137) to getP t C t + P t G t + δ t B t = p t (z)x t (z)+S t q ∗ t (z)v t(z)+B t−1 , (9.136)P ∗t C ∗ t + P ∗t G ∗ t −n δ t B t= p ∗ t (z ∗ )x ∗ t (z ∗ )+ q t(z ∗ )vt ∗ (z ∗ ) − n1 − n S t S tB t−1S t.1 − n(9.137)The equilibrium is characterized by the Euler equations (9.112)—(9.117),the consolidated budget constraints (9.136) and (9.137) with B 0 = G 0 =G ∗ 0 = 0, and the output equations (9.128)—(9.133).From this point on we will consider only on monetary shocks. Tosimplify the algebra, set G t = G ∗ t =0forallt. We employ the samesolution technique as we used in the Redux model. First, solve forthe 0-steady state with zero-international debt and zero-governmentspending, then take a log-linear approximation around that benchmarksteady state.(189)⇒The 0-steady state. The 0-steady state under pricing-to-market isidentical to that in the redux model. Set G 0 = G ∗ 0 = B 0 = 0. Dollarprices of z and z ∗ goods sold at home are identical, p 0 (z) =q 0 (z ∗ ).From the markup rules (9.124) and (9.125), it follows that the law ofone price, p 0 (z) =q 0 (z ∗ )=S 0 q0(z) ∗ =S 0 p ∗ 0(z ∗ ). We also have by PPPP 0 = S 0 P0 ∗ . (9.138)Steady state hours of work, output, and consumption are" # 1/2θ − 1h 0 (z) =y 0 (z) =h ∗ 0(z ∗ )=y0(z ∗ ∗ )=C 0 = C0 ∗ = . (9.139)ρθ

9.2. PRICING TO MARKET 293Table 9.2: Notation for the pricing-to-market modelp t (z)qt ∗(z)p ∗ t (z ∗ )q t (z ∗ )y t (z)x t (z)v t (z)yt ∗ (z ∗ )x ∗ t (z ∗ )vt ∗ (z ∗ )π t (z)πt ∗ (z ∗ )h t (z)h ∗ t (z ∗ )B tBt∗i tδ tW tWt∗G tG ∗ tT tTt∗C tCt∗P tPt∗S tdollar price of home good z in home country.euro price of home good z in foreign country.euro price of foreign good z ∗ in foreign country.dollar price of foreign good z ∗ in home country.home goods output.home goods sold at home.home goods sold in foreign country.foreign goods output.foreign goods sold in foreign country.foreign goods sold in home country.Domestic Þrm proÞts.Foreign Þrm proÞts.Hours worked by domestic individual.Hours worked by foreign individual.Dollar value of nominal bond held by domestic individual.Dollar value of nominal bond held by foreign individual.Nominal interest rate.Nominal price of the nominal bond.Nominal wage in dollars.Nominal wage in euros.Home government spending.Foreign government spending.Home government lump-sum tax receipts.Foreign government lump-sum tax receipts.Home CES consumption index.Foreign CES consumption index.Home CES price index.Foreign CES price index.Nominal exchange rate.

294CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSFrom the money demand functions it follows that the exchange rate isS 0 = M 0. (9.140)M0∗Log-linearizing around the 0-steady state. The log-expansion of (9.114)and (9.115) around 0-steady state values gives 11ˆM t − ˆP t = 1 β² Ĉt +²(1 − β) ˆδ t , (9.141)ˆM t ∗ − ˆP t ∗ = 1 ² Ĉ∗ t + β²(1 − β) [ˆδ t + Ŝt+1 − Ŝt]. (9.142)Log-linearizing the consolidated budget constraints (9.136) and (9.137)with B 0 = G 0 = G ∗ 0 =0gives 12(191-192)⇒Ĉ t = n[ˆp t (z)+ˆx t (z)− ˆP t ]+(1−n)[ˆq ∗ t (z)+Ŝt +ˆv t (z)− ˆP t ]−βˆb t , (9.143)Ĉt ∗ =(1−n)[ˆp ∗ t (z ∗ )+ˆx ∗ t (z ∗ )− ˆP t ∗ ]+n[ˆq t (z ∗ )−Ŝt+ˆv t ∗ (z ∗ )− ˆP t ∗ ]+βnt .1 − nˆb(9.144)Log-linearizing (9.128)—(9.133) givesŷ t (z) = nˆx t (z)+(1− n)ˆv t (z), (9.145)ŷt ∗ (z∗ ) = (1− n)ˆx ∗ t (z∗ )+nˆv t ∗ (z∗ ), (9.146)ˆx t (z) = θ[ ˆP t − ˆp t (z)] + Ĉt, (9.147)11 Taking log-differences of the money demand function (9.114) givesˆM t − ˆP³ ´t = 1 ² [Ĉt−(ln(1−δ t )−ln(1−δ 0 ))]. But ∆(ln(1−δ t )) ' −δ0 δ t−δ 01−δ 0 δ 0= −β ˆδ 1−β t ,which together gives (9.141).12 Write (9.136) as C t = p t(z)x t (z)∆C t = C t − C 0 = ∆the Þrst term is ∆hpt(z)x t(z)P t+ S tqt ∗ (z)v t(z)P t−h i h δtBtP tStqt ∗ (z)vt(z)P t− ∆δt B tP t. It follows thati. The expansion ofiP t+ ∆h ipt(z)x t(z)P t= x 0 (z)[ˆx t +ˆp t − ˆP t ] because P 0 = p 0 (z). The expansionof the second term follows analogously. h i To expand the third term, notingthat P 0 = 1, δ 0 = β, and B 0 =0gives∆ δtB tP t= βB t . After dividing through byC0w = y 0(z), and noting that x 0 (z)/y 0 (z) =n, andv 0 (z)/y 0 (z) =(1 − n), we obtain(9.143).

9.2. PRICING TO MARKET 295ˆv t (z) = θ[Ŝt + ˆP t ∗ t(z)] + Ĉ∗ t , (9.148)ˆx ∗ t (z ∗ ) = θ[ ˆP t ∗ − ˆp ∗ t (z ∗ )] + Ĉ∗ t , (9.149)ˆv t ∗ (z ∗ ) = θ[ ˆP t − Ŝt − ˆp ∗ t (z ∗ )] + Ĉt. (9.150)Log-linearizing the labor supply rules (9.116) and (9.117) and using theprice markup rules (9.124)—(9.125) to eliminate the wage yieldsŷ t (z) = ˆp t (z) − ˆP t − Ĉt, (9.151)ŷt ∗ (z ∗ ) = ˆp ∗ t (z ∗ ) − ˆP t ∗ − Ĉ∗ t . (9.152)Log-linearizing the intertemporal Euler equations (9.112) and (9.113)givesˆP t + Ĉt = ˆδ t + Ĉt+1 + ˆP t+1 , (9.153)ˆP t ∗ t = ˆδ t + Ĉ∗ t+1 + ˆP t+1 ∗ − Ŝt. (9.154)Long-Run ResponseThe log-linearized equations hold for arbitrary t andalsoholdinthenew steady state. By the intertemporal optimality condition (9.112),δ = β in the new steady state which implies ˆδ =0. Notingthatthenominal exchange rate is constant in the new steady state, it followsfrom (9.141) and (9.142)ˆM − ˆP = 1 Ĉ,²(9.155)ˆM ∗ − ˆP ∗ = 1 ² Ĉ∗ . (9.156)By the law-of-one price ˆp(z) = q∗ ˆ (z)+Ŝ. (9.143) and (9.144) become ⇐(194-196)Ĉ = ˆp(z)+ŷ(z) − ˆP − βˆb, (9.157)" #Ĉ ∗ = ˆp ∗ (z ∗ )+ŷ ∗ (z ∗ ) − ˆP nβ∗ + ˆb. (9.158)1 − nTaking a weighted average of the log-linearized budget constraints (9.157)and (9.158) givesĈ w = n[ˆp(z) − ˆP +ŷ(z)] + (1 − n)[ˆp ∗ (z ∗ ) − ˆP ∗ +ŷ ∗ (z ∗ )]. (9.159)

296CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSRecall that world demand for home goods is y(z) = [p(z)/P ] −θ C wand world demand for foreign goods is y ∗ (z ∗ )=[p ∗ (z ∗ )/P ∗ ] −θ C w . Thechange in steady-state demand isŷ(z) = −θ[ˆp(z) − ˆP ]+Ĉw , (9.160)ŷ ∗ (z ∗ ) = −θ[ˆp ∗ (z ∗ ) − ˆP ∗ ]+Ĉw (9.161)(197-198)⇒By (9.151) and (9.152), the optimal labor supply changes byŷ(z) = ˆp(z) − ˆP − Ĉ, (9.162)ŷ ∗ (z ∗ ) = ˆp ∗ (z ∗ ) − ˆP ∗ − Ĉ∗ . (9.163)(9.157)—(9.163) form a system of 6 equations in the 6 unknowns(Ĉ,Ĉ∗ , ŷ(z), ŷ ∗ (z ∗ ), (ˆp(z) − ˆP ), (ˆp ∗ (z ∗ ) − ˆP ∗ ), which can be solved toget 13 β(1 + θ)Ĉ = − ˆb, (9.164)2θĈ ∗ =β(1 + θ)2θµ n ˆb, (9.165)1 − nŷ(z) = β (9.166)2ˆb,ŷ ∗ (z ∗ )=− β µ n ˆb, (9.167)2 1 − nˆp(z) − ˆP = −2θˆb, β (9.168)ˆp ∗ (z ∗ ) − ˆP ∗ = β µ n ˆb. (9.169)2θ 1 − n(199)⇒(200)⇒By (9.164) and (9.165), average world consumption is not affectedĈ w = 0, but the steady-state change in relative consumption isβ(1 + θ)Ĉ − Ĉ∗ = − (9.170)2θ(1 − n)ˆb.13 The solution looks slightly different from the redux solution because the internationallytraded asset is a nominal bond whereas in the redux model it is a realbond.

9.2. PRICING TO MARKET 297From the money demand functions it follows that the steady statechange in the nominal exchange rate isŜ = ˆM − ˆM ∗ − 1 ²hĈ− Ĉ ∗i . (9.171)Adjustment to Monetary Shocks under Sticky PricesConsider an unanticipated and permanent monetary shock at time t,where ˆM t = ˆM, and ˆM t∗ = ˆM ∗ . As in Redux, the new steady state isattained at t +1,sothatŜt+1 = Ŝ, ˆP t+1 = ˆP, and ˆP t+1 ∗ = ˆP ∗ .Date t nominal goods prices are set and Þxedone-periodinadvance.By (9.10) and (9.11), it follows that the general price levels are alsopredetermined, ˆPt = ˆP t∗ =0. The short-run versions of (9.141) and(9.142) areˆM = 1 β² Ĉt +²(1 − β) ˆδ t , (9.172)ˆM ∗ = 1 ² Ĉ∗ t + β²(1 − β) [ˆδ t + Ŝ − Ŝt]. (9.173)Subtracting (9.173) from (9.172) givesˆM t − ˆM ∗ t = 1 ² (Ĉt − Ĉ∗ t ) − β²(1 − β) (Ŝ − Ŝt). (9.174)From (9.153) and (9.154) you getĈ t = ˆδ t + Ĉ + ˆP, (9.175)Ĉ ∗ t = ˆδ t + Ĉ∗ + ˆP ∗ + Ŝ − Ŝt. (9.176)At t +1PPPisrestored, ˆP = ˆP ∗ + Ŝ. Subtract (9.176) from (9.175)to getĈ − Ĉ∗ = Ĉt − Ĉ∗ t − Ŝt. (9.177)The monetary shock generates a short-run violation of purchasing powerparity and therefore a short-run international divergence of real interestrates. The incompleteness in the international asset market results inimperfect international risk sharing. Domestic and foreign consumptionmovements are therefore not perfectly correlated.

298CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSTo solve for the exchange rate take Ŝ from (9.171) and plug into(9.174) to get"1+#β ³ˆMt −²(1 − β)ˆM´t∗ 1 =³Ĉt −²Ĉ∗ t´+Using (9.177) to eliminate Ĉ − Ĉ∗ ,yougetβ² 2 (1 − β)³Ĉ− Ĉ ∗´+ β²(1 − β)Ŝt.Ŝ t =β + ²(1 − β)β(² − 1)h²( ˆMt − ˆM ∗ t ) − (Ĉt − Ĉ∗ t )i . (9.178)This is not the solution because Ĉt − Ĉ∗ t is endogenous. To get thesolution, you have from the consolidated budget constraints (9.143)and (9.144)Ĉ t = nˆx t (z)+(1− n)[Ŝt +ˆv t (z)] − βˆb t , (9.179)Ĉ ∗ t = (1− n)ˆx ∗ t (z ∗ )+n[ˆv ∗ t (z ∗ ) − Ŝt]+β n1 − nˆb t , (9.180)(201-202)⇒and you have from (9.147)—(9.150)ˆx t (z) =Ĉt; ˆx ∗ t (z∗ )=Ĉ∗ t ; ˆv t(z) =Ĉ∗ t ; ˆv∗ t (z∗ )=Ĉt. (9.181)Subtract (9.180) from (9.179) and using the relations in (9.181), youhaveβŜ t =(Ĉt − Ĉ∗ t )+2(1 − n) 2ˆb t . (9.182)Substitute the steady state change in relative consumption (9.170) into(9.177) to get2θ(1 − n)ˆb = −β(1 + θ) [Ĉt − Ĉ∗ t − Ŝt], (9.183)and plug (9.183) into (9.182) to getĈ t − Ĉ∗ t − Ŝt =2θ(1 + θ) [Ĉt − Ĉ∗ t − Ŝt].It follows that Ĉt −Ĉ∗ t −Ŝt =0. Looking back at (9.183), it must be thecase that ˆb = 0 so there are no current account effects from monetaryshocks. By (9.164) and (9.165), you see that Ĉ = Ĉ∗ =0,andby

9.2. PRICING TO MARKET 299(9.155) and (9.156) it follows that ˆP = ˆM, and ˆP ∗ = ˆM ∗ . Money istherefore neutral in the long run.Now substitute Ŝt = Ĉt − Ĉ∗ t back into (9.178) to get the solutionfor the exchange rateŜ t =[²(1 − β)+β]( ˆM t − ˆM t ∗ ). (9.184)The exchange rate overshoots its long-run value and exhibits morevolatility than the monetary fundamentals if the consumption elasticityof money demand 1/² < 1. 14 Relative prices are unaffected by thechange in the exchange rate, ˆp t (z) − ˆq t (z ∗ ) = 0. A domestic monetaryshock raises domestic spending, part of which is spent on foreign goods.ThehomecurrencydepreciatesŜt > 0 in response to foreign Þrms repatriatingtheir increased export earnings. Because goods prices are Þxedthere is no expenditure switching effect. However, the exchange rateadjustment does have an effect on relative income. The depreciationraises current period dollar (and real) earnings of US Þrms and reducescurrent period euro (and real) earnings of European Þrms. This redistributionof income causes home consumption to increase relative toforeign consumption.Real and nominal exchange rates. The short-run change in the realexchange rate isˆP t − ˆP t ∗ − Ŝt = −Ŝt,which is perfectly correlated with the short-run adjustment in the nominalexchange rate.⇐(205)Liquidity effect. If r t is the real interest rate at home, then (1 + r t )=(P t )/(P t+1 δ t ). Since ˆP t =0,itfollowsthatˆr t = −( ˆP + ˆδ t )=−(ˆδ t + ˆM)and (9.175)—(9.172) can be solved to getˆδ t =(1− β)(² − 1) ˆM, (9.185)which is positive under the presumption that ²>0. It follows that⇐(206)14 Obstfeld and Rogoff show that a sectoral version of the Redux model withtraded and non-traded goods produces many of the same predictions as the pricingto-marketmodel.

300CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSˆr t =[²(β − 1) − β] ˆM, (9.186)is negative if ²>1. Now let rt∗ be the real interest rate in the foreigncountry. Then, (1 + rt ∗ )=(Pt ∗ S t )/(Pt+1S ∗ t+1 δ t ), and ˆr t ∗ = Ŝt − [ ˆP ∗ +Ŝ + ˆδ t ]. But you know that ˆP ∗ = ˆM ∗ =0, Ŝ = ˆM, soˆr t∗ =ˆr t + Ŝt.It follows from (9.184) and (9.186) that ˆr t ∗ = 0. The expansion of thedomestic money supply has no effect on the foreign real interest rate.International transmission and co-movements. Since ˆδ t + Ŝ − Ŝt =0,it follows from (9.172) that Ĉt =[²(1 − β)+β] ˆM >0 and from (9.173)that Ĉ∗ t = 0. Under pricing-to-market, there is no international transmissionof money shocks to consumption. Consumption exhibits a lowdegree of co-movement. From (9.181), output exhibits a high-degree ofco-movement, ŷ t =ˆx t = Ĉt =ŷt ∗ =ˆv t ∗ . The monetary shock raises consumptionand output at home. The foreign country experiences higheroutput, less leisure but no change in consumption. As a result, foreignwelfare must decline. Monetary shocks are positively transmittedinternationally with respect to output but are negatively transmittedwith respect to welfare. Expansionary monetary policy under pricingto market retains the ‘beggar-thy-neighbor’ property of depreciationfrom the Mundell—Fleming model.(207-208)⇒The terms of trade. Let P xt be the home country export price indexand P ∗ xt be the foreign country export price indexP xt =P ∗ xt = µZ 1µZ nThehometermsoftradeare,n0[S t q ∗ t (z)] 1−θ dz 1/(1−θ)= n11−θ St q ∗ t ,[q t (z ∗ )/S t ] 1−θ dz ∗ 1/(1−θ)=[(1− n) 11−θ qt ]/S t .P xtτ t = =S t Pxt∗µ 1n 1−θ1 − nS t q ∗ tq t,and in the short run are determined by changes in the nominal exchangerate, ˆτ t = Ŝt. Since money is neutral in the long run, there are no steadystate effects on τ. Recall that in the Redux model, the monetary shock

9.2. PRICING TO MARKET 301caused a nominal depreciation and a deterioration of the terms of trade.Under pricing to market, the monetare shock results in a short-runimprovement in the terms of trade.

302CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSSummary of pricing-to-market and comparison to Redux. Manyof the Mundell—Fleming results are restored under pricing to market.Money is neutral in the long run, exchange rate overshooting is restored,real and nominal exchange rates are perfectly correlated in the short runand under reasonable parameter values expansionary monetary policyis a ‘beggar thy neighbor’ policy that raises domestic welfare and lowersforeign welfare.Short-run PPP is violated which means that real interest rates candiffer across countries. Deviations from real interest parity allow imperfectcorrelation between home and foreign consumption. While consumptionco-movements are low, output co-movements are high andthat is consistent with the empirical evidence found in Chapter 5. Thereis no exchange-rate pass-through and there is no expenditure switchingeffect. Exchange rate ßuctuations do not affect relative prices but doaffect relative income. For a given level of output, the depreciationgenerates a redistribution of income by raising the dollar earnings ofdomestic Þrms and reduces the ‘euro’ earnings of foreign Þrms.In the Redux model, the exchange rate response to a monetary shockis inversely related to the elasticity of demand, θ. The substitutabilitybetween domestic and foreign goods is increasing in θ. Higher valuesof θ require a smaller depreciation to generate an expenditure switchof a given magnitude. Substitutability is irrelevant under full pricingto-market.Part of a monetary transfer to domestic residents is spenton foreign goods which causes the home currency to depreciate. Thedepreciation raises domestic Þrm income which reinforces the increasedhome consumption. What is relevant here is the consumption elasticityof money demand 1/².In both Redux and pricing to market, one-period nominal rigiditiesare introduced as an exogenous feature of the environment. This ismathematically convenient because the economy goes to new steadystate in just one period. The nominal rigidities can perhaps be motivatedby Þxed menu costs, and the analysis is relevant for reasonablysmall shocks. If the monetary shock is sufficiently large however, thebeneÞts to immediate adjustment will outweigh the menu costs thatgenerate the stickiness.

9.2. PRICING TO MARKET 303New International Macroeconomics Summary1. Like Mundell-Fleming models, the new international macroeconomicsfeatures nominal rigidities and demand-determined output.Unlike Mundell-Fleming, however, these are dynamic generalequilibrium models with optimizing agents where tastes andtechnology are clearly spelled out. These are macroeconomicmodels with solid micro-foundations.2. Combining market imperfections and nominal price stickinessallow the new international macroeconomics to address featuresof the data, such as international correlations of consumptionand output, and real and nominal exchange rate dynamics, thatcannot be explained by pure real business cycle models in theArrow-Debreu framework. It makes sense to analyze the welfareeffects of policy choices here, but not in real business cycle models,since all real business cycle dynamics are Pareto efficient.3. The monopoly distortion in the new international macroeconomicsmeans that equilibrium welfare lies below the social optimumwhich potentially can be eliminated by macroeconomicpolicy interventions.4. Predictions regarding the international transmission of monetaryshocks are sensitive to the speciÞcation of Þnancial structureand price setting behavior.

304CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICSProblems1. Solve for effect on the money component of foreign welfare followinga permanent home money shock in the Redux model.(a) Begin by showing that∆U ∗3tµ M∗= −γP ∗ 0 1−² ·ˆP ∗t +β ˆP∗¸1 − βNext, show that ˆP ∗ t= −nŜt andFinally, show that∆U ∗3t ="ˆP ∗ = rn(θ2 − 1)²[r(1 + θ)+2θ]Ŝt.−(θ 2 # µM− 1)∗ 1−²²[r(1 + θ)+2θ] − 1 P0∗ nγŜtThis component of foreign welfare evidently declines followingthe permanent M t shock. Is it reasonable to think that it willoffset the increase in foreign utility from the consumption andleisure components?2. Consider the Redux model. Fix M t = M ∗ t = M 0 for all t. Begin inthe ‘0’ equilibrium.(a) Consider a permanent increase in home government spending,G t = G>G 0 =0. at time t. Show that the shock leads to ahome depreciation ofŜ t =(1 + θ)(1 + r)r(θ 2 − 1)+²[r(1 + θ)+2θ]ĝ,andaneffect on the current account of,What is the likely effect on ˆb?ˆb =(1 − n)[²(1 − θ)+θ 2 − 1]²[r(1 + θ)+2θ + r(θ 2 − 1)]ĝ.

9.2. PRICING TO MARKET 305(b) Consider a temporary home government spending shock in whichG s = G 0 =0fors ≥ t + 1, andG t > 0. Show that the effect onthe depreciation and current account are,Ŝ t =(1 + θ)r²[r(1 + θ)+2θ + r(θ 2 − 1)]ĝt,ˆb =−²(1 − n)2θ(1 + r)r²[r(1 + θ)+2θ + r(θ 2 − 1)]ĝt.3. Consider the pricing-to-market model. Show that a permanent increasein home government spending leads to a short-run depreciationof the home currency and a balance of trade deÞcit for the home country.

306CHAPTER 9. THE NEW INTERNATIONAL MACROECONOMICS

Chapter 10Target-Zone ModelsThis chapter covers a class of exchange rate models where the centralbank of a small open economy is, to varying degrees, committed tokeeping the nominal exchange rate within speciÞed limits commonlyreferred to as the target zone. The target-zone framework is sometimesviewed in a different light from a regime of rigidly Þxed exchange ratesin the sense that many target zone commitments allow for a wider rangeof exchange rate variation around a central parity than is the case inexplicit pegging arrangements. In principle, a target-zone arrangementalso requires less frequent central bank intervention for their maintenance.Our analysis focuses on the behavior of the exchange rate whileit is inside the zone.The target-zone analysis has been used extensively to understandexchange rate behavior for European countries that participated in theExchange Rate Mechanism of the European Monetary System duringthe 1980s where ßuctuation margins ranged anywhere from 2.25 percentto 15 percent about a central parity. The adoption of a commoncurrency makes target-zone analysis less applicable for European issues.However, there remain many developing and newly industrialized countriesin Latin America and Asia that occasionally Þx their exchangerates to the dollar for which the analysis is still relevant. Moreover,there may come a time when the Fed and the European Central Bankwill establish an informal target zone for the dollar—euro exchange rate.Target-zone analysis typically works with the monetary model setin a continuous time stochastic environment. Unless noted otherwise,307

308 CHAPTER 10. TARGET-ZONE MODELSall variables except interest rates are in logarithms. The time derivativeof a function x(t) is denoted with the ‘dot’ notation, úx(t) =dx(t)/dt.In order to work with these models, you need some background instochastic calculus.10.1 Fundamentals of Stochastic CalculusLet x(t) be a continuous-time deterministic process that grows at theconstant rate, η such that, dx(t) =ηdt. LetG(x(t),t)besomepossiblytime-dependent continuous and differentiable function of x(t). Fromcalculus, you know that the total differential of G isdG = ∂G∂x dx(t)+∂G dt. (10.1)∂tIf x(t) is a continuous-time stochastic process, however, the formulafor the total differential (10.1) doesn’t work and needs to be modiÞed.In particular, we will be working with a continuous-time stochasticprocess x(t) called a diffusion process where the growth rate of x(t)randomly deviates from η,dx(t) =ηdt + σdz(t). (10.2)ηdt is the expected change in x conditional on information available att, σdz(t) isanerrortermandσ is a scale factor. z(t) is called a Wienerprocess or Brownian motion and it evolves according to,z(t) =u √ t, (10.3)where u iid ∼ N(0, 1). At each instant, z(t) is hit by an independent drawu from the standard normal distribution. InÞnitesimal changes in z(t)can be thought of asdz(t) =z(t + dt) − z(t) =u t+dt√t + dt − ut√t =ũ√dt, (10.4)where u t+dt√t + dt ∼ N(0,t+ dt) andut√t ∼ N(0,t)deÞne the newrandom variable ũ ∼ N(0, 1). 1 The diffusion process is the continuoustimeanalog of the random walk with drift η. Sampling the diffusion1 Since E[u t+dt√t + dt−ut√t] = 0, and Var[ut+dt√t + dt−ut√t]=t+dt−t = dt,u t+dt√t + dt − ut√t deÞnes a new random variable, ũ√dt, whereũiid∼ N(0, 1).

10.1. FUNDAMENTALS OF STOCHASTIC CALCULUS 309x(t) atdiscretepointsintimeyieldsx(t +1)− x(t) =Z t+1tZ t+1dx(s)Z t+1ds + σ= ηdz(s)tt| {z }z(t+1)−z(t)= η + σũ. (10.5)If x(t) follows the diffusion process (10.2), it turns out that the totaldifferential of G(x(t),t)isdG = ∂G∂x dx(t)+∂G ∂tdt +σ22∂ 2 Gdt. (10.6)∂x2 This result is known as Ito’s lemma. The next section gives a nonrigorousderivation of Ito’s lemma and can be skipped by uninterestedreaders.Ito’s LemmaConsider a random variable X with Þnite mean and variance, and apositive number θ > 0. Chebyshev’s inequality says that the probabilitythat X deviates from its mean by more than θ is bounded by its variancedivided by θ 2P{|X − E(X)| ≥ θ} ≤ Var(X) . (10.7)θ 2If z(t) follows the Wiener process (10.3), then E[dz(t)] = 0 andVar[dz(t) 2 ]=E[dz(t) 2 ] − [Edz(t)] 2 = dt. Apply Chebyshev’s inequalityto dz(t) 2 ,togetP {|[dz(t)] 2 − E[dz(t)] 2 | > θ} ≤ (dt)2θ 2 .Since dt is a fraction, as dt → 0, (dt) 2 goes to zero even faster thandt does. Thus the probability that dz(t) 2 deviates from its mean dtbecomes negligible over inÞnitesimal increments of time. This suggests

310 CHAPTER 10. TARGET-ZONE MODELSthat you can treat the deviation of dz(t) 2 from its mean dt as an errorterm of order O(dt 2 ). 2 Write it asdz(t) 2 = dt + O(dt 2 ).Taking a second-order Taylor expansion of G(x(t),t)gives∆G = ∂G∂x ∆x(t)+∂G ∂t ∆t+ 1 "∂ 2 #G2 ∂x 2 ∆x(t)2 + ∂2 G∂t 2 ∆t2 +2 ∂2 G∂x∂t [∆x(t)∆t]+ O(∆t 2 ), (10.8)where O(∆t 2 ) are the ‘higher-ordered’ terms involving (∆t) k with k>2. You can ignore those terms when you send ∆t → 0.If x(t) evolves according to the diffusion process, you know that∆x(t) = η∆t + σ∆z(t), with ∆z(t) = u √ ∆t, and(∆x) 2 = η 2 (∆t) 2 + σ 2 (∆z) 2 +2ησ(∆t)(∆z) =σ 2 ∆t + O(∆t 3/2 ). Substitutethese expressions into the square-bracketed term in (10.8) toget,∆G = ∂G ∂G ∂ 2 G(∆x(t)) +∂x ∂t (∆t)+σ2 2 ∂x 2 (∆t)+O(∆t3/2 ). (10.9)As ∆t → 0, (10.9) goes to (10.6), because the O(∆t 3/2 )termscanbeignored. The result is Ito’s lemma.10.2 The Continuous—Time Monetary ModelA deterministic setting. To see how the monetary model works in continuoustime, we will start in a deterministic setting. As in chapter 3,all variables except interest rates are in logarithms. The money marketequilibrium conditions at home and abroad arem(t) − p(t) = φy(t) − αi(t), (10.10)m ∗ (t) − p ∗ (t) = φy ∗ (t) − αi ∗ (t). (10.11)2 An O(dt 2 ) term divided by dt 2 is constant.

10.2. THE CONTINUOUS—TIME MONETARY MODEL 311International asset-market equilibrium is given by uncovered interestparityi(t) − i ∗ (t) = ús(t). (10.12)The model is completed by invoking PPPCombining (10.10)-(10.13) you gets(t)+p ∗ (t) =p(t). (10.13)s(t) =f(t)+α ús(t), (10.14)where f(t) ≡ m(t) − m ∗ (t) − φ[y(t) − y ∗ (t)] are the monetary-model‘fundamentals.’ Rewrite (10.14) as the Þrst-order differential equationús(t) − s(t)αThe solution to (10.15) is 3Z ∞= −f(t)α . (10.15)s(t) = 1 e (t−x)/α f(x)dxα t= 1 Z ∞α et/α e −x/α f(x)dx. (10.16)A stochastic setting. The stochastic continuous-time monetary modelistm(t) − p(t) = φy(t) − αi(t), (10.17)m ∗ (t) − p ∗ (t) = φy ∗ (t) − αi ∗ (t), (10.18)i(t) − i ∗ (t) = E t [ ús(t)], (10.19)s(t)+p ∗ (t) = p(t). (10.20)3 To verify that (10.16) is a solution, take its time derivativeús(t) = 1 α et/α · ddtZ ∞= − 1 α f(t)+ 1 α 2 et/α Z ∞= − 1 α f(t)+ 1 α s(t)Therefore, (10.16) solves (10.15).t¸ ·Z ∞¸e −x/α f(x)dx + e −x/α f(x)dx α −2 e t/αte −x/α f(x)dxt

312 CHAPTER 10. TARGET-ZONE MODELSCombine (10.17)-(10.20) to getE t [ ús(t)] − s(t)α= −f(t)α , (10.21)which is a Þrst-order stochastic differential equation. To solve (10.21),mimicthestepsusedtosolvethedeterministicmodeltogetthecontinuoustimeversion of the present-value formulaZ ∞s(t) = 1 e (t−x)/α E t [f(x)]dx. (10.22)α tTo evaluate the expectations in (10.22) you must specify the stochasticprocess governing the fundamentals. For this purpose, we assume thatthe fundamentals process follow the diffusion processdf (t) =ηdt + σdz(t), (10.23)where η and σ are constants, and dz(t) =u √ dt is the standard Wienerprocess. It follows thatf(x) − f(t) ==Z xtZ xtdf (r)drηdr +Z x= η(x − t)+σutσdz(r)q(x − t). (10.24)Take expectations of (10.24) conditional on time t information to getthe prediction ruleE t [f(x)] = f(t)+η(x − t), (10.25)and substitute (10.25) into (10.22) to obtains(t) = 1 αZ ∞t⎡⎢= 1 α ⎣ et/α (f − ηt)e (t−x)α [f(t)+η(x − t)]dxZ ∞Z ∞e −x/α dx +ηe t/α xe −x/α dxt| {z }a⎤⎥t⎦| {z }b= αη + f(t), (10.26)

10.3. INFINITESIMAL MARGINAL INTERVENTION 313which follows because the integral in term (a) is R ∞t e −x/α dx = αe −t/αand the integral in term (b) is R ∞t xe −x/α dx = α 2 e −t/α ( t +1). (10.26) isαthe no bubbles solution for the exchange rate under a permanent freeßoatregime where the fundamentals follow the (η, σ)—diffusion process(10.23) and are expected to do so forever on. This is the continuoustimeanalog to the solution obtained in chapter 3 when the fundamentalsfollowed a random walk.10.3 InÞnitesimal Marginal InterventionConsider now a small-open economy whose central bank is committed tokeeping the nominal exchange rate s within the target zone, s

314 CHAPTER 10. TARGET-ZONE MODELSlemmads(t) = dG[f(t)]= G 0 [f(t)]df (t)+ σ22 G00 [f(t)]dt= G 0 [f(t)][ηdt + σdz(t)] + σ22 G00 [f(t)]dt. (10.28)Taking expectations conditioned on time-t information you getE t [ds(t)] = G 0 [f(t)]ηdt + σ22 G00 [f(t)]dt. Dividing this result throughby dt you getE t [ ús(t)] = ηG 0 [f(t)] + σ22 G00 [f(t)]. (10.29)Now substitute (10.27) and (10.29) into the monetary model (10.21)and re-arrange to get the second-order differential equation in GG 00 [f(t)] + 2ησ 2 G0 [f(t)] − 2ασ G[f(t)] = − 2 f(t). 2 ασ2 (10.30)Digression on second-order differential equations. Consider the secondorderdifferential equation,y 00 + a 1 y 0 + a 2 y = bt (10.31)A trial solution to the homogeneous part (y 00 + a 1 y 0 + a 2 y = 0) isy = Ae λt , which implies y 0 = λAe λt and y 00 = λ 2 Ae λt , andAe λt (λ 2 +√a 1 λ + a 2 ) = 0, for which√there are obviously two solutions,λ 1 = −a 1+ a 2 1 −4a 2and λ2 2 = −a 1− a 2 1 −4a 2. If you let y2 1 = Ae λ1t andy 2 = Be λ2t , then clearly, y ∗ = y 1 + y 2 also is a solution because(y ∗ ) 00 + a 1 (y ∗ ) 0 + a 2 (y ∗ )=0.Next, you need to Þnd the particular integral, y p ,whichcanbeobtained by undetermined coefficients. Let y p = β 0 + β 1 t. Thenyp 00 =0,yp 0 = β 1 and yp 00 + a 1 yp 0 + a 2 y p = a 1 β 1 + a 2 β 0 + a 2 β 1 t = bt.It follows that β 1 = ba 2,andβ 0 = − a 1b.a 2 2Since each of these pieces are solutions to (10.31), the sum of thesolutionsisalsobeasolution.Thusthegeneralsolutionis,y(t) =Ae λ1t + Be λ2t − a 1b+ b t. (10.32)a 2 2 a 2

10.3. INFINITESIMAL MARGINAL INTERVENTION 315Solution under Krugman intervention. To solve (10.30), replace y(t) in(10.32) with G(f), set a 1 = 2η , aσ 2 2 = −2 , and b = aασ 2 2 .TheresultiswhereG[f(t)] = ηα + f(t)+Ae λ 1f(t) + Be λ 2f(t) , (10.33)λ 1 = −ηsησ + 22 σ + 2 > 0, (10.34)4 ασ2 λ 2 = −ηsησ − 22 σ + 2 < 0. (10.35)4 ασ2 To solve for the constants A and B, you need two additional pieces of information.These are provided by the intervention rules. 4 From (10.33),you can see that the function mapping f(t) intos(t) is one-to-one. Thismeans that there is a lower and upper band on the fundamentals, [f, ¯f]that corresponds to the lower and upper bands for the exchange rate[s, ¯s]. When s(t) hits the upper band ¯s, the authorities intervene toprevent s(t) from moving out of the band. Only inÞnitesimally smallinterventions are required. During instants of intervention, ds = 0 fromwhich it follows thatG 0 ( ¯f) =1+λ 1 Ae λ 1 ¯f + λ 2 Be λ 2 ¯f =0. (10.36)Similarly, at the instant that s touches the lower band s, ds =0andG 0 (f) =1+λ 1 Ae λ 1f + λ 2 Be λ 2f =0. (10.37)(10.36) and (10.37) are 2 equations in the 2 unknowns A and B, whichyou can solve to getA =B =e λ 2 ¯f − e λ 2fλ 1 [e (λ 1 ¯f+λ 2 f) − e (λ 1f+λ 2 ¯f)]e λ1f − e λ 1 ¯fλ 2 [e (λ 1 ¯f+λ 2 f) − e (λ 1f+λ 2 ¯f)]< 0, (10.38)> 0. (10.39)4 In the case of a pure ßoat and in the absence of bubbles, you know thatA = B =0.

316 CHAPTER 10. TARGET-ZONE MODELSThe signs of A and B follow from noting that λ 1 is positive and λ 2 isnegative so that e λ 1( ¯f−f) >e λ 2( ¯f−f) . It follows that the square bracketedterm in the denominator is positive.The solution becomes simpler if you make two symmetry assumptions.First, assume that there is no drift in the fundamentals η =0.Setting the drift to zero implies λ 1 = −λ 2 = λ > 0. Second, centerthe admissible region for the fundamentals around zero with ¯f = −fso that B = −A >0. The solution becomesG[f(t)] = f(t)+B[e −λf(t) − e λf(t) ], (10.40)withs2λ =ασ , 2e λ ¯f −λ ¯f− eB =λ[e 2λ ¯f − e−2λ ¯f].Figure 10.1 shows the relation between the exchange rate and thefundamentals under Krugman-style intervention. The free ßoat solutions(t) =f(t) serves as a reference point and is given by the dotted 45-degree line. First, notice that G[f(t)] has the shape of an ‘S.’ TheS-curve lies below the s(t) =f(t) line for positive values of f(t) andvice-versa for negative values of f(t). This means that under the targetzonearrangement, the exchange rate varies by a smaller amount inresponse to a given change in f(t) within[f, ¯f] than it would under afree ßoat.Second, note that by (10.21), we know that E( ús) < 0whenf>0,and vice-versa. This means that market participants expect the exchangerate to decline when it lies above its central parity and theyexpecttheexchangeratetorisewhenitliesbelowthecentralparity.The exchange rate displays mean reversion. Thisispotentiallythe explanation for why exchange rates are less volatile under a managedßoat than they are under a free ßoat. Since market participantsexpect the authorities to intervene when the exchange rate heads towardthe bands, the expectation of the future intervention dampenscurrent exchange rate movements. This dampening result is called theHoneymoon effect.

10.3. INFINITESIMAL MARGINAL INTERVENTION 3170.030.02s0.010-0.03 -0.02 -0.02 -0.01 0.00 0.00 0.01 0.01 0.02 0.02 0.03f-0.01G(f)-0.02s=f-0.03Figure 10.1: Relation between exchange rate and fundamentals underpure ßoat and Krugman interventionsEstimating and Testing the Krugman ModelDeJong [36] estimates the Krugman model by maximum likelihood andby simulated method of moments (SMM) using weekly data from January1987 to September 1990. He ends his sample in 1990 so thatexchange rates affected by news or expectations about German reuni-Þcation, which culminated in the European Monetary System crisis ofSeptember 1992, are not included.We will follow De Jong’s SMM estimation strategy to estimate thebasic Krugman model∆f t = η + σu t ,G t = αη + f t + Ae λ 1f t+ Be λ 2f t,where f = − ¯f, the time unit is one day (∆t =1),andu tiid∼ N(0, 1). λ 1and λ 2 are given in (10.34)-(10.35), and A and B are given in (10.38)

318 CHAPTER 10. TARGET-ZONE MODELSand (10.39). The observations are daily DM prices of the Belgian franc,French franc, and Dutch guilder from 2/01/87 to 10/31/90. Log exchangerates are normalized by their central parities and multiplied by100. The parameters to be estimated are (η, α, σ, ¯f). SMM is coveredin Chapter 2.3.Denote the simulated observations with a ‘tilde.’ You need to simulatedsequences of the fundamentals that are guaranteed to stay withinthe bands [f, ¯f]. You can do this by letting ˆf j+1 = ˜f j + η + σu j andsetting˜f j+1 =⎧⎪⎨⎪⎩¯f if ˆfj+1 ≥ ¯fˆf j+1 if f ≤ ˆf j+1 ≤ ¯ff ifˆfj+1 ≤ ffor j =1,...,M. The simulated exchange rates are given by(10.41)˜s j (η, α, σ, ¯f) = ˜f j + αη + Ae λ 1 ˜f j+ Be λ 2 ˜f j, (10.42)the simulated moments by⎡1 P Mj=3⎤∆˜sM j1 PH M [˜s(η, α, σ, ¯f)] Mj=3∆˜s 2 M j=1 P Mj=3 ∆˜s 3 M j⎢ 1 P .Mj=3 ⎥⎣ ∆˜sM j ∆˜s j−1 ⎦1 P Mj=3∆˜sMj ∆˜s j−2ThesamplemomentsarebasedontheÞrst three moments and the Þrsttwo autocovariances⎡1 P Tt=3⎤∆sT t1 P Tt=3∆s 2 T tH t (s) =1 P Tt=3 ∆s 3 T t⎢ 1 P Tt=3 ⎥⎣ ∆sT t ∆s t−1 ⎦1 P Tt=3∆sTt ∆s t−2with M =20T ,whereT =978. 5The results are given in Table 10.1. As you can see, the estimatesare reasonable in magnitude and have the predicted signs, but they arenot very precise. The χ 2 test of the (one) overidentifying restriction isrejected at very small signiÞcance levels indicating that the data areinconsistent with the model.5 No adjustments were made for weekends or holidays.

10.4. DISCRETE INTERVENTION 319Table 10.1: SMM Estimates of Krugman Target-Zone Model (units inpercent) with deutschemark as base currency.η σ α ¯f χ21Currency (s.e.) (s.e.) (s.e.) (s.e.) (p-value)Belgian 0.697 0.865 1.737 2.641 11.672franc (69.01) (83.98) (327.1) (334.3) (0.001)French 0.007 0.117 6.045 2.44 12.395franc (0.318) (1.759) (1590) (67.88) (0.000)Dutch 2.484 2.240 4.152 5.393 11.35guilder (1.317) (0.374) (146.19) (5.235) (0.001)10.4 Discrete InterventionFlood and Garber [56] study a target-zone model where the authoritiesintervene by placing the fundamentals back in the middle of the bandafter one of the bands are hit. If the band width is β = ¯f − f andeither ¯f or f is hit, the central bank intervenes in the foreign exchangemarket by resetting f = ¯f − β/2. Because the intervention producesadiscretejumpinf, the central bank loses foreign exchange reserveswhen ¯f is hit and gains reserves when f is hit.Letting à ≡ Aeλ 1 ¯f and ˜B ≡ Be λ 2f , rewrite the solution (10.33)explicitly as a function of the bands f and ¯fG(f| ¯f,f)=f + αη + Ãeλ 1(f− ¯f) + ˜Be λ 2(f−f) . (10.43)Impose the symmetry conditions, η =0andf = ¯f. It follows thatλ 1 = −λ 2 = λ = q 2/(ασ 2 ) > 0, and ˜B = −à >0. (10.43) can be ⇐(215)written asG(f|f, ¯f) =f + ˜B h e −λ(f−f) − e −λ( ¯f−f) i . (10.44)Under the symmetry assumptions you need only one extra side-conditionto determine ˜B. We get it by looking at the exchange rate at the instantt 0 that f(t) hits the upper band ¯fs(t 0 )=G[ ¯f|f, ¯f] = ¯f + ˜B[e −λβ − 1]. (10.45)

320 CHAPTER 10. TARGET-ZONE MODELSMarket participants know that at the next instant the authorities willreset f = 0. It follows thatE t0 s(t 0 + dt) =s(t 0 + dt) =G[0|f, ¯f] =0. (10.46)To maintain international capital market equilibrium, uncovered interestparity must hold at t 0 . 6 The expected depreciation at t 0 mustbe Þnite which means there can be no jumps in the time-path of theexchange rate. It follows thatlim s(t 0 + ∆t) =s(t 0 ),∆t→0which implies s(t 0 )=s(t 0 + dt) = 0. Adopt a normalization by settings(t 0 ) = 0 in (10.45). It follows that˜B =−β2[e −λβ − 1] .But if s(t 0 + dt) =G(0|f, ¯f) =0ands(t 0 )=G( ¯f|f, ¯f) =0,thenthere are at least two values of f that give the same value of s so the G—function is not one-to-one. In fact, the G—function attains its extremabefore f reaches f or ¯f and behaves like a parabola near the bands asshown in Figure 10.2.As f(t) approaches ¯f, it becomes increasingly likely that the centralbank will reset the exchange rate to its central parity. This informationis incorporated into market participant’s expectations. When f issufficiently close to ¯f this expectational effect dominates and furthermovements of f towards ¯f results in a decline in the exchange rate. Forgiven variation in the fundamentals within [f, ¯f], the exchange rate underFlood-Garber intervention exhibits even less volatility than it doesunder Krugman intervention.10.5 Eventual CollapseThe target zone can be maintained indeÞnitely under Krugman-styleinterventions because reserve loss or gain is inÞnitesimal. Any Þxed6 If it does not, there will be an unexploited and unbounded expected proÞtopportunity that is inconsistent with international capital market equilibrium.

10.5. EVENTUAL COLLAPSE 3210.03s0.02s=f0.010-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025-0.01G(f)f-0.02-0.03Figure 10.2: Exchange rate and fundamentals under Flood—Garber discreteinterventionsexchange rate regime operating under a discrete intervention rule, onthe other hand, must eventually collapse. The central bank begins theregime with a Þnite amount of reserves which is eventually exhausted.This is a variant of the gambler’s ruin problem. 7The problem that confronts the central bank goes like this. Supposethe authorities begin with foreign exchange reserves of R dollars. Itloses one dollar each time ¯f is hit and gains one dollar each time f ishit. After the intervention, f is placed back in the middle of the [f, ¯f]band, where it evolves according to the driftless diffusion df (t) =σdz(t)until another intervention is required.Let L be the event that central bank eventually runs out of reserves,G be the event that it gains $1 on a particular intervention and G c be7 See Degroot [37].

322 CHAPTER 10. TARGET-ZONE MODELSthe event that it loses a dollar on a particular intervention. 8 In theÞrst round, the probability that f hits ¯f is 1 2 .Thatis,P(Gc )= 1.By2implication, P(G) =1− P(G c )= 1 . It follows that before the Þrst2round starts, the probability that reserves eventually get driven to zeroisPr(L) = 1 2 Pr(L|G)+1 2 Pr(L|Gc ). (10.47)(10.47) true before the Þrst round and is true for any round as long asthe authorities still have at least one dollar in reserves.Let p j be the conditional probability that reserves eventually become0 given that the current level of reserves is j-dollars. For any j ≥ 1,(10.47) can be expressed as the difference equationp j = 1 2 p j+1 + 1 2 p j−1, (10.48)with p 0 =1. 9 Backward substitution gives p 2 =2p 1 − 1, p 3 =3p 1 − 2,p k = kp 1 − (k − 1),...,orequivalently,fork ≥ 2,p k =1− k(1 − p 1 ). (10.49)Since p k is a probability, it cannot exceed 1. Upon rearrangement yougetp 1 =1+ p kk − 1 → 1, as k →∞. (10.50)kbut if p 1 = 1, the recursion in (10.49) says that for any j ≥ 1, p j =1.Translation? It is a sure thing that any Þnite amount of reserves willeventually be exhausted.10.6 Imperfect Target-Zone CredibilityThediscreteinterventionruleismorerealisticthantheinÞnitesimalmarginal intervention rule. But if reserves run out with probability 1,8 G is the event that f hits f, andG c is the event that f hits ¯f.9 Clearly, p 0 = 1 since if j = 0, reserves have been exhausted. If j = 1, thereis a probability of 1 2that reserves are exhausted on the next intervention and aprobability of 1 that the central bank gains a dollar and survives to play again2at which time there will be a probability of p 2 that reserves will eventually beexhausted. That is, for j = 1, p 1 = 1 2 p 0 + 1 2 p 2. Continuingoninthisway,youget(10.48).

10.6. IMPERFECT TARGET-ZONE CREDIBILITY 323there will come a time in any target-zone arrangement when it is nolonger worthwhile for the authorities to continue to defend the zone.This means that the target-zone bands cannot always be completelycredible. In fact, during the twelve years or so that the Exchange RateMechanism of the European Monetary System operated reasonably well(1979—1992), there were eleven realignments of the bands. It would bestrange to think that a zone would be completely credible given thatthere is already a history of realignments.We now modify the target-zone analysis to allow for imperfect credibilityalong the lines of Bertola and Caballero [8]. Let the bands for thefundamentals be [f, ¯f] andletβ = ¯f − f be the width of the band. Ifthe fundamentals reach the lower band, there is a probability p that theauthorities re-align and a probability 1 − p that the authorities defendthe zone.If re-alignment occurs, what used to be the lower band of the oldzone f, becomes the upper band of the new zone [f −β,f]. The realignmentis a discrete intervention that sets f = f − β/2 at the midpointof the new band. If a defense is mounted, the fundamentals are returnedto the midpoint, f = f + β/2. An analogous set of possibilitiesdescribe the intervention choices if the fundamentals reach the upperband. Figure 10.3 illustrates the intervention possibilities.⇐(217)RealignDefendf - > f - > f f + >f >Figure 10.3: Bertola-Caballero realignment and defense possibilities.We begin with the symmetric exchange rate solution (10.44) withη = 0 and an initial symmetric target zone about 0 where f = − ¯f,

324 CHAPTER 10. TARGET-ZONE MODELSλ 1 = −λ 2 = λ = q 2/(ασ 2 ) > 0, and ˜B = −Ã >0.(218)⇒To determine ˜B, suppose that f hits the upper band ¯f at time t 0 .Thens(t 0 )=G( ¯f|f, ¯f) = ¯f + ˜B(e −λβ − 1). (10.51)At the next instant t 0 + dt, the authorities either realign or defends(t 0 + dt) =(G( ¯f + β/2| ¯f, ¯f + β) = ¯f +β2w.p. pG( ¯f − β/2|f, ¯f) = ¯f − β 2w.p. 1 − p.(10.52)To maintain uncovered interest parity at the point of intervention, marketparticipants must not expect jumps in the exchange rate. It followsthat, lim ∆t→0 E t0 s(t 0 +∆t) =s t0 . Using (10.52) to evaluate E t0 s(t 0 +dt)and equating to s(t 0 )givesp"¯f + β 2#+(1− p)"¯f − β 2#= ¯f + ˜B(e −λβ − 1),and solving for ˜B gives˜B = (2p − 1) β 2). (10.53)(e −λβ − 1This solution is a striking contrast to the solution under Krugmaninterventions. ˜B is negative if the target zone lacks sufficient credibility(p > 1 ). This means that the exchange rate solution is an inverted ‘Scurve’.The exchange rate under the discrete intervention rule combined2with low defense credibility is even more volatile than what it would beunder a free ßoat.

10.6. IMPERFECT TARGET-ZONE CREDIBILITY 325Target-zone Summary1. The theory covered in this chapter was based on the monetarymodel where today’s exchange rate depends in part on marketparticipant’s expectations of the future exchange rate. Under atarget zone, these expectations depend on the position of the exchangerate within the zone. As the exchange rate moves fartheraway from the central parity, intervention that manipulates theexchange rate becomes increasingly likely and the expectationof this intervention feeds back into the current value of s(t).2. When the fundamentals follows a diffusion process forf

326 CHAPTER 10. TARGET-ZONE MODELS

Chapter 11Balance of Payments CrisesIn chapter 10 we argued that there is a presumption that any Þxedexchange rate regime must eventually collapse–a presumption that thedata supports. Britain and the U.S. were forced off of the gold standardduring WWI and the Great Depression. More recent collapsesoccurred in the face of crushing speculative attacks on central bank reserves.Some well-known foreign exchange crises include the breakdownof the 1946—1971 IMF system of Þxed but adjustable exchange rates,Mexico and Argentina during the 1970s and early 1980s, the EuropeanMonetary System in 1992, Mexico in 1994, and the Asian Crisis of 1997.Evidently, no ÞxedexchangerateregimehasevertrulybeenÞxed.This chapter covers models of the causes and the timing of currencycrises. We begin with what Flood and Marion [57] call Þrst generationmodels. This class of models, developed to explain balance of paymentscrises experienced by developing countries during the 1970s and1980s. These crises were often preceded by unsustainably large governmentÞscal deÞcits, Þnanced by excessive domestic credit creationthat eventually exhausted the central bank’s foreign exchange reserves.Consequently, Þrst-generation models emphasize macroeconomic mismanagementas the primary cause of the crisis. They suggest that thesizeofacountry’sÞnancial liabilities (the government’s Þscal deÞcit,shorttermdebtandthecurrentaccountdeÞcit) relative to its short runability to pay (foreign exchange reserves) and/or a sustained real appreciationfrom domestic price level inßation should signal an increasinglikelihood of a crisis.327

328 CHAPTER 11. BALANCE OF PAYMENTS CRISESIn more recent experience such as the European Monetary Systemcrisis of 1992 or the Asian crisis of 1997, few of the affected countriesappeared to be victims of macroeconomic mismanagement. Thesecrises seemed to occur independently of the macroeconomic fundamentalsand do not Þt intothemoldoftheÞrst generation models. Secondgenerationmodels were developed to understand these phenomenon. Inthese models, the government explicitly balances the costs of defendingtheexchangerateagainstthebeneÞts of realignment. The government’sdecision rule gives rise to multiple equilibria in which the costsof exchange rate defense depend on the public’s expectations. A shiftin the public’s expectations can alter the government’s cost-beneÞt calculationresulting in a shift from an equilibrium with a low-probabilityof devaluation to one with a high-probability of devaluation. Becausean ensuing crisis is made more likely by changing public opinion, thesemodels are also referred to as models of self-fulÞlling crises.11.1 A First-Generation ModelIn Þrst-generation models, the government exogeneously pursues Þscaland monetary policies that are inconsistent with the long-run maintenanceof a Þxed exchange rate. One way to motivate governmentbehavior of this sort is to argue that the government faces short-termdomestic Þnancing constraints that it feels are more important to satisfythan long-run maintenance of external balance. While this is nota completely satisfactory way to model the actions of the authorities,it allows us to focus on the behavior of speculators and their role ingenerating a crisis.Speculators observe the decline of the central bank’s internationalreserves and time a speculative attack in which they acquire the remainingreserves in an instant. Faced with the loss of all of its foreignexchange reserves, the central bank is forced to abandon the peg andto move to a free ßoat. The speculative attack on the central bank atduring the Þnal moments of the peg is called a balance of paymentsor a foreign exchange crisis. The original contribution is due to Krugman[89]. We’ll study the linear version of that model developed byFlood and Garber [55].

11.1. A FIRST-GENERATION MODEL 329Flood—Garber Deterministic CrisesThe model is based on the deterministic, continuous-time monetarymodel of a small open economy of Chapter 10.2. All variables exceptfor the interest rate are expressed as logarithms–m(t) is the domesticmoney supply, p(t) the price level, i(t) the nominal interest rate, d(t)domestic credit, and r(t) the home-currency value of foreign exchangereserves. From the log-linearization of the central bank’s balance sheetidentity, the log money supply can be decomposed asm(t) =γd(t)+(1− γ)r(t). (11.1)Domestic income is assumed to be Þxed. We normalize units suchthat y(t) =y = 0. The money market equilibrium condition ism(t) − p(t) =−αi(t). (11.2)The model is completed by invoking purchasing-power parity and uncoveredinterest paritys(t) = p(t), (11.3)i(t) = E t [ ús(t)] = ús(t), (11.4)where we have set the exogenous log foreign price level and the exogenousforeign interest rate both to zero p ∗ = i ∗ = 0. Combine (11.2)—(11.4) to obtain the differential equation, ⇐(219)m(t) − s(t) =−α ús(t) (11.5)The authorities establish a Þxedexchangerateregimeatt =0bypegging the exchange rate at its t = 0 equilibrium value, ¯s = m(0).During the time that the Þx isineffect, ús(t) = 0. By (11.5), theauthorities must maintain a Þxed money supply at m(t) =¯s to defendthe exchange rate.Suppose that the domestic credit component grows at the rated(t) ú =µ. The government may do this because it lacks an adequatetax base and money creation is the only way to pay for governmentspending. But keeping the money supply Þxed in the face of expandingdomestic credit means reserves must decline at the rateúr(t) =−γ d(t)1 − γ ú = −µγ1 − γ . (11.6)

330 CHAPTER 11. BALANCE OF PAYMENTS CRISESClearly this policy is inconsistent with the long-run maintenance of theÞxed exchange rate since the government will eventually run out offoreign exchange reserves.Non-attack exhaustion of reserves. If reserves are permitted to declineat the rate in (11.6) without interruption, it is straightforward to determinethe time t N at which they will be exhausted. Reserves at anytime 0

11.1. A FIRST-GENERATION MODEL 331MoneyMoneyd(0)Domestic creditr(0)Reserves0 t A t NtimeFigure 11.1: Time-path of monetary aggregates under the Þx anditscollapse.˜s(t). They will not attack if ¯s >˜s. But if ¯s

332 CHAPTER 11. BALANCE OF PAYMENTS CRISESWhen reserves are exhausted, r(t) =0, and the money supply becomesm(t) =γd(t) =γ[d(0) +Substitute m(t) into(11.8)togetZ t0ú d(u)du] =γ[d(0) + µt].˜s(t) =γ[d(0) + µt]+αγµ. (11.9)Setting ˜s(t A )=¯s = m(0) = γd(0) + (1 − γ)r(0) and solving for thetime of attack givest A =(1 − γ)r(0)γµ− α = t N − α. (11.10)The level of reserves at the point of attack isr(t A )=r(0) − µγ1 − γ t A = µαγ1 − γ> 0. (11.11)Figure 11.1 illustrates the time-path of money and its componentswhen there is an attack. One of the key features of the model is thatepisodes of large asset market volatility, namely the attack, does notcoincide with big news or corresponding large events. The attack comessuddenly but is the rational response of speculators to the accumulatedeffects of domestic credit creation that is inconsistent with the Þxedexchange rate in the long run.One dissatisfying feature of the deterministic model is that the attackis perfectly predictable. Another feature is that there is no transferof wealth. In actual crises, the attacks are largely unpredictable andtypically result in sizable transfers of wealth from the central bank (withcosts ultimately borne by taxpayers) to speculators.AstochasticÞrst-generation model.Let’s now extend the Flood and Garber model to a stochastic environment.We will not be able to solve for the date of attack but wecan model the conditional probability of an attack. In discrete time,

11.1. A FIRST-GENERATION MODEL 333let the economic environment be given bym t = γd t +(1− γ)r t , (11.12)m t − p t = −αi t , (11.13)p t = s t , (11.14)i t = E t (∆s t+1 ). (11.15)Let domestic credit be governed by the random walkd t =(µ − 1 λ )+d t−1 + v t , (11.16)where v t is drawn from the exponential distribution. 2 . Also, assumethat the domestic credit process has an upward drift µ > 1/λ. Attime t, agents attack the central bank if ˜s t ≥ ¯s, where˜s is the shadowexchange rate.Let the publicly available information set be I t and let p t be theprobability of an attack at t + 1 conditional on I t . Then,p t = Pr[˜s t+1 > ¯s|I t ]= Pr[αγµ + m t+1 − ¯s >0|I t ]= Pr[αγµ + γd t+1 − ¯s >0|I t ]· µ ·= Pr αγµ + γ d t + µ − 1 ¸ + v t+1 − ¯s >0|I t¸λ= Pr"v t+1 > 1 γ ¯s − (1 + α)µ − d t + 1 λ |I t#= Pr(v t+1 > θ t |I t )=Z ∞θ tλe −λu du =(e−λθ tθ t ≥ 01 θ t < 0(11.17)where θ t ≡ (1/γ)¯s − (1 − α)µ − d t +(1/λ). The rational exchange rateforecast error isE t s t+1 − ¯s = p t [E t (˜s t+1 ) − ¯s], (11.18)and is systematic if p t > 0.2 A random variable X has the exponential distribution if for x ≥ 0, f(x) =λe −λx . The mean of the distribution is E(X) =1/λ.

334 CHAPTER 11. BALANCE OF PAYMENTS CRISESThus there will be a peso problem as long as the Þx isineffect. By(11.17), we know how p t behaves. Now let’s characterize E t (˜s t+1 )andthe forecast errors. First note thatE t (˜s t+1 ) = αγµ + γE t (d t+1 )= αγµE t·µ − 1 λ + d t + v t+1¸= αγµ + µ − 1 λ + d t +E t (v t+1 ). (11.19)(222)⇒E t (v t+1 ) is computed conditional on a collapse next period which willoccur if v t+1 > θ t . To Þnd the probability density function of v conditionalon a collapse, normalize the density of v such that the probabilitythat v t+1 > θ t is 1 by solving for the normalizing constant φ in1=φ R ∞θ tλe −λu du. This yields φ = e λθt . It follows that the probabilitydensity conditional on a collapse next period isf(u|collapse) =(λeλ(θ t −u)u ≥ θ t ≥ 0λe −λu θ t < 0,andE t (v t+1 )=( R ∞θ tuλe λ(θt−u) du = θ t + 1 λθ t ≥ 0R ∞0 uλe −λu du = 1 λθ t < 0 . (11.20)Now substitute (11.20) into (11.19) and simplify to obtainE t (˜s t+1 )=(¯s +γλθ t ≥ 0(1 + α)γµ + γd t θ t < 0 . (11.21)Substituting (11.21) into (11.18) you get the systematic but rationalforecast errors predicted by the modelE t (s t+1 ) − ¯s =( ptγλθ t ≥ 0(1 + α)γµ + γd t − ¯s θ t < 0 . (11.22)

11.2. A SECOND GENERATION MODEL 33511.2 A Second Generation ModelIn Þrst-generation models, exogenous domestic credit expansion causesinternational reserves to decline in order to maintain a constant moneysupply that is consistent with the Þxed exchange rate. A key featureof second generation models is that they explicitly account for the policyoptions available to the authorities. To defend the exchange rate,the government may have to borrow foreign exchange reserves, raise domesticinterest rates, reduce the budget deÞcit and/or impose exchangecontrols. Exchange rate defense is therefore costly. The government’swillingness to bear these costs depend in part on the state of the economy.Whether the economy is in the good state or in the bad statein turn depends on the public’s expectations. The government engagesin a cost-beneÞt calculation to decide whether to defend the exchangerate or to realign.We will study the canonical second generation model due to Obstfeld[112]. In this model, the government’s decision rule is nonlinear andleads to multiple (two) equilibria. One equilibrium has low probabilityof devaluation whereas the other has a high probability. The costs tothe authorities of maintaining the Þxedexchangeratedependonthepublic’s expectations of future policy. An exogenous event that changesthe public’s expectations can therefore raise the government’s assessmentof the cost of exchange rate maintenance leading to a switch fromthe low-probability of devaluation equilibrium to the high-probabilityof devaluation equilibrium.What sorts of market-sentiment shifting events are we talking about?Obstfeld offers several examples that may have altered public expectationsprior to the 1992 EMS crisis: The rejection by the Danish publicof the Maastrict Treaty in June 1992, a sharp rise in Swedish unemployment,and various public announcements by authorities that suggesteda weakening resolve to defend the exchange rate. In regard tothe Asian crisis, expectations may have shifted as information aboutover-expansion in Thai real-estate investment and poor investment allocationof Korean Chaebol came to light.

336 CHAPTER 11. BALANCE OF PAYMENTS CRISESObstfeld’s Multiple Devaluation Threshold ModelAll variables are in logarithms. Let p t be the domestic price level ands t be the nominal exchange rate. Set the (log) of the exogenous foreignprice level to zero and assume PPP, p t = s t . Output is given by aquasi-labor demand schedule which varies inversely with the real wageiidw t − s t , and with a shock u t ∼ N(0, σu)2y t = −α(w t − s t ) − u t . (11.23)Firms and workers agree to a rule whereby today’s wage was negotiatedand set one-period in advance so as to keep the ex ante real wageconstantw t =E t−1 (s t ). (11.24)Optimal Exchange Rate ManagementWe Þrst study the model where the government actively manages, butdoes not actually Þx the exchange rate. The authorities are assumedto have direct control over the current-period exchange rate.The policy maker seeks to minimize costs arising from two sources.The Þrst cost is incurred when an output target is missed. Notice that(11.23) says that the natural output level is E t−1 (y t )=0. Weassumethat there exists an entrenched but unspeciÞed labor market distortionthat prevents the natural level of output from reaching the sociallyefficient level. These distortions create an incentive for the governmentto try to raise output towards the efficient level. The government setsa target level of output ȳ>0. When it misses the output target, itbears a cost of (ȳ − y t ) 2 /2 > 0.The second cost is incurred when there is inßation. Under PPPwith the foreign price level Þxed, the domestic inßation rate is thedepreciation rate of the home currency, δ t ≡ s t − s t−1 . Together, policyerrors generate current costs for the policy maker `t, accordingtothequadratic loss function`t = θ 2 (δ t) 2 + 1 2 [ȳ − y t] 2 . (11.25)Presumably, it is the public’ desire to minimize (11.25) which it achievesby electing officials to fulÞll its wishes.

11.2. A SECOND GENERATION MODEL 337The static problem is the only feasible problem. Inanidealworld,the government would like to choose current and future values of theexchange rate to minimize the expected present value of future costs⇐(225)E t ∞ Xj=0β j`t+j ,where β < 1 is a discount factor. The problem is that this opportunityis not available to the government because there is no way that theauthorities can credibly commit themselves to pre-announced futureactions. Future values of s t are therefore not part of the government’scurrent choice set. The problem that is within the government’s abilityto solve is to choose s t each period to minimize (11.25), subject to(11.24) and (11.23). This boils down to a sequence of static problemsso we omit the time subscript from this point on.Let s 0 be yesterday’s exchange rate and E 0 (s) be the public’s expectationof today’s exchange rate formed yesterday. The government Þrstobserves today’s wage w =E 0 (s), and today’s shock u, then choosestoday’s exchange rate s to minimize ` in (11.25). The optimal exchangeratemanagement rule is obtained by substituting y from (11.23) into(11.25), differentiating with respect to s and setting the result to zero.Upon rearrangement, you get the government’s reaction functions = s 0 + α θ[α(w − s)+ȳ + u] . (11.26)Notice that the government’s choice of s depends on yesterday’s predictionof s by the public since w =E 0 (s). Since the public knows thatthe government follows (11.26), they also know that their forecasts ofthe future exchange rate partly determine the future exchange rate. Tosolve for the equilibrium wage rate, w =E 0 (s), take expectations of(11.26) to getw = s 0 + αȳθ . (11.27)Tocutdownonthenotation,letλ =α2θ + α 2 .

338 CHAPTER 11. BALANCE OF PAYMENTS CRISESNow, you can get the rational expectations equilibrium depreciationrate by substituting (11.27) into (11.26)⇐(226)δ = αȳθ + λuα . (11.28)The equilibrium depreciation rate exhibits a systematic bias as a resultof the output distortion ȳ. 3 . The government has an incentive to sety =ȳ. Seeing that today’s nominal wage is predetermined, it attemptsto exploit this temporary rigidity to move output closer to its targetvalue. The problem is that the public knows that the government willdo this and they take this behavior into account in setting the wage.The result is that the government’s behavior causes the public to set awage that is higher than it would set otherwise.Fixed Exchange RatesThe foregoing is an analysis of a managed ßoat. Now, we introducea reason for the government to Þx the exchange rate. Assume that inaddition to the costs associated with policy errors given in (11.25), thegovernment pays a penalty for adjusting the exchange rate. Where doesthis cost come from? Perhaps there are distributional effects associatedwith exchange rate changes where the losers seek retribution on thepolicy maker. The groups harmed in a revaluation may differ fromthose harmed in a devaluation so we want to allow for differential costsassociated with devaluation and revaluation. 4 So let c d be the costassociated with a devaluation and c r be the cost associated with arevaluation. The modiÞed current-period loss function is` = θ 2 (δ)2 + 1 2 (ȳ − y)2 + c d z d + c r z r , (11.29)where z d =1ifδ > 0andis0otherwise,andz r =1ifδ < 0 and is zerootherwise. We also assume that the central bank either has sufficient3 This is the inßationary bias that arises in Barro and Gordon’s [7] model ofmonetary policy4 Devaluation is an increase in s which results in a lower foreign exchange valueof the domestic currency. Revaluation is a decrease in s, which raises the foreignexchange value of the domestic currency.

11.2. A SECOND GENERATION MODEL 339reserves to mount a successful defense or has access to sufficient linesof credit for that purpose.The government now faces a binary choice problem. After observingthe output shock u and the wage w it can either maintain the Þx orrealign. To decide the appropriate course of action, compute the costsassociated with each choice and take the low-cost route.Maintenance costs. Suppose the exchange rate is Þxed at s 0 . Theexpected rate of depreciation is δ e =E 0 (s) − s 0 . If the governmentmaintains the Þx, adjustment costs are c d = c r = 0, and the depreciationrate is δ = 0. Substituting real wage w − s 0 = δ e and outputy = −αδ e − u into (11.29) gives the cost to the policy maker of maintainingthe Þx`M = 1 2 [αδe +ȳ + u] 2 . (11.30)Realignment Costs. If the government realigns, it does so according tothe optimal realignment rule (11.26) with a devaluation given byδ = α [α(w − s)+ȳ + u]. (11.31)θAdd and subtract (α 2 /θ)s 0 to the right side of (11.31). Noting thatδ e = w − s 0 and collecting terms givesδ = λ α [αδe +ȳ + u] . (11.32)Equating (11.31) and (11.32) you get the real wagew − s = θδe − α(ȳ + u). (11.33)α 2 + θSubstitute (11.33) into (11.23) to get the deviation of output from thetargetȳ − y =θθ + α 2 [αδe +ȳ + u] . (11.34)Substitute (11.32) and (11.34) into (11.29) to get the cost of realignment⎧⎨`R =⎩θ2(θ+α 2 ) [αδe +ȳ + u] 2 + c d if u>0θ2(θ+α 2 ) [αδe +ȳ + u] 2 + c r if u

340 CHAPTER 11. BALANCE OF PAYMENTS CRISESRealignment rule. A realignment will be triggered if `R 0and2c d > λ[αδ e +ȳ + u] 2 . It will andrevalue if u λ[αδ e +ȳ + u] 2 . The rule can be writtenmore compactly asλ[αδ e +ȳ + u] 2 > 2c k , (11.36)where k = d if u>0andk = r if uū triggers adevaluation. Write (11.36) as an equality, set c k = c d , and solve forthe roots of the equation. You are looking for the positive devaluationtrigger point so ignore the negative root because it is irrelevant. Thepositive root issū = −αδ e 2cd− ȳ +λ . (11.37)Now do the same for negative realizations of u, and throw away thepositive root. The lower trigger point issu = −αδ e 2cd− ȳ −λ . (11.38)The points [u, ū] are those that trigger the escape option. Realizationsof u in the band [u, ū] result in maintenance of the Þxed exchange rate.Figure 11.2 shows the attack points for δ e =0.03 with ȳ =0.01, α =1,θ =0.15, c r = c d =0.0004.

11.2. A SECOND GENERATION MODEL 3410.0160.0140.0120.010.0080.0060.004u0.002u-0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03-0.0020Figure 11.2: Realignment thresholds for given δ e .Multiple trigger points for devaluation.u and ū depend on δ e . But the public also forms its expectationsconditional on the devaluation trigger points. This means that u, ūand δ e must be solved simultaneously.To simplify matters, we restrict attention to the case where the governmentmay either defend the Þx ordevalue the currency. Revaluationis not an option. We therefore focus on the devaluation threshold ū.We will set c r to be a very large number to rule out the possibility of arevaluation. The central bank’s devaluation rule is(δ0 =0 if uū . (11.39)Let P[X = x] be the probability of the event X = x. The expecteddepreciation isδ e = E 0 (δ)

342 CHAPTER 11. BALANCE OF PAYMENTS CRISES= P[δ = δ 0 ]δ 0 +P[δ = δ 1 ]E[(λ/α)(αδ e +ȳ +E(u|u >ū))]= P[u>ū](λ/α)[αδ e +ȳ +E(u|u >ū)].Solving for δ e as a function of ū yieldsδ e =λP(u >ū)1 − λP(u >ū)1[ȳ +E(u|u >ū)] . (11.40)αTo proceed further, you need to assume a probability law governing theoutput shocks, u.Uniformly distributed output shocks. Let u be uniformly distributed onthe interval [−a, a]. The probability density function of u isf(u) =1/(2a) for −a ū) =1/(a − ū). It follows thatP(u >ū) =Z aūZ a(1/(2a))dx =(a − ū), (11.41)2a(a +ū)E(u|u >ū) = x/(a − ū)dx = . (11.42)ū2Substituting (11.41) and (11.42) into (11.40) givesδ e = f δ (ū) =λ(a − ū)2αa⎛⎝ȳ + a+ū21 − λ(a−ū)2a⎞⎠ . (11.43)Notice that δ e involves the square terms ū 2 . Quadratic equations usuallyhave two solutions. Substituting δ e into (11.37) givesū = −αf δ (ū) − ȳ +s2cdλ , (11.44)(229)⇒where f δ (ū) isdeÞned in (11.43). (11.44) has two solutions for ū, eachof which trigger a devaluation. For parameter values a =0.03, θ =0.15,c =0.0004, α =1,ȳ =0.01 solving (11.44) yields the two solutionsū 1 = −0.0209 and ū 2 =0.0030. (11.44) is displayed in Figure 11.3 forthese parameter values.Using (11.43), the public’s expected depreciation associated with ū 1is 2.7 percent whereas δ e associated with ū 2 is 45 percent. The high

11.2. A SECOND GENERATION MODEL 3430.020.0150.010.005-0.03 -0.02 -0.01 0.00 0.01 0.020-0.005Figure 11.3: Multiple equilibria devaluation thresholds.expected inßation (high δ e ) gets set into wages and the resulting wageinßation increases the pain from unemployment and makes devaluationmore likely. Devaluation is therefore more likely under the equilibriumthreshold ū 2 than ū 1 . When perceptions switch the economy to ū 2 ,theauthorities require a very favorable output shock in order to maintainthe exchange rate.There is not enough information in the model for us to say whichof the equilibrium thresholds the economy settles on. The model onlysuggests that random events can shift us from one equilibrium to another,moving from one where devaluation is viewed as unlikely to onein which it is more certain. Then, a relatively small output shock cansuddenly trigger a speculative attack and subsequent devaluation.

344 CHAPTER 11. BALANCE OF PAYMENTS CRISESBalance of Payments Crises Summary1. A Þxed exchange rate regime will eventually collapse. The resultis typically a balance of payments or currency crisis characterizedby substantial Þnancial market volatility and large losses offoreign exchange reserves by the central bank.2. Prior to the 1990s, crises were seen mainly to be the result ofbad macroeconomic management–policies choices that were inconsistentwith the long-run maintenance of the exchange rate.First-generation models focused on predicting when a crisismight occur. These models suggest that macroeconomic fundamentalssuch as the budget deÞcit, the current account deÞcitand external debt relative to the stock of international reservesshould have predictive content for future crises.3. Second-generation models are models of self-fulÞlling criseswhich endogenize government policy making and emphasize theinteraction between the authorities’s decisions and the public’sexpectations. Sudden shifts in market sentiment can weaken thegovernment’s willingness to maintain the exchange rate whichthereby triggers a crisis.

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354 CHAPTER 11. BALANCE OF PAYMENTS CRISES106. Modjtahedi B. 1991. “Multiple Maturities and Time-VaryingRisk Premia in Forward Exchange Markets.” Journal Of InternationalEconomics 30: pp. 69-86107. Hinkle, Lawrence E. and Peter J. Montiel. 1999. Exchange RateMisalignment: Concepts and Measurement for Developing Countries.Oxford: Oxford University Press.108. Mundell, Robert A. 1963. “Capital Mobility and StabilizationPolicy under Fixed and Flexible Exchange Rates.” CanadianJournal of Economics and Political Science 29: pp. 475-85.109. Mussa, Michael. 1976. “The Exchange Rate, the Balance ofPayments, and Monetary and Fiscal Policy Under a Regime ofControlled Floating.” Scandinavian Journal of Economics 78:pp. 229-48.110. Mussa, Michael. 1986. “Nominal Exchange Rate Regimes and theBehavior of Real Exchange Rates: Evidence and Implications.”Carnegie Rochester Conference Series on Public Policy 25: pp.117—213.111. Obstfeld, Maurice. 1985. “Floating Exchange Rates: Experienceand Prospects.” Brookings Papers on Economic Activity 2: pp.369—450.112. Obstfeld, Maurice. 1994. “The Logic of Currency Crises.” CahiersEconomiques et Monetaires Bank of France: pp.189—213.113. Obstfeld, Maurice and Kenneth Rogoff. 1995. “Exchange RateDynamics Redux.” Journal of Political Economy 103: pp.624—660.114. Newey, Whitney and Kenneth D. West. 1987. “A Simple, PositiveSemi-deÞnite Heteroskedasticity and Autocorrelation ConsistentCovariance Matrix.” Econometrica 55: pp. 703-8.115. Newey, Whitney and Kenneth D. West. 1994. “Automatic LagSelection in Covariance Matrix Estimation.” Review of EconomicStudies 61: pp. 631-53.116. Nickell, Stephen J. 1981. “Biases in Dynamic Models with FixedEffects.” Econometrica 49: pp. 1417-1426.

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Author IndexAAkaike, H., 27Andrews, D., 40Arrow, K.J., 107BBackus, D., 144, 149, 183Barro, R.J., 336Bekaert, G., 183Bertola, G., 321Betts, C., 264, 286Beveridge, S., 47Bhargava, A., 43, 44Black, F., 194Blanchard, O., 251Blough, S.R., 50Bowman, D., 62Burnside, C., 126CCaballero, R.J., 321Campbell, J.Y., 54, 92Canzoneri, M.B., 224Cassel, G., 80Cavaglia, S., 184, 185Cecchetti, S.G., 126, 204Chinn,M.D.,184,185Choi, C.Y., 57Christiano L.J., 218Clarida, R., 252Cochrane, J.H., 48,50Cole, H.L., 112, 135Cooley, T., 31, 125, 144Cumby, R.E., 224DDavidson, R., 45De Long, J.B., 194Debreu, G., 105Degroot, M.H., 319DeJong, F., 315Devereux, M.B., 264, 286Diba, B., 224Dornbusch, R., 229, 237Duffie, D., 39EEichenbaum, M., 218, 249Engel, C.M., 172, 203, 211, 226Evans, C., 249, 193F357

358 AUTHOR INDEXFama, E.F., 161, 167,168Faust, J., 50Federal Reserve Bank of New York,2Feenstra, R.C., 285Fisher, R.A., 61Fleming, J.M., 229Flood, R.P., 317, 325, 326Frankel, J.A., 184, 185, 223Frenkel, J.A., 6, 80, 83Friedman, M., 83Froot,K.,184,185GGali, J., 252Garber, P.M., 317, 326Gordon, D.B., 336Gregory, A., 183, 126Grilli, V., 221HHall, A., 54Hamilton, J.D., 23, 45Hansen,L.P.,35,163,177,179Hatanaka, M., 23Hinkle, L.E., 229Hodrick, R.J., 75, 138, 163, 172Huizinga, J., 221IIm, K.S., 51, 61Ingram, B. F., 39Isard, P., 209JJaganathan, R., 179Johansen, S., 23, 62Johnson, H.G., 83KKaminsky, G., 204, 221Kehoe, P.J., 144, 149Kendall, M.G., 57King, R.G., 78, 139, 144Knetter, M.M., 285Krasker, W., 187Krugman, P.R., 311, 326Kydland, F.E., 144, 149LLee, B.S., 39LeRoy, S, 31, 92Levich, R.M., 6Levin, A., 51, 55Lewis, K.K., 188Lin, C.F., 51, 55Lothian, J.R., 220Lucas, R.E., 105MacDonal, R., 92, 223, 229MacKinnon, J.G., 45Maddala, G.S., 51, 61Marion, N., 325Mark, N.C., 97, 98, 172, 177,193McCallum, B.T., 203Meese, R., 97Mehra, R., 179Modjtahedi, B., 177

AUTHOR INDEX 359Montiel, P.J., 229Mundell, R.A., 229NNelson, C.R., 47Newey, W., 37, 38, 39, 54Nickell, S.J., 57OObstfeld, M., 112, 135, 229, 241,264, 333PPapell, D., 58, 218, 223, 249Pederson, T.M., 7Perron, P, 46 54Peruga, R., 204Pesaran, M.H., 51, 61Phillips, P.C.B., 46Plosser, C.I., 139, 144Porter R.D., 92Prescott, E., 75, 125, 138, 144,179QQuah, D., 251SSamuelson, P.A., 82Sarno, L., 62Schwarz, G., 25, 26Schwert, G.W., 63Shiller, R.J., 92Shin, Y., 51, 61Siegel, J., 203Sims, C.A., 249Singleton, K.J., 39, 177Smith, G.W., 126Stein, J., 229Sul, D., 97, 98TTaylor, M.P., 8, 63, 92, 220Telmer, C.I., 183Theil, H., 100Theodoridis, H., 218WWest, K.D., 37, 38, 39, 54Williamson J., 229Wu, S., 51, 61Wu, Y., 58, 172, 193, 223RRebelo, S., 78, 139, 144Rogers, J.H., 211Rogoff, K., 97, 264Rose, A.K., 223

Subject IndexAAbsorption, 16, 230AIC, 25Approximate solutionPricing-to-market model, 292—294Redux model, 274—275Real business cycle model,144—148, 154—158Arrow-Debreu model, 105, 137Ask price, 6Asymptotic distribution, 24Augmented Dickey-Fuller regression,46Autocovariance generating function,73—74Autoregressive process, 24BBalance of paymentsAccounts 18—20Subaccounts, 19Capital account, 19Current account, 19Official settlements account,19TransactionsCredit transactions, 18Debit transaction, 18Balanced growth path, 141Balassa-Samuelson model (begin),214—217Bartlett window, 37Bayes’ Rule, 191Beggar-thy-neighbor policy, 234,384, 300, 301Bias of estimator of ρKendall adjustment univariatecase, 57Nickell adjustment for paneldata, 57BIC, 26Bid price, 6Bootstrap, 29, 58—59Nonparametric, 29Panel unit-root test, 59Panel study of monetarymodel, 98Parametric,Impulse response standarderrors, 29Panel unit-root test, 58Residual bootstrap, 58Brownian motion (Wiener process),306360

SUBJECT INDEX 361CCalibration method, 125Lucas model, 126—132One-country real business cyclemodel, 144—149Two-country real business cyclemodel, 149—158Cash-in-advanceTransactions technology, 113Constraint, 115Causal priority, 26Central limit theorem, 36Certainty equivalence, 147Chebyshev’s inequality, 307Choleski upper triangular decomposition,28Cobb-DouglasConsumption index, 112Price level, 213Production function,140, 151,214Cointegration, 63—67Monetary model, 92Common trend processes, 64Common-time effect in panel unitroot tests, 53Companion form, 43,93Complete markets, 119Complex conjugate, 72Constant elasticity of substitutionindex, 265Constant relative risk aversionutility, 112, 117, 122Convergence, in distribution, 24Convergence, in probability, 23Covariance decomposition, 174Covariance stationarity, 24Asymptotic distribution of OLSestimator of ρ, 42Process, 24Time-series behavior, 41Covered interest parity, 5, 162,197Neutral-band tests, 6-9Crowding out, 236Current account, 17DData generating process for thebootstrap, 58Deep parameters, 125Devaluation, 232, 336, 337, 339Differential equationFirst-order, 309Linear homogeneous, 257Second order, 312Differentiated goods, 264Diffusion process, 306Disequilibrium dynamicsStochastic Mundell-Flemingmodel, 246—248Domestic credit, 231, 234, 251,325, 327, 331, 333Extended by central bank,20Dornbusch model, 237—241EEconometric exogeneity, 26, 33Economic signiÞcance, 217Effective labor units, 139

362 SUBJECT INDEXEfficient capital market, 161Equity premium puzzle, 179Error-components representation,52Escape clause, 338Euler equationsLucas model, 108—109, 111—112, 116, 120, 122Pricing-to-market model, 288Redux model, 271Eurocurrency, 4Excess exchange rate volatility,88Exchange Rate Mechanism of EuropeanMonetary System,305Exchange rate quotationAmerican terms, 2, 79European terms, 2Expected speculative proÞtEstimation of, 170—172Expenditure switching effect, 230,235, 245, 284, 299, 301External balance, 261FFiltering time-series, 67—78Linear Þlters (begin range 1),74—78Hodrick-Prescott Þlter, 75—78Removing non-cyclical components,138First-generation models, 326—332Fisher equation, 268Fisher test—See Maddala—Wu testForeign exchange reserves of centralbank, 20Forward exchange rateLucas model, 123Transactions (outright), 3Contract maturities, 4premium and discount, 4Forward premium bias—see UncoveredInterest Parity,Deviations fromFundamentals traders, 197Fundamentals, economic, 85Futures contracts, 12—15Margin account, 12Yen contract for June 1999delivery, 14Long position, 12Settlement, 12Short position, 12GGambler’s ruin problem, 319Generalized method of moments,35—38, 164Asymptotic distribution, 38Long-run covariance matrix,37Testing Euler equations, 177—179Tests of overidentifying restrictions,38HHalf-life to convergence, 43Real exchange rate, 218

SUBJECT INDEX 363Hall’s general-to-speciÞc method,54Hansen-Hodrick problem, 163Hodrick-Prescott Þlter, 75—79In real business-cycle research,138—141, 148—149, 151,158Honeymoon effect, 314IImperfect competition, 263Imperfectly credible target zones,320—322Inßation premium, 117Inßationary bias, 336Instrumental variables, 37International transmission of shocksMundell-Fleming modelFixed exchange rates, 233Flexible exchange rates, 236Pricing-to-market model, 299Redux model, 284Intertemporal marginal rate ofsubstitution, 173Of money, 176InterventionForeign exchange market, 20Sterilized, 21Unsterilized, 21, 311Ito’s lemma, 307—308JJensen’s inequality, 195KKronecker product, 178LLaw of iterated expectations, 86,93Law-of-one price,Deviations from, 209—212Learning period in peso-problemmodel, 192Liquidity effect, 240, 281, 284,299Log utility, 139, 141, 142Lognormal distribution, 178, 204London Interbank Offer Rate (LI-BOR), 4Long position (exposure), 7Long-horizon panel data regression,97Lucas model, 105—132Risk premium, 172—177MMarkov chain (begin range), 126,133—134Transition matrix, 127, 129,133Markup, price over costs, 290Mean reversion, 49, 314Mean-variance optimizers, 11Measurement in calibrationLucas model, 126Real business cycles, 141, 150,151

364 SUBJECT INDEXMethod of undetermined coefficients,312, 329Dornbusch model, 239Krugman target-zone model,311Lewis peso-problem model,188Stochastic Mundell-Fleming,243Noise-trader model, 201Moment generating function, 10Monetary base, 20Monetary model, 79—103Excess exchange rate volatility,90—92Exchange-rate determination,84—100Long-horizon panel regression,100Of the balance of payments,83—84Peso-problem analysis, 188Target-zone analysisDeterministic continuous time,308—309Stochastic continuous time,309—311Testing present-value form,95Monetary neutralityDornbusch model, 239Pricing to market, 398Redux model, 289Money in the utility function, 265,287Money-demand function, 85Monopolistically competitive Þrm,285, 290Monopoly distortion, 284Moving-average process, 24Moving-average representation ofunit-root time series, 43Multiple equilibria, 341Mundell-Fleming modelStochastic, 241—248Static, 229—236NNational income accounting, 15—18Closed economy, 16Current account, 17Open economy, 16Near observational equivalence inunit-root tests, 50Net foreign asset position, 17Newey-West long-run covariancematrix, 37Noise-Trader Model, 193—202Nominal bond priceLucas model, 118, 123Nontraded goods, 135Normal distribution with meanµ and variance σ 2 ,23OOrdinary least squares (OLS)Asymptotic distribution, 36Out-of-sample prediction and Þt,97Overlapping generations model,194

SUBJECT INDEX 365Overlapping observations, advantagesof, 165Overshooting exchange ratePricing to market, 298Redux model, 279Delayed, 250Dornbusch model, 240Structural VAR, 254Stochastic Mundell-Flemingmodel, 248Overshooting terms of trade, Reduxmodel, 280PPareto optimum—see Social optimumPass through, 266, 285Perfect foresight, 147, 238Permanent—transitory componentsrepresentation of unit-rootprocess, 46—48, 102Peso problem, 186—193, 205, 332Phase diagram, 260Political risk, 5Posterior probabilities, 192Power of a statistical test, 50Present-value formula, 86, 310Pricing nominal contracts, pitfalls,176Pricing to market, 285—301Prior probability, 192Process switching, 311Purchasing-power parity, 80—83,207-208Monetary model, 85Absolute, 209Cassel’s view (begin), 80—81Commodity arbitrage view,82Pricing-to-market model, 292Redux model, 268Relative, 209QQuadratic formula, 147Quadratic optimization problem,145Quantity equation, 117, 121RRational bubble, 87Reaction function, 335Real business cycle model, 137—158One-sector closed economy,139—149Two-country model, 149—159Reduced form, 31Regime shift, 189Representative agent, 106Revaluation, 234, 336Risk aversion, 10, 202Constant absolute, 10Insurance demand, 175Required in Lucas model tomatch data, 182Risk premium compensation,175Risk neutralForward exchange, Lucas model,175

366 SUBJECT INDEXRisk neutrality, 9, 176, 203Risk pooling equilbrium, 112, 117Risk premiumAs explanation of deviationfrom uncovered interestparity, 10Lucas model, 174Lucas state contingent, 127,130Risk sharing, efficient conditions,111SSaddle path solution, 259S-curve, 315, 321Second-generation model, 333—341Segmented goods markets internationally,285Self-fulÞlling crises, 342Shadow prices, 113Short position (exposure), 6Siegel’s paradox, 203SigniÞcance, Statistical versus economic,217Simulated method of moments,38—40Simulated method of momentsAsymptotic distribution, 40Estimating the Krugman model,315Simulating one-country real businesscycle model, 148Simulating two-country real businesscycle model, 158Small country assumption, 229Social Optimum, 105, 109—111Social planner’s problem, 110, 141,142Spectral density function, 73, 75Spectral representationCycles and periodicity, 68Of time series, 68—74Phase shift, 68Spot transactions, 3Static expectations, 230Steady stateOne-country real business cyclemodel, 143—144Pricing-to-market model, 292Redux model, 272—274Two-country real business cyclemodel, 153—154Sticky price adjustment ruleDornbusch model, 238Stochastic Mundell-Flemingmodel, 242Sticky price models, 229—302Stochastic calculus, 306—308Stochastic processContinuous time, 306Diffusion, 306Stochastic trend process, 64Stochastic-difference equationFirst-order in the monetarymodel, 85Nonlinear in real business cyclemodel, 143Second-order in real businesscycle model, 144, 156Strict exogeneity, 33Studentized coefficient, 44Survey expectations, 184—186

SUBJECT INDEX 367Swap transactions, 3TTarget zone, eventual collapse,318Technical change, deterministic,139Technological growthBiased, 216Unbiased, 216Technology shock, 140Termsoftrade,277,280,300Tranquil peg, 7Transversality condition, 86Trend-cycle components decomposition,138Triangular arbitrage, 3Triangular structure of exogenousshocks, 246Trigonometric relations, 6971Turbulent peg, 7UUncovered interest parity, 9—11Deviations from , 162—172Fama decomposition, 167—170Hansen-Hodrick tests, 164—165Monetary model, 85Uniform distribution from a tob, 23Unit rootAnalyses of time series, 40—67Univariate test procedures,44—50Dickey—Fuller test, 45Augmented Dickey-Fuller test,46Finite sample power, Dickey-Fuller test, 52Bhargava framework, 44—45Testing for PPP, 217—222Panel data, 51—63Cross-sectional dependence,53, 58Im-Pesaran-Shin test, 60,225Levin—Lin test, 52—57, 223Maddala—Wu test, 61—62,224Potential pitfalls, 62Testing for PPP, 222Size distortion, Levin-Lintest, 56Size distortion, tests of PPP,226VVariance ratio statistic, 50, 221Vector autoregressionUnrestricted, 24—34Asymptotic distribution ofcoefficient vector, 25Cooley—Leroy critique, 31—34Decomposition of forecasterrorvariance, 29Eichenbaum-Evans 5 variableVAR, 249—250

368 SUBJECT INDEXImpulse response analysis,27Impulse-response standarderrors by parametric bootstrap,29Orthogonalizing the innovations,28StructuralClarida-Gali SVAR (begin),251—256Vector error correction model, 65—67Monetary model, 93Vehicle currency, 2Volatility bounds, 179—183Volatility of exchange rates, stockprices, 89WWelfare analysis, Redux model,282—284Wiener process (Brownian motion),306Wold vector moving-average representation,26, 28