Testing for Speculative Bubbles in Stock Markets A ... - Econometrics

Testing for Speculative Bubbles in Stock MarketsA Comparison of Alternative MethodsUlrich HommJörg BreitungUniversity of BonnAdenauerallee 24-4253113 BonnGermanye-mail: uhomm@uni-bonn.dephone: +49 (0)228 73 39 25University of BonnAdenauerallee 24-4253113 BonnGermanye-mail: breitung@uni-bonn.dephone: +49 (0)228 73 92 011

AbstractIn this paper we investigate the power properties of various test for rational bubbles.We focus on the case where bubble detection is reduced to testing for a changefrom a random walk to an explosive process, where the change point is unknown.We adapt several test procedures, that were originally proposed to test for a changein persistence. In our simulations a Chow type break test exhibits the highest power.We also compare different estimators for the unknown break date and suggest monitoringprocedures for detecting speculative bubbles in real time. As applications weanalyze the Nasdaq and the Hang Seng index.KEYWORDS: speculative bubbles, unit-root test, structural breakJEL: G10, C22IntroductionPhenomena of speculative excesses have long been present in economic history. Galbraith(1993) starts his account with the famous Tulipomania, which took place in the Netherlandsin the 17th century. More recently, the so called dot.com or IT -bubble came to fameduring the end of the 1990s. Enormous increases in stock prices followed by crashes haveled many researchers to test for the presence of speculative bubbles. Among these areShiller (1981) and LeRoy and Porter (1981), who proposed variance bounds tests, West(1987), who designed a two-step test for bubbles, and Froot and Obstfeld (1991), whoconsidered intrinsic bubbles. Moreover, Cuñado, Gil-Alana, and Gracia (2004) and Kruse(2008) employ methods based on fractionally integrated models, while Phillips, Wu, andYu (2009) use sequential unit-root tests. This list is by no means complete. Gürkaynak(2008) provides an overview of different empirical tests on rational bubbles. Here, weadopt the theoretical framework of Phillips et al. (2009) and propose several other testprocedures aiming to improve on the testing power. The forward recursive unit-root testof Phillips et al. (2009) is an attempt to overcome the weaknesses of the approach of Dibaand Grossman (1988), who argue against the existence of bubbles in the S&P 500. Evans(1991) demonstrates that Diba and Grossman’s (1988) tests do not have sufficient powerto effectively detect bubbles that collapse periodically. Phillips et al. (2009) also proposeto use sequences of Dickey-Fuller statistics to estimate the date of the emergence of abubble, i.e. to estimate the date of a regime switch from a random walk to an explosiveprocess.There is a wide literature on tests for a change in persistence of a time series. Forinstance, Kim (2000) and Busetti and Taylor (2004) proposed procedures to test thenull that the time series is I(0) throughout the entire sample against the alternative of achange from I(0) to I(1), or vice versa. In this paper we study the power properties ofthese tests in a different context, the detection of rational bubbles, and compare them to2

the procedure proposed by Phillips et al. (2009). Additionally, two other tests will beincluded in the study. One is based on Bhargava (1986), who tested whether a time seriesis I(1) against an explosive alternative. Bhargava (1986) did not construct his test as atest for a regime switch. However, we will tentatively adapt this test to a regime-switchcontext by applying it sequentially to different subsamples. The other test is a versionof the classical Chow-test. Somewhat unusual, this test will be applied in the context ofunit-root processes. Along with different methods to test for the existence of bubbles weconsider alternative methods to identify the starting date of the bubble.When there is only one regime switch in the sample, the proposed sequential Chowtestand Busetti and Taylor’s (2004) procedure exhibit the highest power. Moreover, abreakpoint estimator derived from the Chow-test turns out to be most accurate. However,this does no longer hold, if there are multiple regime changes due to bubble crashes. Inthat case it is more appropriate to make a slight change of perspective and conceive thebubble tests as monitoring procedures. Basically all procedures could be redesigned asmonitoring procedures, but here we will focus on the sequential Dickey-Fuller test andpropose a simple CUSUM procedure.This paper is organized as follows. In section 1 we present the basic theory of rationalbubbles. In sections 2 and 3, the above mentioned tests and estimation procedures areintroduced. Section 4 presents different models of rational bubbles. We introduce a simplemodel for randomly starting bubbles and review Evans’ (1991) periodically collapsingbubbles. These two models are part of our Monte Carlo analysis in section 5. In section 6we consider monitoring procedures and analyze their properties. Finally, in section 7, thetest and estimation procedures are applied to Nasdaq and Hang Seng index data. Section8 concludes.1 Rational BubblesSpeculative bubbles in stock markets are systematic departures from the fundamentalprice of an asset. Following Blanchard and Watson (1982) or Campbell, Lo and MacKinlay(1997) the fundamental price of the asset is derived from the present value of all expectedpayouts. The return of the stock is defined asR t+1 = P t+1 + D t+1P t− 1, (1)where P t denotes the stock price at period t and D t+1 is the the dividend for the periodt. Assuming risk neutrality, no arbitrage opportunities and a constant expected return(E [R t ] = R for all t), the stock price is obtained asP t = E t [P t+1 + D t+1 ], (2)1 + R3

where E t [·] denotes the expectation conditional on the information at time t. Equation(2) can be solved by forward iteration yieldingP ftP ft =∞∑i=11(1 + R) i E t [D t+i ] . (3)is often referred to as the fundamental stock price. Imposing the transversality condition[]lim E 1k→∞ (1 + R) P k t+k = 0 (4)the (unique) solution of the difference equation is P t = P ft , that is, the existence ofthe bubble is ruled out. Equation (3) states that the fundamental price is equal tothe present value of all expected dividend payments. In this simple framework, the onlyfundamentals determining the stock price are expected dividend payments and the interestrate. However, if (4) does not hold, P ft is not the only price process that solves (2).Consider a process {B t } ∞ t=1with the propertyE t [B t+1 ] = (1 + R)B t . (5)It can easily be verified that adding B t to P t will yield another solution to equation (2).In fact, there are infinitely many solutions to (2). They take the formP t = P ft + B t , (6)where {B t } ∞ t=1is a process that satisfies equation (4). The last equation decomposes theprice into two components: the fundamental component, P ft , and a part that is commonlyreferred to as the bubble component, B t . If a bubble is present in the stock price, equation(5) requires that any rational investor, who is willing to buy that stock, must expect thebubble to grow at a rate that equals the interest rate R. If this is the case and if B tis strictly positive, this builds the stage for speculative investor behavior: A rationalinvestor is willing to buy an “overpriced” stock, since she believes that through priceincreases she will be sufficiently compensated for the extra payment B t . If the bubblecomponent constitutes a large part of the price, then the expectation that it will increaseat rate R means that investors expect price increases that are unrelated to changes infundamentals. If enough investors have this expectation and buy shares, the stock pricewill indeed go up and complete a loop of a self-fulfilling prophecy.Since the fundamental value is not observed, assumptions have to be imposed tocharacterize the time series properties of the fundamental price P ft . A convenient – andnevertheless empirically plausible – assumption is that dividends follow a random walkwith driftD t = µ + D t−1 + u t , (7)where u t is a white noise process. Under this assumption the fundamental price results asP ft= 1 + RR 2 µ + 1 R D t, (8)4

(e.g. Evans 1991). Consequently,if D t follows a random walk with drift, so does P ft . Thisallows us to distinguish the fundamental price from the bubble process that is characterizedby an explosive autoregressive process. Diba and Grossman (1988) argue as follows:In the absence of bubbles observed prices are equal to fundamental prices and are henceintegrated of the same order as dividends (see (3) and (8)). On the other hand, if apositive bubble was present, this would cause the stock price to explode. No matter howoften prices are differenced, the resulting series will not be stationary. Thus, if pricesexhibit explosive behavior while dividends are integrated of order 1, say, this indicatesthe presence of a bubble.2 Test ProceduresConjecturing the presence of a bubble in the NASDAQ, figure 4 in section 7 suggests thatthe explosive feature of the bubble does not dominate the price from the beginning. Itindicates a structural change from a random walk to an explosive process in the mid 90sand then another switch back to a random walk around the year 2000.We start with the case of a single regime switch, where it assumed that the bubbledoes not crash before the end of the sample. The test procedures are based on the timevarying AR(1) modely t = ρ t y t−1 + ɛ t , (9)where ε t is a white noise process with E(ε t ) = 0, E(ε t ) 2 = σ 2 and y 0 = c < ∞. To simplifythe exposition we ignore a possible constant in the autoregression. If the test is appliedto a series of daily stock prices, the constant is usually very small and insignificant. Toaccount for a possible constant in (9), the series may be detrended by running a leastsquaresregression on a constant and a linear time trend. All test statistics presented inthis section are then computed by using the residuals of this regression instead of theoriginal time series. 1Under the null hypothesis y t follows a random walk for all time periods, i.e.,H 0 : ρ t = 1 for t = 1, 2, . . . , T. (10)Under the alternative hypothesis the process starts as a random walk but changes to anexplosive process at an unknown time [τ ∗ T ] (where τ ∗ ∈ (0, 1) and [τ ∗ T ] denotes thegreatest integer smaller than or equal to τ ∗ T ):⎧⎨ 1 for t = 1, . . . , [τ ∗ T ]H 1 : ρ t =(11)⎩ ρ∗ > 1 for t = [τ ∗ T ] + 1, . . . , T.Various statistics have been suggested to test for a structural break in the autoregressiveparameter. Most of the work focus on a change from a nonstationary regime (i.e. ρ t = 1)1 In that case, Brownian motions are replaced by detrended Brownian motions in the limiting distributionof the test statistics (see below).5

to a stationary regime (ρ t < 1) or vice versa. Since these test statistics can easily adaptedto the situation of a change from a I(1) to an explosive process, we first consider varioustest statistics suggested in the literature.a) The Bhargava statisticTo test the null hypothesis of a random walk (ρ t = 1) against explosive alternativesρ t = ρ∗ > 1 for all t = 1, . . . , T , Bhargava 1986 proposed the locally most powerfulinvariant test statistic∑ TB0 ∗ t=1=(y t − y t−1 ) 2∑ Tt=1 (y t − y 0 ) .2Since Bhargava’s (1986) alternative does not incorporate a structural break we will use amodified version of this test statistic. Lets 2 τ =1T − [τT ]T∑t=[τT ]+1(y t − y t−1 ) 2 .To test for a change from I(1) to an explosive process in the symmetric interval τ ∈[τ 0 , 1 − τ 0 ], where τ 0 ∈ (0, 0.5), we consider the statisticsupB(τ 0 ) =sup B τ , where B τ =τ∈[τ 0 ,1−τ 0 ]1s 2 τ (T − [τT ]) 2T∑t=[τT ]+1(y t − y [τT ] ) 2 . (12)Essentially, B τ is the inverse of the statistic B0 ∗ applied to the subsample {y [τT ] , . . . , y T }and divided by T − [τT ]. The statistic may be motivated as follows. Assume that wewant to forecast the value y T ∗ +h at period T ∗ = [τT ]. Since it is assumed that the seriesis a random walk up to period T ∗ the forecast results as ŷ T ∗ +h|T ∗ = y T ∗. The Bhargavastatistic is based on the sum of squared forecast errors for y T ∗ +1, . . . , y T . If the secondpart of the sample is generated by an explosive process, then the random walk forecastbecomes very poor as h gets large. Therefore, this test statistic is supposed to have goodpower against explosive alternatives.The supremum of the statistics B τ is used to cope with the fact that the breakpointis unknown. The asymptotic distribution of this test statistic under null hypothesis wasnot derived in the literature but easily follows from the continuous mapping theorem assupB(τ 0 ) ⇒where ⇒ denotes weak convergence.supτ∈[τ 0 ,1−τ 0 ]{∫1 1−τ(1 − τ) 20}W 2 (r)dr ,b) The Busetti-Taylor statisticBusetti and Taylor (2004) proposed a statistic for testing the hypothesis that a timeseries is stationary against the alternative that it switches from a stationary to an I(1)6

process at an unknown breakpoint. Here we propose a similar test statistic to test thenull hypothesis (10) against the alternative (11): 2supBT (τ 0 ) =sup BT τ , where BT τ =τ∈[τ 0 ,1−τ 0 ]1s 2 0(T − [τT ]) 2T∑t=[τT ]+1(y T − y t−1 ) 2 . (13)Note that BT τ employs the variance estimator s 2 0 based on the entire sample, while theBhargava statistic s 2 τ uses only the observations starting at [τT ]. Another way to illustratethe difference between the two test statistics is to note that the BT statistic is based onthe sum of squared forecast errors of forecasting the final value y T from the periodsy T ∗ + 1, . . . , y T −1 by using the null hypothesis that y t is generated by a random walk.Therefore the BT statistic fixes the target to be forecasted, whereas the Bhargava statisticuses multiple forecast horizons of a fixed forecast interval. Intuitively, one would expectthat the statistic BT τ is more affected by an explosive behavior than B τ . The followingresult for the limiting distribution of supBT can easily be derived,∫ 1}sup BT τ ⇒ sup{(1 − τ) −2 W 2 (1 − r)dr .τ∈[τ 0 ,1−τ 0 ]τ∈[τ 0 ,1−τ 0 ]c) The Kim statisticAnother statistic for testing the I(0) null hypothesis against a change from I(0) to I(1)was proposed by Kim (2000). We will use a slightly modified version of his test statisticyielding:supK(τ 0 ) =supτ∈[τ 0 ,1−τ 0 ]K τ with K τ = (T − [τT ])−2 ∑ Tt=[τT ]+1 (y t − y [τT ] ) 2τ[τT ] −2 ∑ [τT ]t=1 (y t − y 0 ) 2 . (14)The statistic K τ can be interpreted as the scaled ratio of the sum of squared forecast errors.The prediction is made under the assumption that the time series follows a random walk.y 0 is used to forecast y 1 , . . . , y [τT ] (denominator) and y [τT ] is the forecast of y [τT ]+1 , . . . , y T .The limiting distribution is obtained as{ ( ) 2∫ 1τsup K τ ⇒ supW }2 (r − τ)drτ∫ ττ∈[τ 0 ,1−τ 0 ]τ∈[τ 0 ,1−τ 0 ] 1 − τ W .0 2 (r) drd) The Phillips et al. statisticIn the context of (9) to (11), Phillips et al. (2009) suggest to use a sequence of Dickey-Fuller (DF) tests. Let ˆρ τ denote the OLS estimator of ρ and ˆσ ρ,τ the usual estimator forthe standard deviation of ˆρ τ using the subsample {y 1 , . . . , y [τT ] }. 3 The forward recursive2 In their work Busetti and Taylor (2004) considered the process y t = β 0 + µ t + ɛ t , where β 0 is aconstant, ɛ t ∼ iid N(0, σ 2 ), and µ t is a process that is subject to a possible switch from I(0) to I(1).They proposed the statistic ϕ(τ) = ˆσ −2 (T − [τT ]) −2 ∑ Tt=[τT ]+1( ∑Tj=t ˆɛ j) 2where ˆσ 2 = 1 T∑ Tt=1 ˆɛ2 t andˆɛ t are the residuals from OLS-regression of y t on intercept. In the bubble testing framework consideredhere, it seems reasonable to replace ˆɛ t by y t − y t−1 , which leads to (13).3 In their paper, Phillips, Wu, and Yu (2009) apply augmented Dickey-Fuller tests and use a constantin their regression.7

Dickey-Fuller test of H 0 against H 1 is given bysupDF (τ 0 ) =sup DF τ with DF τ = ˆρ τ − 1. (15)τ 0 ≤τ≤1ˆσ ρτNote that the DF statistic is computed for the asymmetric interval [τ 0 , 1]. In simulationexperiments τ 0 will be set to 0.2, in order to start with a sample fraction of reasonablesize. The limiting distribution derived by Phillips at al. (2009) issup DF τ ⇒τ 0 ≤τ≤1supτ 0 ≤τ≤1∫ τW dW 0√ ∫ .τW 0 2e) A Chow-type unit root statistic for a structural breakThe test procedure of Phillips et al. (2009) does not take into account that both underthe null hypothesis given in (10) and under the alternative given in (11) y t is a randomwalk for t = 1, . . . , [τ ∗ T ]. This information about the data generating process can beincorporated in the test procedure by using a Chow-test for a structural break in theautoregressive parameter. Under the assumption that ρ t = 1 for t = 1, . . . , [τT ] andρ t − 1 = δ > 0 for t = [τT ] + 1, . . . , T , the model can be written as∆y t = δ (y t−1 1 {t>[τT ]} ) + ε t , (16)where 1 {·} is an indicator function that equals one when the statement in braces is trueand equals zero otherwise. Correspondingly, the null hypothesis of interest is H 0 : δ = 0which is tested against the alternative H 1 : δ > 0. It is easy to see that the regressiont-statistic for this null hypothesis isDF C τ =T∑∆y t y t−1t=[τT ]+1˜σ τ√T∑yt−12t=[τT ]+1,where˜σ 2 τ = 1T − 2T∑ (∆y t − ̂δ) 2τ y t−1 1 {t>[τT ]}t=2and ̂δ τ denotes the OLS estimator of δ in (16). The Chow-type Dickey-Fuller statistic totest for a change from I(1) to explosive in the interval τ ∈ [τ 0 , 1 − τ 0 ] can be written asStraight forward derivation yields:supDF C(τ 0 ) ⇒supDF C =sup DF C τ . (17)τ∈[τ 0 ,1−τ 0 ]1sup[{W 2 (1) − W 2 (τ) − (1 − τ)}]2√τ∈[τ 0 ,1−τ 0 ] ∫ .1W τ 2 (r)dr8

In finite samples, the critical values for both, the supDFC and the supDF statistic,depend on the initial value of the time series, if the series is not detrended. Therefore,whenever a hypothesis about a time series {y t } T t=0 is tested without detrending, thesestatistics are applied to the transformed series {ỹ t } T t=0 with ỹ t = y t − y 0 .3 Estimation of the break dateAssume that the time series under consideration, {y t } T t=0, can be described by (9) and (11),where τ ∗ is unknown. First, consider the approach that has been proposed by Phillipset al. (2009). In a simple version, the estimate for the starting date of the bubble, ˆτ P ,is given by the smallest value of τ ∈ [τ 0 , 1] for which DF τ is larger than the right-tail5% critical value derived from the asymptotic distribution of the standard Dickey-Fullert-statistic. Therefore, the estimator results as 4ˆτ P = infτ≥τ 0{τ : DF τ > 1.28}. (18)The next estimator for a change point was proposed by Busetti and Taylor (2002). 5It supplements their test for the existence of a structural change from I(0) to I(1). Theidea is to maximize the ratio of the sample variances of the first and second subsample.Intuitively, for the correct break date, the difference between the variance of the secondsubsample (which is assumed to be I(1) under the alternative) and the first sample (whichis assumed to be I(0)) should be maximal. This idea can be adapted for estimating thedate of a change from an I(1) to an explosive process. The estimator results asˆτ BT = argmax τ∈[˜τ,1−˜τ] Λ(τ), where Λ(τ) =(T − [τT ]) −2 T∑[τT∑][τT ] −2∆yt2t [ τT ]+1∆yt2t=1. (19)Finally, we consider an estimator for the break point that is directly related to the supDFCtest. The estimator is given byˆτ DF C = argmax τ∈[˜τ,1−˜τ] DF C τ , (20)where DF C τ is as in equation (17). The idea of this estimator is related to that inLeybourne, Kim, Smith and Newbold (2003). These authors consider the case of a changefrom I(0) to I(1) and propose a consistent estimator for the unknown break point. Notethat this estimator also maximizes the likelihood function with respect to the break date.Bai and Perron (1998) have shown that the ML estimator for the break date is consistent.4 To achieve consistency the significance level has to go to zero as the sample size goes to infinity. Intheir application Phillips et al. (2009) employ log(log(τT ))/100 as critical values. For a sample size ofT = 400 this roughly corresponds to a 4% significance level. The results of our simulation experimentsremain basically unchanged, if these critical values are used.5 As U. Hassler pointed out, Kim et al. (2002) proposed a very similar estimator with Λ(τ) in (19)replaced by [τT ] −1 Λ(τ). In our Monte Carlo analysis we find no noteworthy difference in the performanceof these two estimators. For brevity, we present only the estimator of Busetti and Taylor (2002).9

4 Models for rational bubblesThe simplest example of a process that satisfies (5) is the deterministic bubble, given byB t = (1 + R) t B 0 , where B 0 is an initial value. A somewhat more realistic example, inwhich the bubble does not necessarily grow forever, is taken from Blanchard and Watson(1982). The bubble process is given by⎧⎨π −1 (1 + R)B t + µ t+1 , with probability πB t+1 =(21)⎩µ t+1 , with probability 1 − πwhere {µ t } ∞ t=1 is a sequence of iid random variables with zero mean. In each period,the bubble described in equation (21) will continue, with probability π, or collapse withprobability 1 − π. As long as the bubble does not collapse, the realized return exceedsthe interest rate R as a compensation for the risk that the bubble bursts.Not every process that satisfies (4) is consistent with rationality. For instance, giventhat stock prices cannot be negative, negative bubbles can be excluded: Applying the lawof iterated expectations to (4) yields E t [B t+τ ] = (1 + R) τ B t . If at some time t, B t wasnegative, then, as τ goes to infinity, the expected bubble size would tend towards minusinfinity, implying a negative expected stock price at some future time period. Furthermore,Diba and Grossman (1998) argue that a bubble process cannot start from zero. Assumethat B t = 0 at some time t. Then E t [B t+1 ] = 0, by (4). Assuming nonnegative stockprices, we have B t+1 ≥ 0. Together this implies that B t+1 = 0 almost surely.Although rational bubbles cannot start from zero, they can take a constant positive valuefor some time and start to grow exponentially with some probability π. For instance,consider the following randomly starting bubble:⎧⎨B t−1 + RB t−1θπ t , if B t−1 = B 0B t =for t = 1, . . . , T, (22)⎩(1 + R)B t−1 , if B t−1 > B 0where B 0 is the initial value of the bubble. R is the interest rate and {θ} T t=1is an exogenousiid Bernoulli process with P rob(θ t = 1) = π = 1 − P rob(θ t = 0), and π ∈ (0, 1]. π is theprobability that the bubble starts to grow exponentially in period t+1, under the conditionthat it still is at its initial value, B 0 , in period t. A process generated according to (22)starts at some strictly positive value B 0 and remains at that level until the Bernoulliprocess has taken the value 1. At that point, the process B t makes a jump of size RB 0πand from then on grows at rate R. One can easily verify that such a process satisfiesthe no arbitrage condition for rational bubbles, given in (5). In our Monte Carlo analysiswe will simulate such a bubble process together with a fundamental price P ft generatedaccording to (7) and (8). An example of the resulting price processes P t = P ft + B t isgiven in figure 1 (solid line), which also depicts the fundamental price (dotted line). Inthis example the parameters for the bubble process are π = 0.05, R = 0.05, and B 0 = 1.Following Evans (1991) the dividend process in (7) is simulated with drift µ = 0.0373,10

initial value D 0 = 1.3 and identical normally distributed disturbances with mean zero andvariance σ 2 = 0.1574.In his critique of Diba and Grossman’s (1988) testing approach Evans (1991) formulatedhis periodically collapsing bubble:⎧⎨(1 + R)B t u t+1 , if B t ≤ αB t+1 =⎩[δ + π −1 (1 + R)θ t+1 (B t − (1 + R) −1 δ)] u t+1 , otherwise . (23)Here, δ and α are parameters satisfying 0 < δ < (1+R)α, and {u t } ∞ t=1 is an iid process withu t ≥ 0 and E t [u t+1 ] = 1, for all t. {θ t } ∞ t=1 is an iid Bernoulli process, where the probabilitythat θ t = 1 is π and the probability that θ t = 0 is 1 − π, with 0 < π ≤ 1. R is the interestrate. One can easily verify that the bubble process defined in (23) satisfies (5). Lettingthe initial value B 0 = δ, the bubble increases until it exceeds some value α. Thereafter,it is subject to the possibility of collapse with probability (1 − π), in which case it willreturn to δu t+1 (i.e. to δ in expectation). For our simulations we will follow Evans (1991)and specify u t = exp(ξ t − 1 2 τ 2 ), where ξ t ∼ iid N(0, τ 2 ). The parameter values are set toα = 1, δ = 0.5, τ = 0.05 and R = 0.05. Note that such a periodically collapsing bubblenever crashes to zero. Thus, it does not violate Diba and Grossmans (1988) finding that abubble cannot restart from zero. Evans (1991) demonstrated that Diba and Grossman’s(1988) tests lack sufficient power to detect periodically collapsing bubbles, even if theprobability of collapse is small. A realization of a periodically collapsing bubble is shownin the right panel of figure 2, where π = 0.85. The left panel of figure 2 displays thefundamental price (dotted line), which is generated as above, and the observed price(solid line). The observed price is constructed asP t = P ft + 20B t . (24)As in Evans (1991), the bubble process is multiplied by a factor of 20. This is to ensurethat the variance of the first difference of the bubble component ∆(20B t ) is large relativeto the variance of first difference of the fundamental price ∆P ft .5 Monte Carlo AnalysisWe start our Monte Carlo analysis within our basic framework in (9) – (11). In section 5.1we compute critical values and present the results for the power of the tests. Furthermore,we consider price processes that contain explicitly modeled bubbles. We investigate thepower of the tests to detect randomly starting bubbles (section 5.2) and periodicallycollapsing bubbles (section 5.3). Finally, we evaluate the properties of the break pointestimators in section 5.4.5.1 Testing for a Change from I(1) to explosiveWe use Monte Carlo simulation to calculate critical values for the test statistics supDF,supB, supBT, supK, and supDFC and we investigate the power of the pertaining tests to11

detect a change from I(1) to explosive in finite samples. For the supDF statistic, we startthe forward recursion at a sample fraction of τ 0 = 0.2 (see (15)). For the other statisticswe set τ 0 equal to 0.1 (see (12) – (14) and (17)). To calculate critical values, data isgenerated according to equation (9) with ρ t = 1 (for all t), an initial value y 0 = 0, andgaussian white noise. 10.000 replications are performed for varying sample sizes amongT = 100, T = 200 and T = 400. We apply the test statistics to the original and to thedetrended series, i.e. to the residuals from the regression of y t on a constant and a lineartime trend. The results are reported in table 1.To calculate the empirical power of the tests, we generate data according to (9) and(11). 2000 replications are performed for the sample sizes T = 100, T = 200 and T = 400.We consider varying break points τ ∗ and varying growth rates ρ. The power of the testsis evaluated at a nominal size of 5%, i.e. a test rejects the null hypothesis when thecorresponding statistic is greater than the respective 0.95-quantile in table 1. The resultsare reported in tables 2 and 3.The tests that have the highest power are the supDFC test and the supBT test. Theyperform better than the supDF test in all parameter constellations. The power of thesupB test is comparable with that of the supDF test. Only in a few cases does the supKtest have substantial power. It is the weakest among the considered tests. The advantageof the supDFC test and the supBT test over the supDF test tends to increase as the breakfraction τ ∗ increases. For instance, when T = 400, ρ = 1.03 and τ ∗ = 0.7 the power of thethree tests is approximately equal, while for T = 400, ρ = 1.03 and τ ∗ = 0.9 the supDFtest rejects the null hypothesis in only 50% of the cases, whereas the supDFC test andthe supBT test can reject the null in more than 75% of the cases.5.2 Randomly Starting BubblesWe now investigate how well the tests can detect randomly starting bubbles. We use themodel presented in section 4. That is we simulate the price process as P t = P ft +B t , whereB t is generated as in (22) and P ftfollows (7) and (8). The parameters are also specifiedas in section 4. Since the fundamental price follows a random walk with drift, we detrendthe time series before applying the tests. The results are given in table 4. The null andthe alternative hypothesis are as in equations (9) to (11). Rejection of the null is evidencefor a bubble, since dividends follow a random walk and the fundamental price obeys thepresent value model. The nominal size is 5%. The sample sizes are T = 100 and T = 200.For each parameter constellation 2000 replications of the price process P t = P ft + B t areconducted. Cases in which a bubble remains at its initial value for more than 90% of itsrealizations are excluded from the experiment.Note that for the first row of table 4 we have simulated just the fundamental price.This shows that the tests have correct size.When the sample size equals T = 100, the supDFC test and the supBT test have thehighest rejection frequencies. In most cases they reject the null hypothesis at least 15%12

more often than the supDF test or the supB test. As in section 5, the supK test is thetest with the lowest rejection frequencies. When the sample size is T = 200 and B 0 is0.05, 6 the differences between the tests are less pronounced. Still, the supK test shows theweakest performance among the tests, while the supDFC test and the supBT test havethe highest finite sample power.An interesting observation is that, when T = 100 and B 0 = 2, some of the tests donot show an increase in power, when the probability, that the bubble starts next period,gets higher. This seems odd, since the expected time of the bubble start, which is 1 π , isearlier the higher π is. However, the parameter π also has another effect on the bubbleprocess. As can be seen in equation (22), there is a jump in the bubble process, at thepoint, where it starts to grow explosively. This jump is smaller when π is higher. Forcertain cases the two impacts of π on the bubble process offset each other with regard tothe ability of the tests to detect the bubble. This can be seen for the supDF test, thesupK test, and the supB test.Summing up, the simulation experiments with randomly starting bubbles confirm theresults of section 5.1, that the supDFC test and the supBT test are most powerful amongthe considered tests.5.3 Periodically Collapsing BubblesWe now analyze how well the tests are suited to detect Evans’ (1991) periodically collapsingbubbles. The price process P t = P ft + 20B t is generated as described in section4. The tests are then applied to the detrended series. As in the previous section we takerejection (at the 5% level) of the I(1)-null hypothesis as equivalent to the detection of abubble.Table 5 reports the rejection frequencies of the tests in our Monte Carlo experiments.2000 replications of the price process P t have been performed for a sample size of T = 100and different values for the probability that the bubble lasts (π). Obviously, the finitesample power of the tests decreases when the value of π gets smaller. The more strikingfinding, however, is that the supDF test is clearly superior to the other tests. For instance,when π = 0.85 the supDF test rejects in more than 60% of the cases, while the supK testrejects in 33% of the cases and the other tests have rejection frequencies lower than 10%.There is an intuitive explanation of the superiority of the supDF test. In contrastto the other tests, the supDF test is based on subsamples of the form { }P 1 , . . . , P [τT ] .Presumably the DF t-statistic for such a subsample is large, when the subsample containsa long period, in which the bubble grows steadily, but does not contain the period afterthe peak. A case for which this is clear is provided in figure 3. It shows a realizationof the price process P t , when π = 0.85, and the corresponding plot of the DF τ statistic.DF τ is plotted against the size of the subsample to which it is applied, which is [τT ].As opposed to the supDF test, the idea behind the supK test is to oppose the first part6 When B 0 is set to 2, all tests, except the supK tes, have rejection frequencies close to 1 for π ≥ 0.05.13

of the sample, { P 1 , . . . , P [τT ]}, to the last part,{P[τT ]+1 , . . . , P T}. But when the bubblecollapses frequently, these two parts have similar properties.The supDFC test, the supBT test, and the supB test essentially build on informationfrom the last part of the sample. As the Dickey-Fuller test over the whole sample theyare subject to Evans’ (1991) criticism. When the bubble collapses frequently during thelast part of the sample, explosiveness cannot be detected.Thus among the alternative tests, the supDF test has the highest power when appliedto periodically collapsing bubbles. However, in applications one might have an obviousindication of where a substantial growth phase of the suspected bubble has ended. If thesubsequent observations are excluded from the sample, the supDFC test and the supBTtest would still be most powerful among the tests considered here.5.4 Estimating the Break DateIn this section we investigate the performance of the estimators from section 3 to datestamp the point where a time series switches from I(1) to explosive. In the simulationexperiments equations (9) and (11) serve as a model for the data generating process. Theexperiments were conducted for different break points τ ∗ and for sample sizes T = 200 andT = 400. The slope parameter ρ was set to 1.05. For each parameter constellation 2.000replications were performed. Table 6 reports the empirical mean and standard deviation(in parentheses) of the break point estimates. For the estimator proposed by Phillips, Wuand Yu (2009), ˆτ P , we take the 5% right tailed critical value of the standard asymptoticDickey-Fuller distribution (which equals 1.28) as the signaling threshold for a change fromI(1) to explosive (cf. (18)). Those realizations of the data generating process where thethreshold is not reached do not enter the calculations of the mean value and the standarddeviation of the estimates.The results in table 6 show that for a sample size of T = 200 the estimators do notperform very well: In most cases the mean value of the estimates are far off the true breakpoint and the standard deviations are fairly high. The cases in which the mean value ofthe estimates are close to the true break point seem to be mere coincidence. When thesample size is increased to T = 400 the Chow-style break point estimator, ˆτ DF C , turnsout to be the most reliable estimator. The mean estimated break fraction is closest to thetrue value and the standard deviation is relatively low. ˆτ P shows a tendency to estimatethe break point too early. Among all estimators ˆτ P has the largest standard deviation.When T = 400, the standard deviation of ˆτ P is more than twice as large as the standarddeviation of the other estimators. As to ˆτ BT , the mean of the estimated break points ismostly about 10% of the sample size higher than the true break point.Overall, the best results in the simulation experiments were achieved by the estimatorˆτ DF C . The use of the estimators ˆτ P and ˆτ BT is questionable in samples with no more than400 observations.14

6 Real time monitoringThe test statistics considered in the previous sections are designed to detect speculativebubbles within a historical data set. As argued by Chu, Stinchcombe and White (1996)such test may be highly misleading when applied to an increasing sample. This is dueto the fact that structural break tests are constructed as “one-shot” test procedures, i.e.,the (asymptotic) size of the test is controlled provided that the sample is fixed and thetest procedure is applied only once to the same data set (cf. Chu et al. 1996 and Zeileiset al. 2005). To illustrate the problem involved assume that an investor is interested tofind out whether the stock price is subject to a speculative bubble. Applying the testsproposed in Section 2 to a sample of the last 100 trading days (say) he or she is not ableto reject the null hypothesis of no speculative bubbles. If the stock price continues toincrease in the subsequent days the investor is interested to find out whether the evidencefor a speculative bubble has increased. However, repeating the tests for structural breakswhen new observations become available eventually leads to a severe over-rejection of thenull hypothesis due to the multiple application of statistical tests.Another practical problem is that the tests assume a single structural break from arandom walk regime to an explosive process. As has been demonstrated in our MonteCarlo simulations the tests generally lack power if the bubble bursts within the sample,that is, if there is an additional structural break back to a random walk process. Themonitoring procedure suggested in this section is able to sidestep the problems due tomultiple breaks.Assume that, before the monitoring starts, a training sample of n observations isavailable and that the null hypothesis of no structural break holds for the training sample.Then, in each period a new observation n + 1, n + 2, . . . arrives. As we will argue below,it is important to fix in advance the maximal length of the monitoring interval n + 1, n +2, . . . , N = kn as the critical value depends on N. Following Chu et al. (1996) we considertwo different statistics (detectors):CUSUM:S t n = ˆσ −1tt∑j=n+1y j − y j−1 = ˆσ −1t (y t − y n ) (t > n)rDF: Z t = ˆv −1/2t (ˆρ t − 1) = DF t/n (t > n)where ˆv t = ˆσ 2 t /( ∑ t−1j=1 y2 j ). ˆρ t denotes the OLS estimate of the autoregressive coefficient andˆσ 2 t is some consistent estimator of the residual variance based on the sample {y 0 , . . . , y t }.Note that Chu et al. (1996) suggest a fluctuation test statistic based on the coefficients ˆρ t .Since the coefficient ρ t is equal to unity under the null hypothesis, the rDF is essentiallysimilar to the fluctuation statistic advocated by Chu et al. (1996).15

Under the null hypothesis the functional central limit theorem implies as n → ∞1√ nS [λn]n ⇒ W (λ) − W (1) (1 ≤ λ ≤ k)Z [λn]⇒∫ λ/ √ ∫ λW (r)dW (r) W (r) 2 dr00(1 ≤ λ ≤ k),where W (r) is a Brownian motion defined on the interval r ∈ [0, k]. Our CUSUM monitoringis based on the fact that (see Chu et al. 1996)(lim P √ )|Sn| t > c t t for some t ∈ {n + 1, n + 2, . . . , N} ≤ exp(−a 2 /2), (25)n→∞wherec t = √ a 2 + log(t/n)(t > n).For the significance level of 0.05 we obtain a 2 = 6.Since our null hypothesis is one-sided (i.e. we reject the null hypothesis for largepositive values of Sn) t and Sn t is distributed symmetrically around zero, a one-sided decisionrule is adopted as follows. The null hypothesis is rejected if S t exceeds the threshold c ∗ tthe first time, that is,reject H 0 if S t n > c ∗ t√t for some t > n, (26)where c ∗ t is constructed as c t but using a value of a that corresponds to the significancelevel α ∗ = 2α. If the original significance level is α = 0.05, then we obtain a = 4.6.Such a test sequence has the advantage that if the evidence for a bubble process issufficiently large the test procedure eventually stops before the bubble collapses. Accordingly,such a test procedure sidesteps the problem of multiple breaks due to a possibleburst of the bubble.For the second monitoring using the statistic DF t/n we apply the following rule:where the critical is obtained fromwherereject H 0 if DF t/n > κ N for some t > n, (27)P (sup{Q 1 , . . . , Q t } > κ t ) = α, (28)Q t =t∑s=1W s−1 (W s − W s−1 )√t∑Ws−12s=1and W t denotes a random walk with increments W t − W t−1 ∼ N (0, 1). Table 7 presentscritical values for various values of N and α. It is important to note that κ N is monotonicincreasing and κ N tends to infinity as N → ∞. Thus, the maximal length of the intervalN has to be fixed before starting the monitoring.16

It is easy to see that the decision rule (27) rejects the null hypothesis of no bubblewith an error probability of maximal α. If we arrive at period N without rejectingthe null hypothesis it follows by the decision rule that the implied probability that thewhole sequence of rDF statistics is below the critical level is 1 − α. Therefore, the errorprobability cannot exceed α. To put it differently, the decision rule is equivalent to a testin period N whether the maximum of all DF t/n statistics (n < t ≤ N) exceeds the criticallevel. Hence, by controlling the probability of the final step of the monitoring procedurewe control the error probability for all previous steps at the same time.To investigate the relative performance of the tests, a number of Monte Carlo experimentshave been performed. CUSUM denotes the sequential monitoring procedure basedon (25). The monitoring procedure based on the recursive Dickey-Fuller statistics DF t/nis denoted by rDF. The results are presented in Table 8. The data generating process wassimulated as in (9) and (11) with slope coefficient ρ = 1.03 under the explosive regime.The total sample size was set to N = 200 and N = 400. The size of the training samplewas set to n = 40 and n = 80 respectively. Monitoring boundaries corresponding to a 5%significance level were used. The breakpoint τ ∗ is measured relative to the total samplesize. For each case 5000 replications were performed. We estimated the residual varianceas ˆσ t2 = ˆσ n 2 = (n − 1) ∑ −1 nt=2 (y t − y t−1 ) 2 for t > n. 7 The columns labeled delay reportthe mean number of observations necessary to detect explosiveness relative to the wholesample. Only those cases where the detection occurs after the true breakpoint enter thecalculations. The same applies to the empirical standard deviation of the detection delayreported in the columns labeled σ(delay). Table 8 shows that the power of the CUSUMmonitoring is comparable to that of the rDF procedure. The CUSUM monitoring detectsbubbles slightly later. The fact that the mean and the standard deviation of the detectiondelay decrease as the regime switch occurs later is probably due to the decreasing size ofthe interval where explosiveness can be detected.7 Applications7.1 The Nasdaq Composite Index and the Dot.com BubbleA price movement for which the term bubble is widely used is that of the Nasdaq compositeprice index in the late 90s of the last century. A mere look at the plot of this time seriesappears to justify the name “dot.com bubble” (see figure 4). Phillips et al. (2009) foundempirical evidence that suggested that a rational bubble indeed was present. In thissection we reproduce their findings and also apply the other test presented in section 2.The data we use is the same as that considered by Phillips et al. (2009). The times7 In principal one could update ˆσ 2 as new data arrive during the monitoring. This can lead to powerreduction and a higher detection delay since ˆσ 2 would be deflated by including observations from theexplosive phase of the data generating process. However, in further simulations we found that thedeterioration in performance is not substantial (not shown).17

series of the Nasdaq composite price index and the Nasdaq composite dividend yield aretaken from Datastream International. Phillips et al. (2009) considered monthly Nasdaqdata for an observation period from February 1973 to June 2005. However, if there wasbubble in the Nasdaq, figure 4 suggest that it has crashed in March 2000, when theNasdaq reached its peak. If the observations after the peak are included in the sample,the probability that the bubble tests will detect a bubble will be very low, even if it wasindeed present, as was shown in section 5.3. Therefore we apply the tests to the restrictedsample period from February 1973 to March 2000, which makes T = 326 observations.The Nasdaq composite price index is multiplied by the Nasdaq composite dividend yieldto compute the time series of the total Nasdaq dividends. By use of the Consumer PriceIndex from the Federal Reserve Bank of St. Louis, nominal data are transformed to realdata. Figure 4 shows a plot of the time series for the real Nasdaq price index and the realNasdaq dividends, where the time series have been normalized to 100 at the beginning ofthe observation period. While the price index shows a sharp increase during the 1990sfollowed by an at least equally sharp drop, real index dividends are more or less constant.For the logarithm of these two time series we conduct tests of the constant-I(1) hypothesisagainst the alternative of a switch from I(1) to explosive. If the time series of theNasdaq composite price index turns out to have changed from I(1) to explosive, while theindex dividend series remained constant I(1), this would suggest the presence of bubbles.In the applications we follow Phillips, Wu, and Yu (2009) more closely and use anaugmented Dickey-Fuller (ADF) test that includes an intercept term in the regression,i.e. the regression equation isy t = α + ρy t−1 +p∑φ j ∆y t−j + ɛ t , for t = 1, . . . , [τT ], (29)j=1where y t denotes the logarithm of either the Nasdaq price or dividends and ɛ t is whitenoise. The lag order p is determined by a general-to-specific test sequence. (29) leads tothe augmented DF t-statistic ADF τ . The supremum of ADF τ over τ ∈ [0.2, 1] is referedto as supADF. The pertaining critical values are taken from Phillips et al. (2009).The results are shown in table 9. For the dividend series, none of the tests rejects theconstant I(1) hypothesis at the 5% significance level. Even at the 10% significance levelthe null hypothesis is not rejected. The application of the supADF test leads to resultsthat are very similar to those obtained by Phillips et al. (2009) 8 . The supDFC test andthe supBT test also detect explosivness in the Nasdaq index, while the supB test andthe supK test fail to reject the null hypothesis 9 . This reflects our findings in the MonteCarlo section. Those test which were found to be most powerful support the view that theNasdaq price index has changed from I(1) to explosive, while the dividend series remainedI(1) throughout the whole sample. Under the assumption that the present value modelin equations (2) and (3) is not misspecified, one may conclude that rational bubbles are8 They report a test value of 2.984 for the index price and a value of -1.018 for the index dividends9 We find similar results for the break test, when we apply them to the detrended series18

indeed present in the Nasdaq index.This leads to the question about when the bubble started. First consider the approachtaken by Phillips et al. (2009). Figure 5 contains the plots of the ADF τ statistic for thelog real Nasdaq price index (solid line). ADF τ is the augmented Dickey-Fuller t-statisticfor the subsample {y 1 , . . . , y [τT ] }. Given on the x-axis is the period that corresponds tothe [τT ]-th observation. The estimated bubble start is June 1995, which is the first timethat the ADF τ statistic is greater than the 5% right-tail critical value of the asymptoticdistribution of the standard Dickey-Fuller t-statistic (−0.08). 10We also estimate the time of the bubble emergence using the estimator from theChow-style DF-statistic ˆτ DF C . The maximum of DF C τ is attained at ˆτ DF C = 81, 3%.The corresponding date is February 1995.From a different perspective, we can address the question about when the bubblewould have been detected, if the Nasdaq had been monitored. We assume that themonitoring started in July 1979 with a training sample of 77 monthly observations (fromFebruary 1973 to June 1979) and apply the recursive (augmented) DF procedure. Thecorresponding critical value is 1.468 (see table 9). We find that the bubble would havebeen detected in July 1999 (see figure 5).7.2 The Hang Seng IndexThe Hang Seng Index is the main market indicator at the Hong Kong stock exchange,which ranks among the largest stock exchanges in Asia. In the history of the Hang SengIndex traders have seen many phases of impressive growth followed by steep falls. Forinstance, between October 1983 and October 1987 the index rose by more than 450% andthen lost half of its peak value until July 1989. Similarly, from May 2003 to November 2007the Hang Seng index increased by almost 350%. During the subsequent global financialcrisis it dropped by more than 60% within 15 month. A graph is shown in figure 6 wherethe real Hang Seng index is plotted together with the corresponding real dividends. Bothseries have been normalized to 100 at the beginning of the observation period.Data of the Hang Seng price index and dividend yield are taken from DatastreamInternational. These two series are used to calculate the Hang Seng dividends. In orderto transform nominal into real data we use the composite Consumer Price Index from theHong Kong Census and Statistics Department.We test wether the global financial crisis has been preceded by a bubble in the HangSeng index. If there was a bubble, it has certainly burst in November 2007. Therefore, wedo not include observations following that date. Data on the composite Consumer PriceIndex for Honk Kong is only available from October 1980 on. Thus, for our index anddividend time series, we have 326 monthly observations from October 1980 to November2007 .10 Using log(log(τT ))/100 as critical value for ADF τ (cf. Phillips et al.(2009)) results in almost thesame estimate for the start of the bubble, namely July 1995.19

We apply our tests to the natural logarithm of the Hang Seng index and dividendseries. The results are reported in table 10. Regarding the index price, the only statisticthat is significant is that of the sequential Chow-type DF test, supDFC. This statistic isborderline significant at the 5% level. The other tests cannot reject the random walk nullhypothesis, even at the 10% level. Furthermore, the supDFC statistic for the dividendseries is not significant at the 10% level. Thus, given that the present value model in(2) and (3) is correctly specified, the supDFC test leads to the conclusion that, indeed,a bubble has preceded the outbreak of the global financial crises. The other tests do notdetect an explosive movement in the Hang Seng index and hence do not corroborate theresult of the supDFC test. We ascribe this to the fact that most of those tests have lesspower than the supDFC test as our Monte Carlo analysis has shown.The detection of a bubble in the Hang Seng index by the supDFC test suggests theuse of the pertaining estimator ˆτ DF C to determine the date at which the bubble emerged.The resulting estimate is ˆτ DF C = 0.83 which corresponds to April 2003.8 ConclusionIn this paper the ability of several tests to detect rational bubbles has been investigated.The focus was on rational bubbles in stock markets. A basic standpoint was that explosivestock prices together with nonexplosive dividends imply the presence of bubbles in theprice process. Hence, all tests aimed at detecting a switch to explosiveness in a timeseries. The sequential Dickey-Fuller test, proposed by Phillips, Wu and Yu (2009) servedas a reference point. As opposed to Phillips et al.’s (2009) procedure the other tests wereconceived as tests for a structural break. In a simple simulation framework it was shownthat a Chow-type Dickey-Fuller test and a modified version of Busseti and Taylor’s (2004)test have higher finite sample power than the test of Phillips et al. (2009), especially whenthe change from I(1) to explosive occurs late in the sample. This result was confirmed insimulation experiments with randomly starting bubbles.With regard to the estimation of the time of the bubble emergence, different proceduresto estimate the point of change from I(1) to explosive were considered. It turned outthat, in finite samples, Phillips et al.’s (2009) estimator is downward biased. Moreover,this estimator has a very high standard deviation. It would be more reliable to use theestimator ˆτ DF C , which directly relates to the Chow-type Dickey-Fuller test.The consideration of Evans’ (1991) periodically collapsing bubbles showed the weaknessof break tests in bubble detection. For these tests it is crucial that the bubble doesnot crash within the last part of the sample.The sequential Dickey-Fuller test of Phillips et al. (2009) can also be conceived as amonitoring procedure. We tabulated critical values for different sizes of the monitoringsample. Moreover, we propose a simple CUSUM monitoring scheme. The power and thedetection delay of the CUSUM monitoring are comparable with those of the recursive DFprocedure.20

In the applications, Phillips et al.’s (2009) result that bubbles are present in theNasdaq index was confirmed by our sequential Chow-type Dickey-Fuller test as well asby the modified version of Busetti and Taylor’s (2004) test. The estimation of the dateof the bubble emergence indicated that the explosive behavior in the Nasdaq started inthe first half of the year 1995. The supDFC test also detected a bubble in the Hang Sengindex. Our estimate for the start of this bubble is April 2003.21

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Table 1: Upper Tail Critical ValuesSample size Quantiles Test statisticssupDF supDFC supK supBT supB(a) Critical values without detrendingT=100 0.90 2.1944 1.5422 28.4766 2.0004 2.84980.95 2.5671 1.9290 40.0557 2.5396 3.32130.99 3.2778 2.5950 74.8903 3.8404 4.3459T=200 0.90 2.2145 1.5709 29.1150 1.9459 3.03530.95 2.5841 1.9206 39.6741 2.4691 3.54630.99 3.2055 2.6022 72.9779 3.7759 4.8781T=400 0.90 2.2374 1.5527 29.4079 1.9496 3.05200.95 2.5980 1.9155 40.7870 2.4397 3.59350.99 3.2227 2.5882 75.7367 3.7150 4.9529(b) Critical values with detrendingT=100 0.90 0.0599 0.9071 24.294 1.8448 2.35140.95 0.3206 1.2864 33.729 2.3928 2.77200.99 0.7871 2.0408 55.935 3.7456 3.6201T=200 0.90 0.1006 0.9108 24.948 1.7766 2.48050.95 0.3402 1.2709 34.067 2.2800 2.96360.99 0.7753 2.0436 59.077 3.6151 3.9984T=400 0.90 0.1192 0.9542 26.692 1.7661 2.62400.95 0.3562 1.3451 36.402 2.3116 3.16170.99 0.8264 2.0168 62.729 3.5697 4.3906Notes: The critical values are estimated by simulation of (9) - (10) with 10.000 replications.24

Table 2: Empirical powerBreak pointTest statistics(a) Power for T = 100supDF supDFC supK supBT supBτ ∗ = 0.7 ρ = 1.02 0.171 0.311 0.094 0.267 0.180ρ = 1.03 0.293 0.479 0.144 0.441 0.298ρ = 1.04 0.431 0.632 0.221 0.593 0.433ρ = 1.05 0.562 0.741 0.328 0.721 0.541τ ∗ = 0.8 ρ = 1.02 0.108 0.240 0.079 0.195 0.125ρ = 1.03 0.174 0.366 0.099 0.324 0.176ρ = 1.04 0.260 0.485 0.129 0.444 0.251ρ = 1.05 0.353 0.584 0.173 0.558 0.326τ ∗ = 0.9 ρ = 1.02 0.069 0.147 0.062 0.115 0.068(b) Power for T = 200ρ = 1.03 0.086 0.215 0.066 0.173 0.073ρ = 1.04 0.109 0.291 0.073 0.245 0.076ρ = 1.05 0.140 0.367 0.079 0.324 0.071τ ∗ 0.7 1.02 0.441 0.630 0.234 0.608 0.4851.03 0.669 0.811 0.477 0.801 0.7011.04 0.820 0.899 0.770 0.897 0.8311.05 0.894 0.945 0.896 0.945 0.892τ ∗ 0.8 1.02 0.269 0.496 0.156 0.460 0.3161.03 0.467 0.688 0.249 0.673 0.5161.04 0.636 0.799 0.445 0.792 0.6661.05 0.748 0.869 0.692 0.869 0.758τ ∗ 0.9 1.02 0.108 0.313 0.080 0.268 0.1141.03 0.178 0.465 0.102 0.430 0.1831.04 0.284 0.590 0.134 0.562 0.2501.05 0.385 0.686 0.183 0.676 0.325Notes: The empirical power is computed at a nominal size of 5%. The simulations are conducted with2, 000 replications of (9) and (11). The true (simulated) breakpoint τ ∗ is measured relative to the samplesize. ρ is the autoregressive slope parameter under the explosive regime.25

Table 3: Empirical power cont.Break pointTest statistics(c) Power for T = 400supDF supDFC supK supBT supBτ ∗ = 0.7 ρ = 1.02 0.825 0.901 0.766 0.901 0.864ρ = 1.03 0.939 0.970 0.945 0.970 0.951ρ = 1.04 0.978 0.989 0.979 0.989 0.980ρ = 1.05 0.994 0.998 0.995 0.998 0.994τ ∗ = 0.8 ρ = 1.02 0.644 0.815 0.471 0.809 0.707ρ = 1.03 0.848 0.927 0.822 0.931 0.871ρ = 1.04 0.938 0.972 0.933 0.972 0.941ρ = 1.05 0.971 0.986 0.977 0.987 0.973τ ∗ = 0.9 ρ = 1.02 0.299 0.582 0.116 0.565 0.361ρ = 1.03 0.498 0.762 0.273 0.763 0.548ρ = 1.04 0.660 0.860 0.540 0.863 0.681ρ = 1.05 0.771 0.910 0.720 0.915 0.753Notes: The empirical power is computed at a nominal size of 5%. The simulations are conducted with2, 000 replications of (9) and (11). The true (simulated) breakpoint τ ∗ is measured relative to the samplesize. ρ is the autoregressive slope parameter under the explosive regime.26

Table 4: Rejection frequencies in the case of randomly starting bubblesInitial value π Test statisticssupDF supDFC supK supBT supBRejection frequencies when the sample size is T = 100B 0 = 2 no bubble 0.053 0.046 0.050 0.049 0.0410.02 0.481 0.654 0.073 0.636 0.5140.05 0.522 0.737 0.078 0.730 0.5560.10 0.515 0.851 0.081 0.837 0.5890.25 0.493 0.919 0.093 0.902 0.5870.50 0.503 0.943 0.097 0.924 0.5901.00 0.502 0.946 0.097 0.930 0.587Rejection frequencies when the sample size is T = 200B 0 = 0.05 0.01 0.580 0.656 0.112 0.651 0.5870.02 0.695 0.767 0.151 0.768 0.6990.05 0.893 0.945 0.211 0.948 0.8910.10 0.980 0.993 0.221 0.995 0.976Notes: The tests are applied to series generated as P t = P f t + B t , with P f t as in (8) and B t as in (22).2000 replications are performed. The “no-bubble” hypothesis is rejected when the test statistic exceedsthe 95% quantile from table 1. B 0 is the initial value of the bubble. π is the probability that the bubbleenters the phase of explosive growth.27

Table 5: Rejection frequencies in the case of periodically collapsing bubblesSample size π Test statisticssupDF supDFC supK supBT supBT=100 no bubble 0.053 0.046 0.050 0.049 0.0410.999 0.902 0.899 0.043 0.931 0.2900.990 0.887 0.593 0.220 0.646 0.2900.950 0.764 0.179 0.390 0.213 0.1460.850 0.624 0.072 0.330 0.073 0.0420.750 0.531 0.053 0.303 0.044 0.0320.500 0.365 0.025 0.233 0.028 0.0270.250 0.253 0.023 0.210 0.021 0.033Notes: The tests are applied to series generated as P t = P f t + 20B t , with P f t as in (8) and B t as in (23).2000 replications are performed. The “no-bubble” hypothesis is rejected when the test statistic exceedsits 95% quantile from table 1. π is the probability that the bubble does not collapse in the next period.28

Table 6: Estimated break dates when ρ = 1.05Break point Sample Size T=200 Sample Size T=400ˆτ P ˆτ DF C ˆτ BT ˆτ P ˆτ DF C ˆτ BTτ ∗ = 0.4 0.4840 0.4616 0.6564 0.4258 0.4232 0.4901(0.1558) (0.1037) (0.1334) (0.1049) (0.0456) (0.0601)τ ∗ = 0.5 0.5498 0.5582 0.7636 0.4987 0.5207 0.5933(0.1873) (0.0980) (0.1137) (0.1410) (0.0436) (0.0616)τ ∗ = 0.6 0.6089 0.6493 0.8577 0.5688 0.6208 0.6991(0.2185) (0.0862) (0.0692) (0.1803) (0.0441) (0.0621)τ ∗ = 0.7 0.6554 0.7388 0.8967 0.6355 0.7207 0.8080(0.2523) (0.0747) (0.0149) (0.2216) (0.0427) (0.0548)τ ∗ = 0.8 0.6800 0.8186 0.8990 0.6937 0.8143 0.8903(0.2875) (0.0594) (0.0036) (0.2603) (0.0329) (0.0185)τ ∗ = 0.9 0.6299 0.8701 0.8978 0.7145 0.8890 0.8996(0.3208) (0.0775) (0.0071) (0.3032) (0.0512) (0.0023)Notes: The table reports mean values and standard deviations (in parentheses) of the break point estimatorsˆτ P , ˆτ DF C and ˆτ K . These values as well as the true breakpoint τ ∗ are expressed as fraction of thesample size T . Data is generated according to equation (9) and (11) for the sample size T = 200 andT = 400. The autoregressive slope parameter under the explosive regime is ρ = 1.05. In each case 1000replications are performed.29

Table 7: Critical Values for Fluctuation Monitoring (rDF)N 50 75 100 120 150 200 250 300α0.10 2.2877 2.3529 2.4073 2.4267 2.4520 2.4853 2.5163 2.53020.05 2.5987 2.6679 2.7242 2.7331 2.7633 2.7934 2.8181 2.83250.01 3.1577 3.2286 3.2835 3.2971 3.3229 3.3316 3.3485 3.3596N 400 500 600 750 1000 1500 2000 2500α0.10 2.5753 2.5979 2.6164 2.6345 2.6657 2.7017 2.7186 2.73350.05 2.8636 2.8809 2.9006 2.9123 2.9353 2.9691 2.9931 3.00950.01 3.3904 3.4108 3.4335 3.4579 3.5006 3.5152 3.5329 3.5470Notes: The table shows critical values for different sample sizes N and significance levels α. For thesimulation of the critical values 10, 000 replications of the DGP in (9) and (10) were carried out.30

Table 8: Performance of Monitoring ProceduresrDFCUSUMBreakpoint size/ power delay σ(delay) size/ power delay σ(delay)a) size of training sample n = 40, size of total sample kn = 200no break 0.0412 — — 0.0358 — —τ ∗ = 0.2 0.9756 0.2886 0.1748 0.9712 0.3081 0.1626τ ∗ = 0.4 0.9314 0.2503 0.1371 0.9198 0.2601 0.1354τ ∗ = 0.6 0.8086 0.2040 0.0969 0.7758 0.2139 0.0961τ ∗ = 0.8 0.4560 0.1256 0.0464 0.3900 0.1288 0.0473b) size of training sample n = 80, size of total sample kn = 400no break 0.0332 — — 0.0286 — —τ ∗ = 0.2 0.9996 0.1420 0.0970 0.9996 0.1572 0.0928τ ∗ = 0.4 0.9972 0.1339 0.0896 0.9966 0.1454 0.0912τ ∗ = 0.6 0.9850 0.1263 0.0781 0.9816 0.1381 0.0805τ ∗ = 0.8 0.8300 0.0978 0.0469 0.7870 0.1058 0.0480Notes: The data is generated as a random walk switching to explosive at the relative break point τ ∗ (cf.(9) and (11)). The number of replications is 5, 000. The size/ power is computed at the 5% significancelevel. In the column delay we report the average time needed to detect the regime switch relative tothe sample size kn. Detections that occur prior to the true break (type I error) are removed from thecalculation. The column σ(delay) contains standard deviations of the detection delay relative to thesample size.31

Table 9: Testing for an explosive root in the Nasdaq indexTest StatisticssupADF supDFC supK supBT supBlog price index 3.0467 4.5874 12.5203 7.8722 3.0242log dividends -0.9974 -0.2086 5.0756 0.1438 2.0057Upper tail critical values1% 2.094 2.5882 75.7367 3.715 4.95925% 1.468 1.9155 40.787 2.4397 3.593510% 1.184 1.5527 29.4079 1.9496 3.0520Notes: Table 9 reports the values of the test statistics applied to the log real Nasdaq price index anddividends. The table also shows the corresponding critical values. Those for the supADF statistic aretaken from Phillips et al. (2009). The others correspond to our simulation results.32

Table 10: Testing for an explosive root in the Hang Seng indexTest StatisticssupADF supDFC supK supBT supBlog price index -0.1538 1.933 4.6439 1.3842 2.6096log dividends 0.2644 1.1578 40.6668 0.7147 2.4152Upper tail critical values1% 2.094 2.5882 75.7367 3.715 4.95925% 1.468 1.9155 40.787 2.4397 3.593510% 1.184 1.5527 29.4079 1.9496 3.0520Notes: Table 10 reports the values of the test statistics applied to the log real Hang Seng price index anddividends. The table also shows the corresponding critical values. Those for the supADF statistic aretaken from Phillips et al. (2009). The others correspond to our simulation results.33

Price and Fundamental component25020015010050pricefundamental component00 20 40 60 80 100PeriodFigure 1: Simulated price series with randomly starting bubble component34

Price and Fundamental component35030025020015010050pricefundamental componentBubble component3002502001501005000 20 40 60 80 100Period00 20 40 60 80 100PeriodFigure 2: Simulated price with fundamental component (left) and bubble component(right)35

Price3503002502001501005000 20 40 60 80 100PeriodDF τ−Statistic210−1−2−3−40 20 40 60 80 100Size of Sample PeriodFigure 3: Price Containing Periodically Collapsing Bubble (left) and DF τ -Statistic (right)36

12001000Real Nasdaq PriceReal Nasdaq Dividends8006004002000Jan70 Jan80 Jan90 Jan00 Jan10PeriodFigure 4: Real Nasdaq Price and Dividends (normalized)37

432July 199910−1−2June 1995−3−4Jan75 Jan80 Jan85 Jan90 Jan95 Jan00 Jan05 Jan10Figure 5: ADF τ Statistics for (log) real Nasdaq prices (solid line), with 5% right tailcritical value from asymptotic distribution of the standard Dickey-Fuller t-statistic (dottedline) and the finite sample 5% right tail critical value for the supADF statistic with T=400observations (dash-dotted line)38

700600500Real Hang Seng Price IndexReal Hang Seng Dividends4003002001000Jan80 Jan85 Jan90 Jan95 Jan00 Jan05 Jan10PeriodFigure 6: Real Hang Seng Index and Dividends (normalized)39