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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,