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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)