ECON217_HW_UnitRoot_Key

ECON217_HW_UnitRoot_Key - H.W(unitroot 1 Consider a...

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1 H.W. (unitroot) 1. Consider a Difference Stationary Model X t - X t-1 = β + ε t , where ε t is a stationary ARMA term and X 0 = α . a. Rewrite X t as a function of α , β and ε i . b. Verify that the variance of X t is t-dependent and ε t has a permanent effect on X t . c. Show that the variance of the forecast error X t+h - E(X t+h I t ) increases as h increases. d. Show that we cannot obtain a stationary series by removing a trend term from X t . e. Which data transformation should be used to achieve stationarity? Ans. Consider the difference stationary model X t - X t-1 = β + ε t , where ε t is a stationary ARMA term. To simplify the discussion, let us assume ε t is iid~(0, σ 2 ) and the initial value of X t is given by X 0 = α , then X t = α + β t + 1 t i i ε = Σ .1 The intercept α depends on the initial condition X 0 . The variance of X t is given by V(X t ) = t σ 2 . That is, the variance of X t is t-dependent. As X t+h = α + β (t+h) + 1 t h i i ε + = Σ 2, the effect of ε t shows up in X t+h for h > 0. thus, ε t has an permanent impact on {X t }. The forecast error is X t+h - E(X t+h I t ) = 1 t h i t i ε + = + Σ 3 and its variance is V[X t+h - E(X t+h I t )] = h σ 2 . Thus, the prediction error is unbound as the forecast horizon increases. When we remove the constant and the trend from the data, we have X t = 1 t i i ε = Σ 4, which again is nonstationary (WHY?). To achieve stationarity, we have to first difference the data; that is Y t X t - X t-1 is
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2 stationary. 2. Consider a Trend stationary model X t = α + β t + ε t , where ε t is a stationary ARMA term. a. Verify that the variance of X t is t invariant and the effect of ε t on X t dissipates asymptotically. b. Show that the variance of the forecast error X t+h - E(X t+h I t ) is (asymptotically) constant. c. Show that differencing the data removes the trend and, at the same time, introduces a moving average unit root. d. Which data transformation should be used to achieve stationarity?
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