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40056

# 40056 - 2 Z 3 = E Z 4 | Z 1 Z 2 Z 3 is the minimum mean...

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STATISTICS 4005 ASSIGNMENT 6 Due date: April 16, 2004 1. For an AR(1) model with n = 100 , φ 0 . 4, and μ 11 . 2, the lag 1 sample autocorrelation of the residuals is 0.5. Should we consider this unusual? 2. For an MA(1) model with n = 100 , θ 0 . 4, and μ 11 . 2, the lag 1 sample autocorrelation of the residuals is 0.5. Should we consider this unusual? 3. Based on a series of length n = 200, we fit an AR(2) model and obtain residual autocorrelations of ˆ r 1 = 0 . 13 , ˆ r 2 = 0 . 13, and ˆ r 3 = 0 . 12. If ˆ φ 1 = 1 . 1 and ˆ φ 2 = - 0 . 8, do these residual autocorrelations support the AR(2) specification? Individually? Jointly? 4. Suppose we have observations Z 1 , Z 2 , Z 3 and we want to get a function h ( Z 1 , Z 2 , Z 3 ) to predict Z 4 . Show that h ( Z 1 , Z
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Unformatted text preview: 2 , Z 3 ) = E ( Z 4 | Z 1 , Z 2 , Z 3 ) is the minimum mean square error predictor. 5. Consider the AR(1) model: Z t = 10 + 0 . 9 Z t-1 + a t where a t ∼ NID (0 , 4). Suppose that Z 200 = 90 and Z 199 = 80. (a) Find the forecasts ˆ Z 200 (1) and ˆ Z 200 (10). (b) Calculate the 95% prediction intervals for Z 201 and Z 210 . (Remark: If X is a standard normal random variable, P ( X ≤ 1 . 96) = . 975) (c) Update your forecast for Z 210 given Z 201 = 100. (d) Let e t ( k ) = Z t + k-ˆ Z t ( k ) be the k-step ahead forecast error at the time of forecast origin t . Find Corr [ e 201 (2) , e 201 (5)] ....
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