4 Problem 4 1 point Let X t denote the unique stationary solution of the

4 problem 4 1 point let x t denote the unique

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4. Problem 4 (1 point) Let { X t } denote the unique stationary solution of the autoregressive equations X t = φX t - 1 + Z t , t = 0 , ± 1 , ... (1) where { Z t } ∼ WN (0 , σ 2 ) and | φ | > 1. Then we know that X t is given by X t = - X j =1 φ - j Z t + j (2) Hence, this AR model in is not causal. But now, define a new sequence W t = X t - 1 φ X t - 1 (3) and show that W t WN (0 , σ 2 W ) and write σ 2 W as some function of σ 2 and φ . Conclude that { X t } is the unique stationary solution to the causal AR equations X t = 1 φ X t - 1 + Z t , t = 0 , ± 1 , ... (4) 5. Problem 5 (2 point) (a) Calculate the autocovariance function γ ( · ) of the stationary time series Y t = Z t + θ 1 Z t - 1 + θ 12 Z t - 12 , { Z t WN (0 , σ 2 ) (b) Recall the football data from the homework 1. Compute and plot the sample autocovariances ˆ γ ( · ) of {∇∇ 12 X t } for lags one through twenty. (c) Equate ˆ γ (1) , ˆ γ (11), and ˆ γ (12), from part (b) with γ (1) , γ (11), and γ (12) from (a), estimate θ, θ 12 , and σ 2 . 3
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