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400531

# 400531 - ψ k k = 1 2 3 if the process is expressed in the...

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STATISTICS 4005 ASSIGNMENT 3 Due date: February 26, 2004 1. Consider the AR(2) process Z t = 0 . 7 Z t - 1 - 0 . 12 Z t - 2 + a t where a t ’s are independently, identically distributed as N (0 , 1). (a) Find the autocovariances γ 0 , γ 1 and γ 2 . (b) Is the process { Z t } stationary? Why? (c) By considering the following recursive formula P k = A P k - 1 where P k = " ρ k ρ k - 1 # , A = " 0 . 7 - 0 . 12 1 0 # , derive the general form of the autocorrelation function ρ k for k = 0 , 1 , 2 , 3 ,... 2. (a) Find the range of b so that the following AR(2) process Z t = Z t - 1 + b Z t - 2 + a t is stationary, where { a t } is a white noise process. (b) Find the autocorrelations ρ 1 and ρ 2 for part (a) when b = - 0 . 5. 3. Suppose that a random process { X t } ( t = 0 , ± 1 , ± 2 ,... ) is deﬁned by X t - 0 . 6 X t - 1 = Z t + 0 . 4 Z t - 1 where { Z t } is a white noise process with V ar ( Z t ) = 1 and Z t is independent of X t - 1 ,X t - 2 ,... . (a) Find the values of
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Unformatted text preview: ψ k , k = 1 , 2 , 3 ,... if the process is expressed in the form X t = ψ ( B ) Z t where ψ ( B ) = 1 + ψ 1 B + ψ 2 B 2 + ψ 3 B 3 + ... . (b) Find the autocovarianace function γ k , k = 0 , 1 , 2 , 3 ,... of the process { X t } . 4. Let a t ∼ NID (0 ,σ 2 a ). Suppose the ARMA(3,2) process Z t-φ 1 Z t-1-φ 2 Z t-2-φ 3 Z t-3 = a t-θ 1 a t-1-θ 2 a t-2 is also written as Z t = ∞ X j =0 ψ j a t-j . Take ψ j = 0 for j < 0. Show that for k > 1, the autocovariance function of the process γ k is given by γ k = 3 X j =1 φ j γ k-j-σ 2 a 2 X j = k θ j ψ j-k ....
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