A3 - STA 4005B ASSIGNMENT 3 Due date 2 Let at N ID(0 a 1...

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STA 4005B ASSIGNMENT 3 Due date: February 29, 2008 Let a t NID (0 2 a ). 1. Consider the MA(4) process Z t = a t + θ 1 a t - 1 + θ 2 a t - 2 + θ 3 a t - 3 + θ 4 a t - 4 where a t NID (0 2 a ). (a) Find V ar ( Z t ) in terms of σ 2 a and the θ ’s. (b) Find the autocorrelation function ρ k , k = 0 , 1 , 2 ,... of the process in terms of θ ’s. 2. Consider the model Z t = 0 . 6 Z t - 1 + a t + 0 . 4 a t - 1 + 0 . 7 a t - 2 (a) Find the value of the ψ k , k = 1 , 2 , 3 ,... if the process is written in the form Z t = a t + ψ 1 a t - 1 + ψ 2 a t - 2 + ψ 3 a t - 3 .... (b) Find the autocovariance function for { Z t } . 3. Consider the AR(2) process Z t = 0 . 5 Z t - 1 - 0 . 06 Z t - 2 + a t where a t ’s are independently, identically distributed as N (0 , 1). (a) Find the value of the ψ k , k = 1 , 2 , 3 ,... if the process is written in the form Z t = (1 + X k =1 ψ k B k ) a t (b) Find the autocovariances
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