1. Let (Y) be an ARMA(1,1) process, i.e. a stationary process such that for all t...

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1. Let (Y) be an ARMA(1,1) process, i.e. a stationary process such that for all t Z, the following recursion holds Yt 0.5Y 1et + 0.25et1, - where (et) are iid mean 0 and variance o = 64. (a) Show that the process is causal, and find the first three terms in the linear representation, i.e. find yo, 1, 2 in Y = Voet + viet-1+V2et2+ . . .. (b) Use the Yule-Walker approach to find the autocovariance values yo and 1 for this process. (c) Now using a recursion, compute 2, 3. And ultimately, all Yk for k > 2. (d) Find the first three terms in an invertible representation of (Yt), i.e., find 0, T1, T2 such that et = +1+2Yt2+.... 1. Let (Y) be an ARMA(1,1) process, i.e. a stationary process such that for all t Z, the following recursion holds Yt 0.5Y 1et + 0.25et1, - where (et) are iid mean 0 and variance o = 64. (a) Show that the process is causal, and find the first three terms in the linear representation, i.e. find yo, 1, 2 in Y = Voet + viet-1+V2et2+ . . .. (b) Use the Yule-Walker approach to find the autocovariance values yo and 1 for this process. (c) Now using a recursion, compute 2, 3. And ultimately, all Yk for k > 2. (d) Find the first three terms in an invertible representation of (Yt), i.e., find 0, T1, T2 such that et = +1+2Yt2+....