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Question 3 [42] Consider the ARMA(1, 1) model, replace with Xt- , X X, = $X,)+ a, - 0at, a; ~ N (0, 2) X and its moving average (MA(co)) representation X1 = a, + w jai-1+ 12ar-2+ ... where the w weights can be found from " (B) bp (B) = 0, (B). Find (a) the corresponding w weights, (b) the variance of the process yo. 3 33 3 (c) the autocovariance of the process y , and (d) its autocorrelation function (ACF) pi. (e) The forecast error based on the minimum mean square error /-step ahead forecast, X, (1) = E [X,# | X,, Xm-1, . . .] at forecast origin n is given by en (1 ) = X.H - X. (1) = Ev ,anti-1. (i) Determine the formula that gives the /-step ahead forecasts X,, (!). (3) (ii) Evaluate the first five non-zero w weights of the model when expressed as an MA(co) and hence, (5) (iii) show that Var[ X. (1)] = Var [en (1)]. (5) (iv) Find the covariance between forecast errors from same origin but with different lead times, i.e. Cou [en (1), en-k (1)] , in terms of ca, $ and 0. (6)