Question 3 Consider the ARMA(1, 1) model, X = X + a , a ~...

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Question 3 Consider the ARMA(1, 1) model, X = φX + a – θαι, a₁ ~ N N (0,0²) and its moving average (MA()) representation X₁ = a₁ +₁a₁-1 + 4/₂9₁-2 + ·· where they weights can be found from y (B) p (B)=0q (B). Find (a) the corresponding y weights, (b) the variance of the process y, (c) the autocovariance of the process y, and (d) its autocorrelation function (ACF) pk. (6) (7) (7) (3) (e) The forecast error based on the minimum mean square error 1-step ahead forecast, X, (1) = E [Xn+1 | Xn, Xn−1, ...] at forecast origin n is given by 1-1 en (l) = Xn+ − X, (l) =j@nh-j. j=0 [42] in terms of o2, and 0. (i) Determine the formula that gives the 1-step ahead forecasts Â₁ (1). (3) (ii) Evaluate the first five non-zero y weights of the model when expressed as an MA (x) and hence, (5) (iii) show that Var[ X₁ (1)] = Var [en (1)]. (5) (iv) Find the covariance between forecast errors from same origin but with different lead times, i.e. Cov [en (1), en-k (1)], (6) Question 3 Consider the ARMA(1, 1) model, X = φX + a – θαι, a₁ ~ N N (0,0²) and its moving average (MA()) representation X₁ = a₁ +₁a₁-1 + 4/₂9₁-2 + ·· where they weights can be found from y (B) p (B)=0q (B). Find (a) the corresponding y weights, (b) the variance of the process y, (c) the autocovariance of the process y, and (d) its autocorrelation function (ACF) pk. (6) (7) (7) (3) (e) The forecast error based on the minimum mean square error 1-step ahead forecast, X, (1) = E [Xn+1 | Xn, Xn−1, ...] at forecast origin n is given by 1-1 en (l) = Xn+ − X, (l) =j@nh-j. j=0 [42] in terms of o2, and 0. (i) Determine the formula that gives the 1-step ahead forecasts Â₁ (1). (3) (ii) Evaluate the first five non-zero y weights of the model when expressed as an MA (x) and hence, (5) (iii) show that Var[ X₁ (1)] = Var [en (1)]. (5) (iv) Find the covariance between forecast errors from same origin but with different lead times, i.e. Cov [en (1), en-k (1)], (6)