Property 3.7 Truncated Prediction for ARMA For ARMA(p, q) models, the truncated predictors for m = 1,2, . .., are (3.92) where of = X, for l s t n, and W. = (B)i, 4 for l st sn.(a) Fit an AR(2) to x, using linear regression as in Example 3.18. (b) Assuming the fitted model in (a) is the true model, find the forecasts over a four- week horizon, co.., for m = 1, 2, 3, 4, and the corresponding 95% prediction intervals. 3.11 Consider the MA(1) series where w, is white noise with variance ora, (a) Derive the minimum mean-square error one-step forecast based on the infinite past, and determine the mean-square error of this forecast. (b) Let ," , be the truncated one-step ahead forecast as given in (3.92). Show that Compare the result with (a), and indicate how well the finite approximation works in this case. 3.12 In the context of equation (3.63), show that. if y (0) 2 0 and y(4) -> 0ash -> 09. then I , is positive definite. 3.13 Suppose & is stationary with zero mean and recall the definition of the PACE