Consider the MA(1) series xt = wt + wt1, where wt is white noise with variance 2 w. (a) Derive the minimum mean-square error one-step

Consider the MA(1) series xt = wt + θwt−1, where wt is white noise with variance σ2 w.

(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 xen n+1 be the truncated one-step-ahead forecast as given in (3.92). Show that E

(xn+1 − xen n+1)

2

= σ2(1 + θ2+2n).

Compare the result with (a), and indicate how well the finite approximation works in this case.