Consider the series xt = wtwt1, where wt is a white noise process with mean zero and variance 2 w. Suppose we consider the problem

Consider the series xt = wt−wt−1, where wt is a white noise process with mean zero and variance σ2 w. Suppose we consider the problem of predicting xn+1, based on only x1, . . . , xn. Use the Projection Theorem to answer the questions below.

(a) Show the best linear predictor is xn n+1 = − 1 n + 1 Xn k=1 k xk.

(b) Prove the mean square error is E(xn+1 − xn n+1)

2 = n + 2 n + 1 σ2 w.