Verify (7.18) and (7.19) for the mean-squared prediction error MSE in

(7.11). Use the orthogonality principle, which implies MSE = E



(yt − X∞

r=−∞

β0 rxt−r)yt



and gives a set of equations involving the autocovariance functions. Then, use the spectral representations and Fourier transform results to get the final result. Next, consider the predicted series ybt = X∞

r=−∞

β0 rxt−r, where βr satisfies (7.13). Show the ordinary coherence between yt and ybt is exactly the multiple coherence (7.20).