Prove the following results about conditional means, forecasts, and forecast errors:
(a) Let W be a random variable with mean μW and variance σ2W and let c be a constant. Show that

(b) Consider the problem of forecasting Yt using data on Yt-1, Yt-2,... Let ft-1 denote some forecast of Yt, where the subscript t - 1 on ft-1 indicates that the forecast is a function of data through date t - 1. Let E[(Yt - ft-1)2|Yt-1, Yt-2,...] be the conditional mean squared error of the forecast ft-1, conditional on Y observed through date t - 1. Show that the conditional mean squared forecast error is minimized when ft-1 = Yt|t-1, where Yt|t-1 = E(Yt|Yt-1, Yt-2,...).
(c) Let ut denote the error in Equation (14.14). Show that cov(ut, ut-j) = 0 for j ‰  0.

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E[ (W-c)]= ow+ (Hw-c)*. Mt)+