15.5 Prove the following results about conditional means, forecasts, and forecast errors:

a. Let W be a random variable with mean mW and variance s2 w, and let c be a constant. Show that E31W - c2 24 = s2 w + 1mW - c2 2.

b. Consider the problem of forecasting Yt, using data on Yt - 1, Yt - 2, c.

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 E31Yt - ft - 12 2

 Yt - 1, Yt - 2, c4 be the conditional mean squared error of the forecast ft - 1, conditional on values of 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 = E1Yt  Yt - 1, Yt - 2, c2.

(Hint: Review Appendix 2.2.)

c. Let ut denote the error in Equation (15.12). Show that cov1ut, ut - j2 = 0 for j  0. [Hint: Use Equation (2.28).]