15.12 Consider the stationary AR(1) model Yt = b0 + b1Yt−1 + ut, where ut is i.i.d.

with mean 0 and variance s2 u . The model is estimated using data from time periods t = 1 through t = T, yielding the OLS estimators b n

0 and b n

1. You are interested in forecasting the value of Y at time T + 1—that is, YT +1. Denote the forecast by Y n

T + 1T = b n

0 + b n

1YT.

a. Show that the forecast error is YT + 1 - Y n

T + 1T = uT +1 - 31b n

0 - b02 +

(b n

1 - b1)YT].

b. Show that uT + 1 is independent of YT.

c. Show that uT + 1 is independent of b n

0 and b n

1

d. Show that var1YT + 1T - Y n

T + 1T2 = s2 u + var31b n

0 - b02 + 1b n

1 - b12YT].