Introductory Econometrics 4th Edition by Jeffrey Wooldridge

تمرين 19

Suppose that y follows the model
yt = + 1zt-1 + ut
ut = ut-1 + et
E(et|lt-1) = 0
where It-1 contains y and z dated at t - 1 and earlier.
(i) Show that E(yt+1|It) = (1 - ) + yt + 1zt - 1zt-1. (Hint: Write ut-1 = yt-1 - - 1zt-2 and plug this into the second equation; then, plug the result into the first equation and take the conditional expectation.)
(ii) Suppose that you use n observations to estimate , 1, and . Write the equation for forecasting yn+ 1.
(iii) Explain why the model with one lag of z and AR(1) serial correlation is a special case of the model
yt = 0 + yt-1 + 1zt-1 + y2zt-2 + et.
(iv) What does part (iii) suggest about using models with AR(1) serial correlation for forecasting