Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 20

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?

Step 1 of 4

(i)

According to the hints given in the problem, one can get the following equation,

According to the assumption,, and since are all in , it can be written as:

Obtain the required answer by adding one to the time index everywhere.