Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 11

Consider the regression model Y t = 0 + 1 X t + u t where u, follows the stationary AR(1) mode with
. with mean 0 and variance and |
1 | 1, the regressor X t follows the stationary AR(1) mode with e t i.i.d. with mean 0 and variance and | 1 | 1, and e t is independent of for all t and i.
a. Show that var and var( X t ) =
.
b. Show that cov and cov
.
c. Show that corr and corr
.
d. Consider the terms and f T in Equation (15.14).
i. Show that where is the variance of X and
, is the variance of u.
ii. Derive an expression for f