16.6 Consider the regression model Yt = b0 + b1Xt + ut, where ut follows the stationary AR(1) model ut = f1ut - 1 + u 

t with u 

t i.i.d. with mean 0 and variance s2 u and f1  6 1; the regressor Xt follows the stationary AR(1) model Xt = g1Xt - 1 + et with et i.i.d. with mean 0 and variance s2e and  g  6 1; and et is independent of u 

i for all t and i.

a. Show that var(ut) =

s2 u

1 - f21 and var(Xt) =

s2e 1 - g21

.

b. Show that cov(ut, ut - j) = fj1 var(ut) and cov(Xt, Xt - j) = gj1 var(Xt).

c. Show that corr(ut, ut - j) = fj1 and corr(Xt, Xt - j) = gj1

.

d. Consider the terms s2v and fT in Equation (16.14).

i. Show that s2v

= s2 Xs2 u, where s2 X is the variance of X and s2 u is the variance of u.

ii. Derive an expression for f .