15.6 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 0 f1 0 6 1; the regressorXt follows the stationary AR(1)

model Xt = g1Xt - 1 + et with et i.i.d. with mean 0 and variance s2e and 0 g 0 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 - g2 1 .

b. Show that cov(ut, ut - j) = f j 1var(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 ƒT in Equation (15.14).
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  .