15.9 Consider the “constant-term-only” regression model Yt = b0 + 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.

a. Show that the OLS estimator is b n

0 = T -1gTt

= 1Yt.

b. Show that the (infeasible) GLS estimator is b nGLS 0 =

(1 - f1) - 1(T - 1) - 1gTt

= 2(Yt - f1Yt - 1). [Hint: The GLS estimator of b0 is (1 - f1)-1 multiplied by the OLS estimator of a0 in Equation

(15.23). Why?]

c. Show that b nGLS 0 can be written as b nGLS 0 = (T - 1)-1gT - 1 t = 2 Yt +

(1 - f1)-1(T - 1)-1(YT - f1Y1). [Hint: Rearrange the formula in (b).]

d. Derive the difference b n

0 - b nGLS 0 and discuss why it is likely to be small when T is large.