ML Estimation of Linear Regression Model with AR(1) Errors and Two Observations. This is based on Magee (1993). Consider the regression model yi = xiβ + ui, with only two observations i = 1, 2, and the nonstochastic |x1| = |x2| are scalars. Assume that ui ∼ N(0, σ2) and u2 = ρu1+

with |ρ| < 1. Also,  ∼ N[0, (1 − ρ2)σ2] where  and u1 are independent.

(a) Show that the OLS estimator of β is (x1y1 + x2y2)/(x21

+ x22

).

(b) Show that the ML estimator of β is (x1y1 − x2y2)/(x21

− x22

).

(c) Show that the ML estimator of ρ is 2x1x2/(x21

+ x22

) and thus is nonstochastic.

(d) How do the ML estimates of β and ρ behave as x1 → x2 and x1 → −x2? Assume x2 = 0.

Hint: See the solution by Baltagi and Li (1995).