Springer Texts in Business and Economics

ML Estimation of Linear Regression Model with AR(1) Errors and Two

Observations. This is based on Baltagi and Li (1995).

a. The OLS estimator of " is given by

22

"ols = Y^ xiyi/X)x2 = (y1x1 C y2x2)/ (x1 + x2) . i=1 i=1

b. The log-likelihood function is given by logL = — log 2л — log с2 — (1/2) log(1 — p2) — (u2 — 2pu1u2 + u2) /2o2(1 — p2); setting @logL/@o2 = 0 gives сі2 = (u2 — 2pu1u2 + u2) /2(1 — p2); setting @logL/@p = 0 gives po2(1 — p2) + u1u2 + p2u1u2 — pu2 — pu1 = 0; substituting 02 in this last equation, we get p = 2u1u2/ (u2 + u2); setting @logL/@" = 0 gives u2(x2 — px1) + u1 (x1 — px2) = 0; substituting p in this last equation, we get (u1x1 — u2x2) (u2 — u2) = 0.Notethatu1 = u2 impliesap of ±1,and this is ruled out by the stationarity of the AR(1) process. Solving (u1x1 —u2x2) = 0 gives the required MLE of ":

"mle = (y1x1 — y2x2)/ (x1 — x^ .

c. By substituting "mle intou1 andu2, onegetsir 1 = x2(x1y2 —x2y1)/ (x2 — x2) and u2 = x1(x1y2 — x2y1)/ (x1 — x|), which upon substitution in the expression for p give the required estimate of p : p = 2x1x2/ (x2 + x2).

d. If x1 ! x2, with x2 ф 0, then p! 1. For "mle, we distinguish between two cases.

(i) For yi = y2, " mle! y2/(2x2), which is half the limit of " ols! yilx2. The latter is the slope of the line connecting the origin to the observation

(x2,y2).

(ii) For y1 ф y2, "mle! ±1, with the sign depending on the sign of x2, (yi — y2), and the direction from which x1 approaches x2. In this case, "ols! y/x2, where y = (y1 + y2)/2. This is the slope of the line connecting the origin to (x2, y).

Similarly, if x1 ! —x2, with x2 ф 0, then p! — 1. For "mle, we distinguish between two cases:

(i) For y1 = — y2, "mle! y2/(2x2), which is half the limit of "ols!

y2/x2.

(ii) For y1 ф —y2, "mle! ±1, with the sign depending on the sign of x2, (y1 + y2), and the direction from which x1 approaches —x2. In this case, "ols! (y2 — y1)/2x2 = (y2 — y1)/(x2 — x1), which is the standard formula for the slope of a straight line based on two observations. In conclusion, "ols is a more reasonable estimator of " than "mle for this two-observation example.