Asymptotic Normality of the Least Squares Estimator

As was noted earlier, the first step in obtaining FGLS is calculating LS. There­fore the properties of FGLS depend on the properties of LS. The least squares estimator is consistent in this model if assumption В of Theorem 6.1.2 is satisfied because in this case assumption A is satisfied because of Theorem

5.2.3. In fact, Theorem 5.2.3 states that assumption A is satisfied even when u follows a more general process than AR( 1). So in this subsection we shall prove only the asymptotic normality of LS, and we shall do so for a process of u more general than AR(1) but only for the case of one regressor (that is, К = 1) in Model 6. (Anderson, 1971, p. 585, has given the proof in the case of К regressors. He assumed an even more general process for u than the one we assume in the following proof.)

Theorem 6.3.1. Assume К = 1 in Model 6. Because X is a Г-vector in this case, we shall denote it by x and its ah element by xt. Assume

(A) limr_„ ^ = c, # 0.

(B) u, = 2y“_02"-0|ф;1 < °°> where (ej are i. i.d. with Ее, = 0 and

Then fTifi — /J) — N(0, c, 2c2), where c2 — limr_„, Г-‘х’Гии’х.

image430

Proof. We need only prove Г-1/2х’и —* jV(0, c2) because then the theorem follows from assumption A and Theorem 3.2.7 (iii) (Slutsky). We can write

But V(A2) = T 1 x’ xer 2(2JL „+1ф^. Therefore A2 can be ignored if one takes N large enough. We have

Подпись:

image432

(6.3.2)

^Аи+Ліг+А13.

But V(An) = T~1 a2NM2(1?L 1ІФ/І)2, which goes to 0 as Г—»°° for a fixed N. The same is true for AI3. The theorem follows by noting that 2jL0<£/*t+/ satisfies the condition for x, in Theorem 3.S.3.

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