Estimation of p

Because 2 defined in (5.2.9) depends on a2 only through a scalar multiplica­tion, fio defined in (6.1.3) does not depend on a2. Therefore, in obtaining FGLS (6.2.1), we need to estimate only p. The most natural estimator of p is

2 Й/-ІЙ/

р=Ц—. (6.3.3)

Й?_,

t-г

where й, = у, — xlfi. The consistency of pis straightforward. We shall prove its asymptotic normality.

Using Uf = put- + e„ we have

Подпись: JT(P~P) =Подпись: (6.3.4)Tt%™+6′

1 t ’

•* г-2

where

4, – 4= У О» – + (6-3.5)

and

*2 = J. i2 U – Д>’*г-.]2 + (fi – Д>’*-іЧг-. • (6-3.6)

If we assume that НтпГ Г_1Х’Х is a finite nonsingular matrix, it is easy to show that both Д, and Д2 converge to 0 in probability. For this, we need only the consistency of the LS estimator fi and — fi) = 0(1) but not the
asymptotic normality. Therefore, by repeated applications of Theorem 3.2.7 (Slutsky), we conclude that 4Т(р — p) has the same limit distribution as

T

image435

Ik*-‘

Hence the asymptotic normality of p follows from the result of Section 5.4.

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