# Efficiency of Least Squares Estimator

It is easy to show that in Model 6 the least squares estimator fi is unbiased with its covariance matrix given by

FJe = (X’X)-,X’XX(X’X)-1. (6.1.5)

Because GLS is BLUE, it follows that

(X’X)-1X’XX(X’X)-1 ё (X’X-^X)-1, (6.1.6)

which can also be directly proved. There are cases where the equality holds in

(6.1.6) , as shown in the following theorem.

Theorem 6.1.1. Let X’X and X both be positive definite. Then the following statements are equivalent.

(A) (X’X)-1X’XX(X’X)-‘ = (X’X-‘X)-‘.

(B) XX = XB for some nonsingular B.

(C) (Х’ХГ’Х’ = (X’X-‘X^X’X-1.

(D) X = HA for some nonsingular A where the columns of H are К characteristic vectors of X.

(E) X’XZ = 0 for any Z such that Z’X = 0.

(F) X = ХГХ’ + Z0Z’ + <t2I for some Г and 0 and Z such that Z’X = 0. Proof. We show that statement A => statement В => statement C:

Statement A =>X’XX – X’X(X, X-,X)-‘X’X

=> x’X,/2[i – x-|^х(х/х-,х)-,х/х-,/а]Х,/2х = о

=> X1/2x = X_1/2XB for some В using Theorem 14 of Appendix 1

=> XX = XB for some В => (X’X)-,X’XX = В

=> В is nonsingular because X’XX is nonsingular => statement В

=> X’X-‘X – (B’)-*X’X and X’X"1 = (В’Г’Х’

=> statement C.

Statement C => statement D can be easily proved using Theorem 16 of Appendix 1. (Anderson, 1971, p. 561, has given the proof.) Statement D=* statement A and statement В => statement E are straightforward. To prove statement E => statement B, note

statement E => X’X(Z, X) = (0, X’XX)

=> X’X = (0, X’XXXZfZ’Z)-1, X(X’X)-‘]’

= X’XXIX’XJ-‘X’ using Theorem 15 of Appendix 1 => statement В because X’XX(X’X)-1 is nonsingular.

Fora proof of the equivalence of statement F and the other five statements, see Rao (1965).

There are situations in which LS is equal to GLS. For example, consider

y = X(/? + v) + u, (6.1.7)

where /lisa vector of unknown parameters and u and v are random variables with £u = 0, Ev = 0, £uu’ = <r2I, Erv’ = Г, and £uv’ = 0. In this model, statement F of Theorem 6.1.1 is satisfied because £(Xv + u)(Xv + u)’ = ХГХ’ + <t2I. Therefore LS = GLS in the estimation of fi.

There are situations in which the conditions of Theorem 6.1.1 are asymptotically satisfied so that LS and GLS have the same asymptotic distribution. Anderson (1971, p. 581) has presented two such examples:

Уі = Рі+ ^ + Pit2 + ■ • • + PxtK~l + u, (6.1.8)

and

y, = fix COS Xxt + /?2 cos lyt + . . ■+ Pk cos kilt + u„ (6.1.9)

where in each case (u,} follow a general stationary process defined by (5.2.42). [That is, take the y, of (5.2.42) as the present u,.]

We shall verify that condition В of Theorem 6.1.1 is approximately satisfied for the polynomial regression (6.1.8) with K= 3 when {и,} follow the stationary first-order autoregressive model (5.2.1). Define X = (x,, x2, x3), where the fth elements of the vectors x,, x2, and x3 are xu = 1, хь = t, and x3l = t2, respectively. Then it is easy to show XX = XA, where X is given in (5.2.9) and

(1 ~pf 0 -2p

0 (1 – pf 0

0 0 (1 – pf

The approximate equality is exact except for the first and the last rows.

We have seen in the preceding discussion that in Model 6 LS is generaly not efficient. Use of LS in Model 6 has another possible drawback: The covariance matrix given in (6.1.5) may not be estimated consistently using the usual formula <t2(X’X)-1. Under appropriate assumptions we have pJim a2 = lim Г-1 trMX, where M = I-X(X’X)-‘X’. Therefore plim ^(X’X)"1 is generally different from (6.1.5). Furthermore, we cannot unequivocally determine the direction of the bias. Consequently, the standard t and F tests of linear hypotheses developed in Chapter 1 are no longer valid.

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