# Generalized Least Squares Estimator

Because 2 is positive definite, we can define 2“1/2 as HD – 1/2H’, where H is an orthogonal matrix consisting of the characteristic vectors of 2, D is the diago­nal matrix consisting of the characteristic roots of 2, and D-1/2 is obtained by taking the (—i)th power of every diagonal element of D (see Theorem 3 of Appendix 1). Premultiplying (6.1.1) by 2~1/2 yields y* = X*fl + и*,

where y* = X-1/2y, X* = 2-1/2X, and u* = X_I/2u. Note that Ем* — 0 and £11*11*’ = I, so (6.1.2) is Model 1 except that we do not assume the elements of u* are i. i.d. here. The generalized least squares (GLS) estimator of fl in model (6.1.1), denoted 0G, is defined as the least squares estimator of fi in model (6.1.2); namely,

fia-(**’X*rlX*’y* (6.1.3)

– (x’x-‘xr’x’x-y

Using the results of Section 1.2, we obtain that EfiG = ft and

КД0 = (Х’Х-,Х)-*. (6.1.4)

Furthermore, GLS is the best linear unbiased estimator of Model 6. (Note that in Section 1.2.5 we did not require the independence of the error terms to prove that LS is BLUE.)