Seemingly Unreleted Regression Model

The seemingly unrelated regression (SUR) model proposed by Zellner (1962) consists of the following N regression equations, each of which satisfies the assumptions of the standard regression model (Model 1):

y.-X/A+u,, 1=1,2______ ,N, (6.4.1)

where y, and u, are T-vectors, X, is a T X Kt matrix, and fit is a A, -vector. Let utt bethetthelementofthevectoru,-.Thenweassumethat(uu, Цц,. . . , uM)is an i. i.d. random vector with Eu„ = 0 and Cov (ий, Uj,) = a0. Defining у = (УЇ, У2. • • • ,У*)’> Р^іРиРг,- • • ,0’nY, u = (ui, u^,. . . ,ui,)’, and X = diag (Xj, X2,. . . , ХД we can write (6.4.1) as

у = Xfi + u. (6.4.2)

This is clearly a special case of Model 6, where the covariance matrix of u is given by

£uu’ = ft = 2 ® Iy, (6.4.3)

where 2 = {(Ту) and ® denotes the Kronecker product (see Theorem 22 of Appendix 1).

This model is useful not only for its own sake but also because it reduces to a certain kind of heteroscedastic model if 2 is diagonal (a model that will be discussed in Section 6.S) and because it can be shown to be equivalent to a certain error components model (which will be discussed in Section 6.6).

The GLS estimator of/f is defined by fiG = (X’ft^XJ-‘X’ft^y. Because of

(6.4.3) we have ft-1 = 2-1 ® I, using Theorem 22 (i) of Appendix 1. In the special case in which X, = X2 = . . . = XN, we can show fio = fi as follows: Denoting the common value of X, by X and using Theorem 22 (i) of Appendix 1 repeatedly, we have

Pa = [X'(2-> ® ЦХГ’Х’ф-1 © I)y

= [(I ® X’X2_1 ® Щ ® X)]-‘(I ® X’X2-‘ ® l)y

= (2-1 ® Х’ХГЧЇ-1 ® X’)y

= [I ® (X’X)-‘X’]y

“(x’x^x’y.

The same result can be also obtained by using Theorem 6.1.1. Statement E is especially easy to verify for the present problem.

To define FGLS, we must first estimate X. A natural consistent estimator of its ijth element is provided by where fi(- — yt — X,/?, are the

least squares residuals from the zth equation. These estimates are clearly consistent as T goes to *> while N is fixed. Because of the special form of fl-1 given in the preceding paragraph, it is quite straightforward to prove that FGLS and GLS have the same asymptotic distribution (as T —*°°) under general assumptions on u and X. The limit distribution of — Д) or 4T{Pf~ P) is N[0, (lim 7’-1Х, й_1Х)_1]. Suitable assumptions on u and X can easily be inferred from Theorem 3.5.4.

FGLS is generally unbiased provided that it possesses a mean, as proved in a simple, elegant theorem by Kakwani (1967). The exact covariance matrix of FGLS in simple situations has been obtained by several authors and compared with that of GLS or LS (surveyed by Srivastava and Dwivedi, 1979). A particu­larly interesting result is attributable to Kariya (1981), who obtained the following inequalities concerning the covariance matrices of GLS and FGLS in a two-equation model with normal errors:

(6.4.5)

where r = rank[X,, X2].