Often, the situation arises that the economic theory underlying the phe- nomenon studied requires that the vector 8 obey certain restrictions;
moreover, it would not be unusual that such restrictions apply to param- eters belonging to more than one equation. Implementing this procedure entails estimating the parameter vector 8, subject to such restrictions or, otherwise, using restricted least squares theory to test their validity. Indeed, we may also use the inference theory connected with restricted estimators to test the validity of some of the overidentifying restrictions routinely im- posed on the GL8EM. We shall return to this topic at the end of this chapter.
We now investigate the following problem. Consider the GL8EM in C8F, i.e.,
w = Q8+r,
and let H be n x K, K = L::lim;+Gi ) such that rank(H) =n,
(1.85)
(1.86) and its elements are known; let h be n x 1 with known elements, and let it be desired to estimate the parameters of Eq. (1.85) subject to
H8=h. (1.87)
Remark 19. The rank condition in Eq. (1.86) does not constitute a restric- tion on the generality of the results we are about to obtain. For example, if H* is (n+T) X K of rank nand h* is (n+T) X 1 and we impose
the restrictions H*8 =h* then, to the extent that these restrictions form a consistent system of equations, they can be reduced to
which is simply Eq. (1.87) in conjunction with Eq. (1.86).
Now the 28L8 restricted estimator can be defined as the one that solves the problem
min~(w- Q8)'(w - Q8) {; T
subject to
H8 =h.
The first order conditions to this problem are found by setting to zero the derivatives of
A= (l/T)(w - Q8)'(w - Q8)+2>"(h - H8) (1.88) with respect to 8 and the vector of Lagrange multipliers >.. These condi- tions imply
(1.89) where 82SLSR, 8 2 S L S denote, respectively, the restricted and unrestricted 28L8 estimators of 8 .
The restricted 38L8 estimator can easily be obtained by noting that 38L8 may be thought to be derived by minimizing with respect to 8
given a prior consistent estimator of q>. Thus, the restricted 38L8 estimator can be obtained by solving the first order conditions of a La- grangian problem similar to that in Eq. (1.88). The derivatives with respect to 8 and>. of
imply
A= (l/T)(w - Q8)'<i>~1(w- Q8)+2>"(h - H8) (1.90)
8A 88 8A 88
-(2/T)(w - Q8)'<i>-lQ - 2>.'H=0, 2(h - H8)' = o.
Solving for 15 and ..\ we find
(1.91 ) The first equation in Eq. (1.91) gives the estimator that minimizes, with respect to 15,
1 1- 1
T(W - QI5) ep- (w - QI5),
subject to HI5 = h. Formally, the GL8EM in canonical structural form (C8F) can be thought of, as a system of GLMs; thus, the mechanics of operating with restricted estimators are identical with those in the context of the standard GLM. The latter problem is treated extensively elsewhere, see Dhrymes (1978).11
Remark 20. The perceptive reader will have noticed, from the previous section, that the 28L8 estimatoris not essentially a systemwide estima- tor. It was presented as such only for simplicity and economy of exposition, and for ease of comparison with the 38L8 estimator which isessentially a systemwide estimator. What is meant by that is that 38L8 "naturally"
obtains an estimator for all the parameters of the system (i.e., the elements of the vector 15) simultaneously. The 28L8 estimator, on the other hand, is "naturally" a single equation procedure, i.e., it obtains estimators of the subvectors, l5.i , of 15 one at a time-i.e., it estimates the parameters of the structural equations, one equation at a time. To the extent that the restrictions imposed by H "cut across" equations, i.e., the rows of H act on parameters appearing in more than one structural equation, the "single equation" character of 28L8 is destroyed. One may ask, then, why use the restricted estimator in Eq. (1.89) instead of that in Eq. (1.91). There may be many possible responses to that, but we need not discuss the merits or disadvantages of the two procedures. We simply treat the matter as a conceptual problem, and one to which we have provided a solution. It is neither desirable nor necessary, at this juncture, to enter upon a discourse on the possible applications of either procedure; we leave this to the judge- ment of the user, given the properties (as well as the vulnerabilities) of the two sets of estimators.
11The difference here is that we have divided the minimand by T,while in the GLM context we did not do so. Itis necessary, in the GLSEM context, since the justification for this estimation procedure is essentially asymptotic. Ifwe fail to divide by T, sample information embodied in Q' Q ,or Q'<I>-lQ ,which grows with T, will ultimately overwhelm or render irrelevant the "information" in the constraint H/5= h, which does not. This is evident from the second equation of Eq. (1.91) which shows that if we fail to divide by T,the Lagrangian multiplier displayed therein would be ill defined, in the limit.