Corollary 1. In the context of Proposition 3, suppose that
5.10 The GNLM with Restrictions
In dealing with restrictions on the parameter space we are really not follow- ing any procedures that are substantially different from those employed in the previous section; the reader should recall that we had earlier specified the admissible parameter space to be given by 8 C Rk. This immme- diately implies that the admissible space is "restricted" since we do not specify 8 = Rk . The difference in this section is that the restrictions are explicitly stated, so that more structure is imparted on the problem. Basi- cally, we begin as before by specifying that the admissible space is some, perhaps very large, compact subset of Rk ; let this subset be denoted by 8, as in previous discussion. When we impose restrictions, say of the form '1'( (}) = 0 , we deal with therestricted admissible space
8* = {(): () E 8 and r((})= O}, (5.120) where 'I' is an s -element, vector valued, twice continuously differentiable function. The effect of the restrictions is to reduce the dimension of the admissible parameter space. For example, in the case of linear restrictions, where the consequences are seen most clearly, a restriction of the form A(}=a , where A is s xk ,of rank s, means that the restricted admissible space lies in a (k - s) -dimensional subspace of Rk •Of course, we continue to maintain that (}O, the true parameter vector, is an interior point of 8* .We further note that the set
R={():r((})=O} (5.121)
is closed, in the sense that if {(}n :n ~ I} is a sequence in R, with limit
()* then ()* E R. Since the admissible parameter space obeys
8* = 8nR, (5.122)
it follows that 8* is also compact; evidently, we still maintain that (}o E 8* ,and that it is an interior point of 8* as well. The estimation problem is to find the ML estimator within this restricted space, i.e. through the operation SUPOE8* LT ( (} ) . Define the Lagrangian
ST((}) = Lt ( (})+A'r((}), (5.123) where A is a vector of Lagrange multipliers, and derive the first order conditions as
r((}) O. (5.124)
A solution to this system of equations will yield the restricted ML estimator, iJT ,satisfying the condition
sup LT(B)
IIEe'
(5.125) Since 8* is compact, and the other relevant conditions hold, it follows by the discussion in previous sections that iJT is strongly consistent, i.e.
iJT ~. BO .To determine its limiting distribution we have recourse to the devise of expanding the first order conditions, using the mean value theorem of calculus, thus obtaining
(5.126)
where B* and B** are intermediate points between iJT and BO. Under the hypothesis that the restrictions are valid, r(BO) =0, and the limiting distribution is given by33
(5.127) Since LT is defined after divisionby T, we have that,inthis discussion as well asin the previous section,
r,:;:,T8LT(Bo) 1 ~ *
V1- 8B = vT LJw't,
t=l
where w*t is as defined in the last equation of Eq. (5.112), and
. (8LT 0) ]
All = )~mooEllo 8B8B (B) = -C*, [of Eq. (5.117)
33See Dhrymes (1989), Corollary 6, p. 243.
From Eq. (5.118) we see that
1 T
1mL w*t ~ N(O, C*).
vT t=l
It follows therefore, on the assumption that r(80) = 0 is a valid set of restrictions, that
Moreover,
N(O,w), W
e.,ô:Bl l (5.129)
Consider now the special case where there are no cross parameter re- strictions, i.e.
r(8) = [~~~:n = O.
Inthis context
(5.130)
a(8)
[
OT!(¢)
R(8) = _r_ = ---a¢
a8 0 oT2(a)0] - [RIO]0 R
2 '
oa
(5.131) and the covariance matrix of the limiting distribution becomes
[ Wll
W = 0
o o
where
o ] o o '
W44
(5.132)
The preceding discussion has established
(5.133)
Proposition 9. In the context of Proposition 8 suppose, in addition, that the parameters of the model obey the restrictions r(8) , where r is an
s-element vector, continuously34 differentiable and such that R(eO) (8r /8())(eO) is nonsingular. Then, the following statements are true:
1. The restricted ML (RML) estimator obeys, supBT ( () )
OE8
sup[LT ( () )+A'r(())]
OE8
LT(OT)+A'r(OT) = sup LT(());
OE8*
ii. the RML estimator is strongly consistent and its limiting distribution is given by
vT(jjT - eO) '" N(O, BllC*Bii), where BllC* B ll is as defined in Eq. (5.129);
iii. the RML estimators of r/J and a are asymptotically mutually in- dependent;
iv. the Lagrange multiplier A and the RML estimator of () are asymp- toticallymutually independent;
v. in the special case where
r(e) = [~~?~~] ,
in addition to iv, the Lagrange multiplier corresponding to restric- tions on r/J is asymptotically independent of the Lagrange multi- plier corresponding to restrictions on a .
Remark 12. The reader has no doubt already noted the similarity be- tween linear and nonlinear least squares procedures, whether restricted or unrestricted. Although the means by which we arrive at the results are rather different, the results themselves are remarkably similar. Thus, looking at the covariance matrix of the limiting distribution of the ML estimator of e,as exhibited for example in Eq. (5.119), we see that it is mutatis mutandis identical with that of the feasible Aitken estimator.
To produce maximal correspondence between this result and the system of GLM Yt. = Xt.B+Ut. we may write the covariance matrix C-1 of Eq.
(5.119) as C-1
[
, ] -1
. 1 8g 1 8g
)~oo T (8r/J) (~- 0 Ir) (8r/J)
9 = vec(G), 9 = (gd, t=1,2, ... ,T.
34Continuity of its first derivatives is the most convenient simple requirement;
continuity of second derivatives could simplify somewhat a rigorous proof in mak- ing it simpler to use a residual in a certain Taylor series expansion.
If all (T) observations on the it h equation of the set of GLM above is written out we have Y.i = Xi (3.i + U.i, where (3.i contains the ele- ments of the it h column of B, not known a priori to be null. Putting X* = diag(X1 ,X2 ,X3 , . . . ,Xm ) we have that the covariance matrix of the limiting distribution of the feasible Aitken estimator is given by
[i~moo (~X~(L-lQ9!r)X*)r1
Comparing this with the representtion of C-1 we see that identifying X*
with ~ gives a complete correspondence between the two representations.
Moreover, a look at the expression for \[In, in the special case of item v, shows the equivalence between the covariance matrices of the limiting distribution of the restricted and unrestricted estimators. This becomes obvious if, in addition to the identifications above, we identify R1 with the matrix of restrictions in the linear case.