Corollary 1. In the context of Proposition 3, suppose that
6.2.1 Identification and Consistency of NL2SLS
Itwould be futile to attempt to characterize identification here in the same manner as is done for the GLSEM; although it is a necessary condition that none of the equations of the GNLSEM be equivalent to a linear com- bination of other structural equations, it is clearly not a sufficient one. By far, the most essential character of identification in this context is that the limit of STi , whether in probability, a.c., or in quadratic mean be a con- trast function, i.e. be nonnegative and assume its global minimum (i.e.
zero) if and only if ().i = ()~.
Let us now formalize this discussion; first, suppose
[
I ( I ) -1 I ]
S .(())= ~ (Y.i - g.i) Wi Wi Wi Wi (Yãi - g.i)
Tt 2T T T T
a.c. orP K. (() . ()O)
----7 ' l - . t " 2 •
For the limit above to be well defined, we require of the matrix Wi
(A.l) rank(Wi ) = ki, (W;WdT)~'M ii >0,
in an appropriate mode. 18 Second, if the convergence above is taken as given, how do we define identification for the it h equation? Assuming, in addition to (A.l),
(A.2) STi((}) a.~pK,((}.i ,(}~) ,uniformly in e;
(A.3) Ki((}.i,(}~) ~ 0, and K i((}\l ),(}~) = K i((}\2),(}~) , if and only if (}\1) = (}\2) .
we see that the (P or a.c.) limit of STi is a contrast function; thus, if the conditions above hold, the NL2SLS estimator is a minimum contrast estimator. By Propositions 3, 4, and Corollary 1of Chapter 5, we may conclude, therefore, that this estimator converges a.c. or P to the true parameter vector, according as STi converges to K, a.c. or P. Finally, we come to the ultimate question: what must be true about the structural error process, i.e. the vector sequence U.i, and the vector sequence g.i for the consistency and identification results above to hold? From the definition of the function K, we see that we require the existence of the limits of
(6.86) in one mode of convergence or another. We require, therefore the assump- tions
(A.4) (W; g.dT) a.~p f.i((}.i ,(}~) uniformly for ()Ee;
(A.5) (W; u.dT) p~c. 0, as wellas the technical assumption
(A.6) the admissible parameter space e is compact, and the true param- eter point, ()~,is an interior point of e.
Note that in the context of this discussion (A.1)and (A.4) imply (A.2). As a matter of research strategy, as well as exposition, it is best if assumptions such as (A.2) are avoided, in that they refer to synthetic entities; it is preferable to confine assumptions to more primary entities such as those in (A.l) and (A.4). In any event, utilizing (A.l) through (A.6) and assuming the mode of convergence is a.c., 19 we have that
(6.87)
18For example, if we view the elements of Wi as nonrandom and {Wi: T :0:: ki}
as a matrix sequence in some space, say X, convergence as an ordinary limit (OL) is appropriate; if its elements are random, but (asymptotically) indepen- dent of the structural error process, convergence either in probability or a.c. is appropriate.
19Although this may appear unduly strong to the reader, we note that by Kol- mogorov's zero-one law, a sequence of independent variables either converges a.c., or does not converge at all.
uniformly for e.i E e ,where
Evidently K, :::: 0 , and the parameters of the it h equation are identified if and only if
Moreover,
and the strong consistency of the NL2SLS estimator follows from the iden- tification condition.
In the preceding discussion we have proved
Theorem1. Consider the GNLSEM ofEq. (6.82) under assumptions (A.I) through (A.6) of this section; then, the following statements are true:
i. the parameter vector in the it h equation, is identified;
ii. the NL2SLS estimator of that structural parameter vector, iii,obeys {j .'1, ~. eO.1,"
Remark 6. In the preceding it is assumed (implicitly) that, if dynamic, the GNLSEM is stable, and that the exogenous variables of the model are "well-behaved". All that is meant to be conveyed by these provisos is that nothing in the exogenous variables, and/or the dynamic aspects of the model, invalidates any of the six basic assumptions made above. We do not propose, however, to examine these issues. Notice, further, the complete equivalence between the assumptions here and in the GLSEM. Assump- tion (A.I) has an exact counterpart in the GLSEM, as does (A.3). In the GLSEM, (A.4) corresponds to the statement that (W: Zi/T) converges, where Zi is the matrix of observations on the (right hand) explanatory variables; or, equivalently, to the statement that (W: ZitJ.i/T) converges uniformly for 8.i in the admissible parameter space! Similarly, (A.5) and (A.6) have exact counterparts in the GLSEM. Thus, the conceptual differ- ences of solving the estimation problem in the context of the GLSEM and the GNLSEM with linear error terms are rather miniscule.
Remark 7. Although the estimator above is known as "NL2SLS", it is not clear what the "two stages" are. Itis actually better described as lim- ited information nonlinear simultaneous equations (LINLSE) instrumental variables (IV) estimator, based on an ill-specified model.20 In the linear
20 It is regrettable that the term "limited information" has become somewhat
case, the stages in question are quite apparent, at least in the conceptual framework, if not in the computational procedure. Inpoint of fact, the es- timator above is just one within a class of estimators determined by the choice of the "instrument" matrix Wi.