Corollary 1. In the context of Proposition 3, suppose that
6.4.1 Reformulation of GMM as NL2SLS and NL3SLS
This method was introduced into the literature by Hansen (1982) as a novel estimation procedure for nonlinear models. As an estimation procedure, however, GMM is a rather minor modification of the methods examined in the previous two sections, in the context of nonlinear models with additive structural errors. Its framework differs from that employed in NL2SLS and NL3SLS in two respects: first, the set of "instruments" by which iden- tification and estimation is carried out is left as a primitive in NL2SLS and NL3SLS; in practice, the class of admissible instruments is taken to be the space spanned by the predetermined variables of the model in question. In that context, given the standard i.i.d. assumption regarding the structural error terms, all predetermined variables may serve as instruments.
In the GMM context, which was inspired by the "rational expectations"
approach, "instruments" are defined within the model through assertions that some variables are uncorrelated with ("orthogonal" to) the structural errors; second, the errors are asserted to be strictly stationary25 (as are the dependent variables and instruments); thus, the proof of consistency, and asymptotic normality is obtained under more general conditions, in GMM, than was the case in the earlier literature.
Abstracting from these motivational aspects, the GMM is nothing more than the GNLSEM with additive errors, treated in bascially the same man- ner as NL2SLS and NL3SLS. There is no difficulty, in the latter context, with a specification that the error process is stationary; what such an as- sertion would mean is that the class of instruments should be confined to some space spanned by the exogenous variables of the model, which are asserted to be independent or uncorrelated with the error process. Thus, the major difference between GMM and NL2SLS, NL3SLS is the man- ner in which "instruments" are defined or rationalized, or motivated in the two strands of this literature and nothing more.
We shall briefly outline the procedure as given in Hansen, and then recast the problem in the framework of the previous two sections. Without loss
25Hansen is not explicit as to what stationarity is being assigned, strict sta- tionarity or covariance stationarity. One is left to infer the precise meaning from the context, see also the discussion below.
of generality, write the econometric model as26
Ytã g(Yt., Xtã;8)+Ut.= gtã(8)+Utã, t = 1,2, ...,T,
Ztã h(Yt., Xt.,8), t = 1,2, ... ,T. (6.104) Hansen's assumptions imply that the U - and Z-processes are jointly (strictly) stationary.27 The question now arises as to what we are to as- sume regarding the "predetermined" variables, Xtã = (lL-1,pd ,where Ptã
is the vector of exogenous variables and lL-1 = (Yt-h Yt-2ã, ... ,Yt-k.) . Actually, far from adding generality and/or complicating the arguemnt it is simpler to assume the exogenous variables to be (strictly) sta- tionary than to allow them to be arbitrary nonstochastic sequences. Ifwe allow the latter, we destroy the (strict) stationarity property of the class of predetermined variables.28 Itis further assumed that29
E(Ut. ®zd' = 0, for all t. (6.105) Thereafter the problem is defined as a minimum chi-squared problem with
26In Hansen, the model is written as uc. = F(Xt.;B) and zô.= G(Xt.;B) .In Hansen's notation xô. stands for (yt., Xt.) in our notation, the distinction be- tween predetermined and jointly dependent variables being muted. It is assumed that {Xt. : t 2': 1} is a (strictly) stationary process, a term to be defined below. Note that this implies that Ut. and zô. are jointly strictly stationary processes.
27 A stochastic sequence {Xn :n 2':1} defined on the probability space (n, A, P) is said to be strictly stationary, if for every k, P(A(lằ) =P(A(k+1)) ,where A(l) = {w: (X1,X2 , . . . ) E E} and A(k+l) = {w: (Xk+l,Xk+2, ... ) E E}, for any E E B(ROO) • It is also said to be ergodic if and only if every invariant set relative to it has P-measure either zero or one. For greater detail on these and related issues, see Dhrymes (1989) Ch. 5, pp. 338ff, especially pp. 357ff.
Ergodicity, however, is not generally familiar to econometricians, and we will refrain from using it unless the context makes it is absolutely necessary.
Hansen actually does not specify the probability characteristics of the U - and
Z-processes; it is merely stated that {xn : n 2': 1} is a stationary process;
since the author defines Un = F(xn ;B), Zn = G(xn ;B) ,for suitably measurable functions, and subsequent arguments imply that the Z - and U-processes are jointly stationary, one has to conclude that the stationarity in question must be strict stationarity since a measurable transformation of a weakly stationary process is not necessarily weakly stationary.
28Incidentally this is the "reason" why it is not only convenient, which it cer- tainly is, but almost imperative that the "instruments" be asserted to be (strictly) stationary. In this fashion, all variables in the model are stationary, thus sim- plifying matters considerably. On the other hand, in the context of rational expectations models, the underlying theory implies that certain variables are uncorrelated (orthogonal in the appropriate Hilbert space context) with certain other variables.
29These are the "orthogonality" conditions which define the "instruments" of the model.
370 6. Topics in NLSE Theory
weighting matrix J;JT, i.e. one obtains an estimator of 0, say OT by the operation
where
inf ST(Y,X;O) =ST(OT),
lIEe (6.106)
(6.107) Itis further assumed that JT~. J , where J is an appropriate nonstochas- tic matrix. This is a formulation which is, mutatis mutandis, identical to that given in Dhrymes (1969), Amemiya (1974), Jorgenson and Laffont (1974) and others, which may be seen as follows: put
1 , 1 ,
Tvec(Z U) = T(IQ9 Z )(y - g), and note that
(6.108)
ST where30
1 , , ,
2T2 (y - g) (IQ9Z)WT(IQ9 Z )(y - g), W T = JTJr,
Z y
(zd, Y = (yd, G(O) = (gdO)), X vec(Y), u = vec(U), 9 = vec[G(O)].
(xd, U (ud,
(6.109) Itis evident that, apart from the scalar term T-2 ,the objective function of GMM is the one considered, e.g., in Eq. (6.98) above, with
D = (IQ9Z) W T (IQ9Z')!
Since, for every T, the matrix WT has the nonsingular decomposition WT = P;PT we may proceed as follows: Consider the system y = 9+u
and transform on the left by PT(IQ9Z') to obtain
PT(IQ9Z')(y - g)= PT(IQ9 Z')u; (6.110) the method of GMM consists of applying nonlinear least squares to the system in Eq. (6.110); but this is exactly the nonlinear 2SLS or 3SLS framework. The difference is, as we had noted earlier, that in the latter
"instruments" is left as a primitive context, while in the former "instru- ments" is something that is defined by the rational expectations framework.
30The objective function Sr has been divided by two for notational conve- nience only.
Thus, the formal or estimation aspects of the GMM are neither more nor less general than what we had studied in the preceding two sections, and the particularly complex manner in which its formulation is stated in Hansen (1982) is a definite barrier to a clear understanding of what it entails.