# Other Estimators as Special Cases of GMM

It is remarked in the introduction that GMM estimation encompasses many esti­mators of interest in econometrics, and so provides a very convenient framework for the examination of various issues pertaining to inference. In this section, we justify this statement and illustrate it using maximum likelihood estimation. Many econometric estimators are obtained by optimizing a scalar of the form

X N (9). (11.31)

t=1

If Nt(0) is differentiable then the estimator, 0, is the value which solves the asso­ciated first order conditions

T

X dNt(0)/d9 = 0. (11.32)

t=1

Equation (11.32) implies that 0 is equivalent to the MM estimator based on the population moment condition

Е[ЭЩ00)/Э0] = 0. (11.33)

Since dNt(00)/d0 is a (p x 1) vector it can be recalled from Section 3 that 0 is also the GMM estimator based on (11.33).

As an illustration, we now derive the population moment condition implicit in the GMM interpretation of maximum likelihood estimation. Suppose the condi­tional probability density function of the continuous stationary random vector vt given {vt-1, Vt-2,…} is p(v; 00, VM) where V-1 = (V-1, vU,… VU). The maximum likelihood estimator (MLE) of 00 based on the conditional log likelihood function is the value of 0 which maximizes,

Lt (9) = X ln{ p(vt; 9, Vt-1)}.

t=1

This fits within our framework with Nt(0) = ln{p(vt; 0, Vt-1)} and so the MLE can be interpreted as a GMM estimator based on the population moment condition

E[dln{p(v; 0, Ум)}/Э0] = 0. (11.35)

Since MLE is derived from a perfectly valid estimation principle in its own right, it is reasonable to question whether there is any value to this GMM inter­pretation. The advantage of the GMM interpretation is that it focuses attention specifically on the information used in estimation, and thereby facilitates an analy­sis of the consequences of misspecification. For example, the implementation of MLE requires a specific assumption about the distribution of the data. In many cases economic theory does not provide such information and so it is natural to be concerned about the consequences of choosing the wrong distribution. The GMM interpretation reveals that the estimator is still consistent provided (11.35) holds when expectations are taken with respect to the true distribution. Further­more, Theorem 2 can be used to deduce the asymptotic distribution of the MLE in misspecified models23 for which (11.35) holds.