The previous chapter focused on m-estimation, including ML and NLS estimation. Now we consider a much broader class of extremum estimators, those based on method of moments (MM) and generalized method of moments (GMM).
The basis of MM and GMM is specification of a set of population moment conditions involving data and unknown parameters. The MM estimator solves the sample moment conditions that correspond to the population moment conditions. For example, the sample mean is the MM estimator of the population mean. In some cases there may be no explicit analytical solution for the MM estimator, but numerical solution may still be possible. Then the estimator is an example of the estimating equations estimator introduced briefly in Section 5.4.
In some situations, however, MM estimation may be infeasible because there are more moment conditions and hence equations to solve than there are parameters. A leading example is IV estimation in an overidentified model. The GMM estimator, due to Hansen (1982), extends the MM approach to accommodate this case.
The GMM estimator defines a class of estimators, with different GMM estimators obtained by using different population moment conditions, just as different specified densities lead to different ML estimators. We emphasize this moment-based approach to estimation, even in cases where alternative presentations are possible, as it provides a unified approach to estimation and can provide an obvious way to extend methods from linear to nonlinear models.