LARGE SAMPLE PROPERTIES OF GENERALIZED METHOD OF

IN THIS PAPER we study the large sample properties of a class of generalized method of moments (GMM) estimators which subsumes many standard econometric estimators. To motivate this class, consider an econometric model whose parameter vector we wish to estimate. The model implies a family of orthogonality conditions that embed any economic theoretical restrictions that we wish to impose or test. For example, assumptions that certain equations define projections or that particular variables are predetermined give rise to orthogonality conditions in which expected cross products of unobservable disturbances and functions of observable variables are equated to zero. Heuristically, identification requires at least as many orthogonality conditions as there are coordinates in the parameter vector to be estimated. The unobservable disturbances in the orthogonality conditions can be replaced by an equivalent expression involving the true parameter vector and the observed variables. Using the method of moments, sample estimates of the expected cross products can be computed for any element in an admissible parameter space. A GMM estimator of the true parameter vector is obtained by finding the element of the parameter space that sets linear combinations of the sample cross products as close to zero as possible. In studying strong consistency of GMM estimators, we show how to construct a class of criterion functions with minimizers that converge almost surely to the true parameter vector. The resulting estimators have the interpretation of making the sample versions of the population orthogonality conditions as close as possible to zero according to some metric or measure of distance. We use the metric to index the alternative estimators. This class of estimators includes the nonlinear instrumental variables estimators considered by, among others, Amemiya [1, 2], Jorgenson and Laffont [24], and Gallant [11].2 There the