Asymptotic Properties of GMM Estimators

For the discussion of consistency of the GMM estimator it is convenient to start from a basic consistency theorem for the large class of M-estimators defined by either maximizing or minimizing a certain objective function subject to the constraint given by the parameter space (this is the definition used by Gouriéroux and Monfort, 1995a, p. 209). The term ‘M-estimator’ was introduced by Huber (1964) as an abbreviation for minimization estimators. The class of M-estimators also includes maximization approaches like ML and pseudo ML. Accordingly, Amemiya (1985, p. 105) introduces M-estimators as ‘maximum-likelihood-like’ estimators although this seems to be a rather irritating translation having in mind the substantially different approaches summarized under the name M-estimation. Amemiya also uses the terms M-estimator and extremum estimator completely equivalent while other authors, e.g. Newey and McFadden (1994), restrict the latter designation to a subgroup of M-estimators with a quadratic form objective function. These authors only consider estimators optimizing a sample average as belonging to the class of M-estimators.