Generalized Method of Moments (GMM)

Overview

The Generalized Method of Moments extends MoM to handle overidentified models where more moment conditions than parameters are available.

Setup

Given \(r > p\) moment conditions \(E[g(X, \theta)] = 0\), the GMM estimator minimizes:

\[ \hat{\theta}_{GMM} = \arg\min_{\theta} \left[\frac{1}{n} \sum_{i=1}^n g(x_i, \theta)\right]^T W \left[\frac{1}{n} \sum_{i=1}^n g(x_i, \theta)\right] \]

where \(W\) is a positive-definite weighting matrix.

Optimal Weighting

The optimal \(W\) is the inverse of the asymptotic variance of the moment conditions, yielding the most efficient GMM estimator.