Tests for Relevance and Redundancy of Moment Conditions

In the paper, we consider generalized method of moments (GMM) and propose statistics to test relevance, conditional relevance and redundancy of moment conditions, as well as they modifications to check if moment conditions help with identification of a particular set of unknown parameters. The statistics have, under the null hypothesis and when the estimator of moments covariance matrix converges at the rate square root of T, a chi-squared limiting distribution. We show that, though the concepts of redundancy and conditional relevance are very close, necessary and sufficient conditions for them may differ, which happens in the case of partial redundancy and relevance