Testing Finite Moment Conditions for the Consistency and the Root-N Asymptotic Normality of the GMM and M Estimators

TL;DR

This paper develops a statistical test to verify if the moment conditions necessary for the consistency and asymptotic normality of GMM and M estimators are satisfied, with applications to retail market data.

Contribution

It introduces a novel testing method for moment conditions crucial for estimator validity, supported by simulations and real data application.

Findings

The test effectively controls size across various data generating processes.

Zero market shares can violate moment conditions, affecting estimator properties.

Simulation results confirm the theoretical properties of the proposed test.

Abstract

Common approaches to inference for structural and reduced-form parameters in empirical economic analysis are based on the consistency and the root-n asymptotic normality of the GMM and M estimators. The canonical consistency (respectively, root-n asymptotic normality) for these classes of estimators requires at least the first (respectively, second) moment of the score to be finite. In this article, we present a method of testing these conditions for the consistency and the root-n asymptotic normality of the GMM and M estimators. The proposed test controls size nearly uniformly over the set of data generating processes that are compatible with the null hypothesis. Simulation studies support this theoretical result. Applying the proposed test to the market share data from the Dominick's Finer Foods retail chain, we find that a common \textit{ad hoc} procedure to deal with zero market shares in analysis of differentiated products markets results in a failure to satisfy the conditions for both the consistency and the root-n asymptotic normality.