Common approaches to statistical inference for structural and reduced-form parameters in empirical economic analysis are based on the  root-n asymptotic normality of the GMM and M estimators. The canonical  root-n asymptotic normality  for these classes of the estimators requires at least the second moment of the score to be bounded . In this article, we present a method of testing this condition for the asymptotic normality of the GMM and M estimators. Our test has a uniform size control over the set of data generating processes  compatible with the root-n asymptotic normality . 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 ad hoc procedure to deal with zero market shares results in a failure of the root-n asymptotic normality 
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 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 ad hoc procedure to deal with zero market shares results in a failure to satisfy the conditions for both the consistency and the root-n asymptotic normality .
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