  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.
