A general diagnostic of the normal approximation in GMM models

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Abstract This paper proposes a diagnostic method for evaluating the quality of the normal approximation in generalized method of moments (GMM) models through sampling from a quasi-Bayesian parameter distribution. GMM estimates are consistent and asymptotically normal under certain regularity conditions, and researchers often assume normality when conducting inference. However, the literature has identified several violations to the normal approximation, such as in cases with weak instruments or parameters on boundaries. We apply our diagnostic and find meaningful deviations from normality in three well-cited papers, which include examples of well-known violations. We also illustrate one example where the normal approximation works well. Our method is convenient to implement using Markov chain Monte Carlo (MCMC) algorithms and serves as a sanity check for researchers before reporting GMM estimates and standard errors. It enables visualization of the quasi-Bayesian distribution, quantification of deviations from normality, and reporting of alternative estimates and credible sets.