Counterexamples and Sufficient Conditions: Comments on "Optimally-Transported Generalized Method of Moments"

원문 초록

We comment on the optimally-transported generalized method of moments (OTGMM) of Schennach and Starck (2026) and give counterexamples to Theorems 2-6. In the small-error analysis, the assumptions do not ensure consistency or asymptotic normality (Theorems 2-3). In the large-error analysis, Theorem 4 states the equivalence between the OTGMM estimator and the GMM estimator with modified moments, and Theorems 5-6 assert consistency and asymptotic normality. Through counterexamples, we show that the equivalence in Theorem 4 fails and a component of the Lagrange multiplier has different limits under the two estimators (Theorems 5-6). We also point out that, under misspecification, the population value selected by OTGMM depends on the metric. We give sufficient conditions and establish results following the original motivation for OTGMM. Our corrections to Theorems 4-6 change their assumptions or interpretation more substantially than those to Theorems 2 and 3. Furthermore, we correct Assumption 16 and the test's null distribution.