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

Abstract

We comment on the optimally-transported generalized method of moments (OTGMM) estimator proposed by Schennach&Starck (2026a) and give counterexamples to Theorems 2-6 under their stated assumptions. First, the assumptions used in the small-error analysis are insufficient for consistency in Theorem 2 and asymptotic normality in Theorem 3. Next, we consider the large-error analysis, in which Theorem 4 states that the OTGMM estimator is equivalent to a GMM estimator with modified moments. We show that in a scalar model, Theorem 4 selects a value that differs from the unique OTGMM minimizer and violates the OTGMM sample moment restriction. In an overidentified model satisfying the assumptions used in Theorems 5 and 6, the first component of the Lagrange multiplier has different probability limits under the OTGMM estimator and the GMM estimator with modified moments. Under misspecification, the population value selected by OTGMM depends on the transport metric and on which variables may be adjusted. We give a sufficient condition under which solutions of the modified moment equations also solve the original constrained problem at a given parameter value, and separate conditions for consistency and asymptotic normality of the OTGMM estimator. We also show that Assumption 16 does not imply the matrix bound used in the supplemental proofs and replace Assumption 16 with a matrix condition that yields the bound.