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

Masahiro Kato. 2026. Preprint (SSRN 7208218). 日本語版

This study comments on the optimally-transported generalized method of moments (OT-GMM) proposed by Schennach and Starck. We give counterexamples to Theorems 2 through 6 under the assumptions the original paper states, and then supply sufficient conditions under which the results hold, treating consistency and asymptotic normality separately.

The problem

In GMM, when the moment conditions outnumber the parameters, no parameter value may satisfy all of them at once. OT-GMM responds by letting the variables carry errors of the least mean-square magnitude needed to satisfy every moment condition simultaneously, with the magnitude measured by optimal transport, instead of reweighting the data as generalized empirical likelihood does. The original paper states the guarantees for this estimator as theorems: consistency, asymptotic normality, and an equivalence with a GMM estimator built on modified moments. Whether those theorems follow from the stated assumptions alone is the question this comment takes up.

The approach

We check each theorem on concrete models that satisfy the stated assumptions. In what the original paper calls the small-error analysis, we construct examples in which the assumptions hold and yet consistency or asymptotic normality fails. In the large-error analysis, we use a scalar model and an overidentified model, and compare the OT-GMM minimizer directly with the solution of the modified moment equations that Theorems 4 through 6 refer to. For each result, we then derive additional conditions that fill the gap the counterexample exposes.

Main results and conditions

In the small-error analysis, the stated assumptions are insufficient for consistency in Theorem 2 and for asymptotic normality in Theorem 3.

The center of the large-error analysis is Theorem 4, which states that the OT-GMM estimator is equivalent to a GMM estimator with modified moments. In a scalar model, the value Theorem 4 selects differs from the unique OT-GMM minimizer and violates the OT-GMM sample moment restriction. In an overidentified model satisfying the assumptions of Theorems 5 and 6, the first component of the Lagrange multiplier has different probability limits under OT-GMM and under the GMM estimator with modified moments. Under misspecification, the population value OT-GMM selects depends on the transport metric and on which variables may be adjusted.

We also give the missing conditions: 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 under which the OT-GMM estimator is consistent and asymptotically normal. Assumption 16 does not imply the matrix bound used in the supplemental proofs, so we replace it with a matrix condition that yields the bound.

Relation to earlier work

OT-GMM was proposed by Schennach and Starck, motivated by giving GMM results a logical interpretation even when the overidentification test rejects; rather than reweighting the data, it allows the variables to carry the smallest errors, measured by optimal transport, that make all moment conditions hold (2026, Optimally-Transported Generalized Method of Moments). This comment examines which assumptions support each of the paper's guarantees, locates the gaps between the stated assumptions and the proofs through counterexamples, and supplies conditions that close them. Estimation built on moment conditions also connects this comment to another study of ours, which constructs asymptotically unbiased synthetic control methods by moment matching (accepted at Journal of Causal Inference, Asymptotically Unbiased Synthetic Control Methods by Moment Matching).

Where it applies

The reader we have in mind uses OT-GMM in estimation or studies its theoretical properties. The counterexamples and the sufficient conditions are organized theorem by theorem, so they give a starting point for checking which conditions an application has to verify.

Paper and materials

BibTeX

@misc{otgmm-comments,
  author       = {Masahiro Kato},
  title        = {Counterexamples and Sufficient Conditions: Comments on "Optimally-Transported Generalized Method of Moments"},
  year         = {2026},
  url          = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7208218},
}