When Moving the Data Does Not Fix the Model

A method that moves observations

Economic models often imply moment conditions: restrictions requiring certain averages to equal zero. Optimally-transported generalized method of moments, or OTGMM, chooses model parameters and adjusted observations so those restrictions hold, while minimizing the average squared distance between adjusted and observed data.

In his preprint, Counterexamples and Sufficient Conditions: Comments on “Optimally-Transported Generalized Method of Moments”, Masahiro Kato examines this method as presented by Susanne Schennach and Vincent Starck. The supplied arXiv version, revised on November 4, 2026, challenges the assumptions supporting five theorems and offers conditional corrections. Its mathematical counterexamples are constructed models, not evidence about how frequently problems occur in economic datasets.

Tiny adjustments, wrong answers

One part of the original analysis concerns adjustments that become small as the sample grows. Kato constructs a smooth model in which the stated assumptions nevertheless fail to guarantee consistency—the property that estimates approach the true parameter as more observations arrive.

In that construction, changing at most two observations by vanishing amounts can satisfy the sample restrictions at an incorrect parameter more cheaply than at any parameter near the truth. Along a sequence of increasing sample sizes, the probability that every globally optimal estimate remains away from the true value stays bounded away from zero.

The mechanism is unusually sensitive moment functions: near selected observations, a tiny movement can produce a large change in a sample restriction. Kato’s correction imposes a uniform bound on this sensitivity, alongside other assumptions. His normal-limit result requires more: it assumes a particular convergence rate and uniform local approximations, rather than deriving them from consistency alone.

An equation is not an optimum

A separate criticism concerns the general analysis, which does not require small adjustments. The original method selects, for each observation, the nearest solution of an optimization first-order condition, then builds modified moment equations from those selections.

Kato shows that solving these equations need not solve the original minimum-cost problem. First-order conditions identify stationary candidates; selecting the nearest candidate does not necessarily minimize the relevant Lagrangian expression or deliver the cheapest feasible adjustments.

In a one-observation example, the modified equations produce a transport cost about 40.6 percent above the global minimum. This is a mathematical illustration, not an estimated economic effect. Another constructed model gives different large-sample limits for one Lagrange multiplier under genuine OTGMM and the modified-moment estimator. That undermines treating their joint statistical guarantees as interchangeable.

Kato supplies sufficient conditions for minimum-cost adjustments at a fixed parameter, including a tractable case with affine moment functions and unrestricted adjusted observations. Finding the global OTGMM solution still requires minimizing the resulting cost across parameter values.

What does the closest model mean?

Even successful optimization leaves an interpretation problem. Under misspecification—when no parameter makes the observed population satisfy the restrictions—the selected parameter can depend on the distance metric and which variables may move.

Kato illustrates this with two variables whose adjusted means must equal one common parameter. When their observed means differ, making adjustments to one variable more expensive pulls the selected parameter toward that variable’s mean. Fixing one variable entirely forces the parameter to equal its mean.

A separate measurement-error example shows that OTGMM, using only the ordinary least-squares moment condition, leaves the data unchanged and returns the usual attenuated slope. Kato explicitly notes that this example is just identified, unlike the overidentified setting motivating OTGMM. Its lesson is narrower: transport alone does not recover a latent coefficient without additional restrictions connecting observed data, latent variables and measurement error.

Replication without a global guarantee

Kato reports reproducing the coefficients on log highway kilometers in the original study’s two main empirical specifications: approximately 0.39935 and 0.65295, matching the published rounded values of 0.40 and 0.65.

That reproduction does not certify global optimality. For the first specification, using 66 observations and keeping the dependent variable and constant fixed, he establishes a strict local minimum with transport cost about 60.3 percent above a lower-cost feasible point. The highway coefficient barely changes, but several control coefficients reverse signs. Calculations from nearby starting values finish close to this inferior minimum.

Conditional repairs, not blanket rejection

The preprint also corrects the proposed zero-multiplier test. Under its stated conditions, the limiting chi-squared distribution has degrees of freedom equal to the number of moments minus the number of parameters, rather than the number of moments alone.

Kato’s conclusions distinguish three requirements: finding a genuine global optimum, establishing its statistical behavior, and connecting its target to an economically meaningful parameter. Reproducing coefficients or satisfying numerical equations does not, by itself, establish all three.