Optimally-Transported Generalized Method of Moments

TL;DR

This paper introduces an optimal transport-based GMM that allows for minimal errors in variables to satisfy moment conditions, improving interpretability especially when overidentification tests fail.

Contribution

It develops a new GMM variant using optimal transport and Wasserstein metrics, addressing overidentification issues and enhancing interpretability of results.

Findings

Method corroborates previous findings under weaker assumptions

Provides insights into the error structure of variables

Addresses overidentification rejection issues in GMM

Abstract

We propose a novel optimal transport-based version of the Generalized Method of Moment (GMM). Instead of handling overidentification by reweighting the data to satisfy the moment conditions (as in Generalized Empirical Likelihood methods), this method proceeds by allowing for errors in the variables of the least mean-square magnitude necessary to simultaneously satisfy all moment conditions. This approach, based on the notions of optimal transport and Wasserstein metric, aims to address the problem of assigning a logical interpretation to GMM results even when overidentification tests reject the null, a situation that cannot always be avoided in applications. We illustrate the method by revisiting Duranton, Morrow and Turner's (2014) study of the relationship between a city's exports and the extent of its transportation infrastructure. Our results corroborate theirs under weaker assumptions and provide insight into the error structure of the variables.