Generalized Optimal Transport

TL;DR

This paper introduces a flexible framework for deriving sharp bounds on economic parameters using optimal transport and kernel moments, applicable under complex data restrictions and structural assumptions.

Contribution

It develops a general method to compute sharp bounds with arbitrary joint distribution restrictions and structural conditions, using duality and approximation theory.

Findings

Provides a finite-dimensional reformulation of an infinite-dimensional optimization problem.

Yields a $ ext{sqrt}(n)$-consistent estimator for sharp bounds.

Achieves a $ ext{sqrt}(n)$ convergence rate in empirical optimal transport with Lipschitz cost.

Abstract

Many causal and structural parameters in economics can be identified and estimated by computing the value of an optimization program over all distributions consistent with the model and the data. Existing tools apply when the data is discrete, or when only disjoint marginals of the distribution are identified, which is restrictive in many applications. We develop a general framework that yields sharp bounds on a linear functional of the unknown true distribution under i) an arbitrary collection of identified joint subdistributions and ii) structural conditions, such as (conditional) independence. We encode the identification restrictions as a continuous collection of moments of characteristic kernels, and use duality and approximation theory to rewrite the infinite-dimensional program over Borel measures as a finite-dimensional program that is simple to compute. Our approach yields a consistent estimator that is $\sqrt{n}$-uniformly valid for the sharp bounds. In the special case of empirical optimal transport with Lipschitz cost, where the minimax rate is $n^{2/d}$, our method yields a uniformly consistent estimator with an asymmetric rate, converging at $\sqrt{n}$ uniformly from one side.