Partial Identification of Policy-Relevant Treatment Effects with Instrumental Variables via Optimal Transport

TL;DR

This paper introduces a novel optimal transport-based framework for partially identifying policy-relevant treatment effects using instrumental variables, providing sharp bounds and efficient estimation methods.

Contribution

It formulates PRTE partial identification as a Constrained Conditional Optimal Transport problem, reducing it to one-dimensional problems with closed-form bounds and developing new estimation procedures.

Findings

Bounds are substantially tighter than existing moment-relaxation methods.

The framework accommodates high-dimensional covariates and various types of instruments.

Estimation procedures achieve parametric rates and asymptotic normality.

Abstract

Policy-Relevant Treatment Effects (PRTEs) are generally not point-identified under standard Instrumental Variable (IV) assumptions when the instrument generates limited support in treatment propensity. We show that PRTE partial identification in the generalized Roy model can instead be formulated as a Constrained Conditional Optimal Transport (CCOT) problem over the joint conditional law of the potential outcome and the latent resistance. The resulting multidimensional CCOT problem reduces analytically to separable one-dimensional OT problems with product costs, yielding sharp closed-form bounds and avoiding direct solution of the original high-dimensional CCOT problem. We also develop estimation and inference procedures for these bounds: for discrete instruments, we use a Double Machine Learning (DML) approach based on Neyman-orthogonal scores that accommodates high-dimensional covariates while achieving the parametric $\sqrt{n}$ rate and asymptotic normality; for continuous instruments, we explicitly characterize the corresponding nonparametric convergence rates. The framework accommodates covariates, discrete and continuous instruments, and extensions to general treatment settings. In simulations and a bed-net subsidy application, the resulting bounds are substantially tighter than the moment-relaxation method.