Essays on Optimal Transport Theory and Causal Inference: A Theoretical and Empirical Approach

Abstract

This dissertation aims to study the causal inference from both theoretical and empiricalperspectives. Specifically, the first two chapters seek to extend causal inference methods for problems that are underdeveloped in current methodologies, i.e., addressing non-overlap support and misreporting of outcome variables. The third chapter explores the use of COVID- 19 as an instrumental variable from an empirical perspective to identify the causal impact of transportation on air pollution. The first chapter explores identifying average treatment effects for the treated in the region where the covariate distributions across treatment and control groups have non-overlap support. We make a natural domain shift assumption for the non-overlap region based on the optimal transport theory. We study the identification of average treatment effects for the nonoverlap region and propose three-step estimators of the average treatment effect and quantile treatment effect for the treated in the non-overlap region. We establish the consistency and asymptotic normality of the proposed estimators under high-level assumptions on the estimator of the optimal transport map. Three examples of the estimator of the optimal transport map are studied in detail and are shown to satisfy the high-level assumptions under primitive conditions. We investigate the finite sample performance of our estimator and Wald inference via simulation.