# Estimation of distributional effects of treatment and control under selection on observables: consistency, weak convergence, and applications

## TL;DR

This paper develops methods for estimating the distribution of potential outcomes under treatment and control, using propensity score weighting, and establishes their theoretical properties and practical applications.

## Contribution

It introduces a weighted empirical process approach for distributional estimation and proves its weak convergence, enabling new nonparametric tests for treatment effects.

## Key findings

- Weak convergence of the weighted empirical process to Gaussian process
- Consistent estimation of ATE and QTE distributions
- Finite sample properties demonstrated via simulations

## Abstract

In this paper the estimation of the distribution function for potential outcomes to receiving or not receiving a treatment is studied. The approach is based on weighting observed data on the basis on estimated propensity score. A weighted version of empirical process is constructed and its weak convergence to bivariate Gaussian process is established. Results for the estimation of the Average Treatment Effect (ATE) and Quantile Treatment Effect (QTE) are obtained as by-products. Applications to the construction of nonparametric tests for the treatment effect and for the stochastic dominance of the treatment over control are considered, and their finite sample properties and merits are studied via simulation.