Estimation and Inference for Distribution Functions and Quantile Functions in Treatment Effect Models

In this paper, we propose inverse probability weighted estimators for the distribution functions of the potential outcomes of a binary treatment under the unconfoundedness assumption. We also apply the inverse mapping on the distribution functions to obtain the quantile functions. We show that the proposed estimators converge weakly to zero mean Gaussian processes. A simulation method based on the multiplier central limit theorem is proposed to approximate these limiting Gaussian processes. The estimators in the treated subpopulation are shown to share the same properties. To demonstrate the usefulness of our results, we construct Kolmogorov-Smirnov type tests for stochastic dominance relations between the potential outcomes and Monte-Carlo simulation results confirm the theoretical findings of our tests. The proposed stochastic dominance tests are applied to evaluate the effect of a job training program on incomes, and we find that job training has a positive effect on real earnings.Hypothesis testing, stochastic dominance, treatment effects, propensity score.