Sharp Design-based Inference for Quantiles of Individual Treatment Effects

Abstract

Understanding the distribution of individual treatment effects (ITEs) is a central problem in causal inference. In general, however, this distribution is only partially identifiable from observed data because it depends on the unidentifiable joint distribution of the two potential outcomes. In this paper, we develop design-based inference for the distribution of ITEs whose validity relies solely on the randomization of treatment assignment, treating the potential outcomes as arbitrary fixed constants. Building on recent randomization tests for ITE quantiles, we propose tests based on either a rank-based Kolmogorov--Smirnov (KS) statistic or the usual KS statistic, with the latter being more conservative in the presence of ties. We show that, under certain regularity conditions, the confidence bounds obtained by test inversion are asymptotically no worse than the sharp bounds on ITE quantiles given the marginal distributions of the potential outcomes. We further derive a closed-form expression for the confidence bounds from the KS-based randomization test and show that they can be equivalently constructed by combining sharp bounds on ITE quantiles with uncertainty of empirical marginal distributions. This equivalence connects randomization-based inference with partial identification and clarifies the special role of the KS statistic in achieving sharp inference. Under additional distributional assumptions, we characterize the power of the proposed randomization tests under local alternatives. Finally, we discuss how to incorporate covariates to further sharpen inference. We illustrate the properties of the proposed methods using both simulations and a real-data application.