Nonparametric Inference of Heterogeneous Treatment Effects with Two-Scale Distributional Nearest Neighbors

Understanding heterogeneous treatment effects (HTE) plays a key role in many contemporary causal inference applications arising from different areas. Most of the existing works have focused on the estimation of HTE. Yet the statistical inference aspect of the problem remains relatively undeveloped. In this paper we investigate the inference of HTE in a nonparametric setting for randomized experiments. We formulate the problem as two separate nonparametric mean regressions, one for control group and the other for treatment group. For each mean regression, we extend the tool of $k$-nearest neighbors to the framework of distributional nearest neighbors (DNN). We show that the DNN estimator has two equivalent representations of L-statistic and U-statistic, where the former endorses easy and fast implementation, and the latter enables us to obtain higher-order asymptotic expansion of bias and establish the asymptotic normality. To reduce the finite sample bias of DNN, we further suggest a new method of two-scale distributional nearest neighbors (TDNN). Under some regularity conditions, we show through delicate higher-order asymptotic expansions that the TDNN heterogeneous treatment effect estimator is asymptotically normal. We further establish the consistency of the variance estimates of the TDNN estimator with both jackknife and bootstrap, enabling user-friendly inference tools for heterogeneous treatment effects. The theoretical results and appealing finite-sample performance of the suggested TDNN method are illustrated with several simulation examples and a children's birth weight application.