Inference on Individual Treatment Effects in Nonseparable Triangular Models

TL;DR

This paper advances the statistical inference of individual treatment effects in nonseparable triangular models with binary treatments and instruments, by establishing asymptotic properties, bias correction, and confidence bands for density estimation.

Contribution

It extends existing methods by deriving asymptotic normality, proposing valid standard errors, and constructing uniform confidence bands for ITE density estimation.

Findings

Asymptotic normality of the density estimator is established.

New standard errors that account for ITE estimation errors are proposed.

Uniform confidence bands for the ITE density are developed using bootstrap methods.

Abstract

In nonseparable triangular models with a binary endogenous treatment and a binary instrumental variable, Vuong and Xu (2017) established identification results for individual treatment effects (ITEs) under the rank invariance assumption. Using their approach, Feng, Vuong, and Xu (2019) proposed a uniformly consistent kernel estimator for the density of the ITE that utilizes estimated ITEs. In this paper, we establish the asymptotic normality of the density estimator of Feng, Vuong, and Xu (2019) and show that the ITE estimation errors have a non-negligible effect on the asymptotic distribution of the estimator. We propose asymptotically valid standard errors that account for ITEs estimation, as well as a bias correction. Furthermore, we develop uniform confidence bands for the density of the ITE using the jackknife multiplier or nonparametric bootstrap critical values.