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.

翻译：在具有二元内生处理变量和二元工具变量的非可分三角模型中，Vuong与Xu（2017）在秩不变假设下确立了个体处理效应（ITE）的识别结果。基于其方法，Feng、Vuong与Xu（2019）提出了一种利用估计ITE的ITE密度一致核估计量。本文建立了Feng、Vuong与Xu（2019）密度估计量的渐近正态性，并指出ITE估计误差对估计量的渐近分布具有不可忽略的影响。我们提出了考虑ITE估计的渐近有效标准误及偏差校正方法。此外，基于刀切乘子法或非参数自举临界值，我们构建了ITE密度的均匀置信带。