In nonseparable triangular models with a binary endogenous treatment and a binary instrumental variable, Vuong and Xu (2017) show that the individual treatment effects (ITEs) are identifiable. Feng, Vuong and Xu (2019) show that a kernel density estimator that uses nonparametrically estimated ITEs as observations is uniformly consistent for the density of the ITE. In this paper, we establish the asymptotic normality of the density estimator of Feng, Vuong and Xu (2019) and show that despite their faster rate of convergence, ITEs' estimation errors have a non-negligible effect on the asymptotic distribution of the density estimator. We propose asymptotically valid standard errors for the density of the ITE that account for estimated ITEs as well as bias correction. Furthermore, we develop uniform confidence bands for the density of the ITE using nonparametric or jackknife multiplier bootstrap critical values. Our uniform confidence bands have correct coverage probabilities asymptotically with polynomial error rates and can be used for inference on the shape of the ITE's distribution.

翻译：在具有二元内生处理和二元工具变量的非可分离三角模型中,Vuong和Xu(2017年)显示个别处理效果(ITE)是可以辨认的。Feng、Vuong和Xu(2019年)显示,使用非对称估计的ITE值作为观测与ITE密度一致的内核密度估计值的内核密度估计值显示,使用非对称估计 ITE 的密度标准误差与ITE 密度一致。在本文中,我们用非对称或jknife 增殖靴式临界值为ITE 密度建立无症状常值(2019年),并显示,尽管ITEs的估计误差速度更快,但对密度估测器的无症状分布具有不可忽略的效果。我们建议对ITE的密度使用不完全有效的标准误差,并且可以用来在ITE值的形状上进行折射。