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和二元工具变量的非可分离三角模型中,Vuong和Xu(2017年)根据等级不变假设设定了个人治疗效果的识别结果(ITEs),Feng、Vuong和Xu(2019年)提出统一一致的ITE密度内核估测器,使用估计的ITE。在本文件中,我们建立了Feng、Vuong和Xu(2019年)密度估测器的无症状常态,并表明ITE估算误差对估测器的无症状分布具有不可忽略的影响。我们提出了说明ITE估算值的标准误差,以及偏差校正。此外,我们用千刀叶乘数或非参数式靴式关键值为ITE密度开发了统一的置信带。