Inference on Individual Treatment Effects in Nonseparable Triangular Models

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 ITEs distribution.