  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.
