  In this paper, we develop a new censored quantile instrumental variable
(CQIV) estimator and describe its properties and computation. The CQIV
estimator combines Powell (1986) censored quantile regression (CQR) to deal
with censoring, with a control variable approach to incorporate endogenous
regressors. The CQIV estimator is obtained in two stages that are non-additive
in the unobservables. The first stage estimates a non-additive model with
infinite dimensional parameters for the control variable, such as a quantile or
distribution regression model. The second stage estimates a non-additive
censored quantile regression model for the response variable of interest,
including the estimated control variable to deal with endogeneity. For
computation, we extend the algorithm for CQR developed by Chernozhukov and Hong
(2002) to incorporate the estimation of the control variable. We give generic
regularity conditions for asymptotic normality of the CQIV estimator and for
the validity of resampling methods to approximate its asymptotic distribution.
We verify these conditions for quantile and distribution regression estimation
of the control variable. Our analysis covers two-stage (uncensored) quantile
regression with non-additive first stage as an important special case. We
illustrate the computation and applicability of the CQIV estimator with a
Monte-Carlo numerical example and an empirical application on estimation of
Engel curves for alcohol.
