A Comparison of two Quantile Models with Endogeneity

This paper analyzes estimators based on the instrumental variable quantile regression (IVQR) model (Chernozhukov and Hansen, 2004, 2005, 2006) under the local quantile treatment effects (LQTE) framework (Abadie et al., 2002). I show that the quantile treatment effect (QTE) estimators in the IVQR model are equivalent to LQTE for the compliers at transformed quantile levels. This transformation adjusts for differences between the subpopulation-specific potential outcome distributions that are identified in the LQTE model. Moreover, the IVQR estimator of the average treatment effect (ATE) corresponds to a convex combination of the local average treatment effect (LATE) and a weighted average of LQTE for the compliers. I extend the analysis to more general setups that allow for partial failures of the LQTE assumptions, non-binary instruments, and covariates. The results are illustrated with two empirical applications.