A Comparison of Two Quantile Models With Endogeneity

This article studies the relationship between the two most-used quantile models with endogeneity: the instrumental variable quantile regression (IVQR) model (Chernozhukov and Hansen 2005) and the local quantile treatment effects (LQTE) model (Abadie, Angrist, and Imbens 2002). The key condition of the IVQR model is the rank similarity assumption, a restriction on the evolution of individual ranks across treatment states, under which population quantile treatment effects (QTE) are identified. By contrast, the LQTE model achieves identification through a monotonicity assumption on the selection equation but only identifies QTE for the subpopulation of compliers. This article shows that, despite these differences, there is a close connection between both models: (i) the IVQR estimands correspond to QTE for the compliers at transformed quantile levels and (ii) the IVQR estimand of the average treatment effect is equal to a convex combination of the local average treatment effect and a weighted average of integrated QTE for the compliers. These results do not rely on the rank similarity assumption and therefore provide a characterization of IVQR in settings where this key condition is violated. Underpinning the analysis are novel closed-form representations of the IVQR estimands. I illustrate the theoretical results with two empirical applications.