A Comparison of Two Quantile Models With Endogeneity. Issue 2 (2nd April 2020) - Wüthrich, Kaspar
Abstract: 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.