On Quantile Treatment Effects, Rank Similarity,and Variation of Instrumental Variables

This paper develops a nonparametric framework to identify and estimate distributional treatment effects under nonseparable endogeneity. We begin by revisiting the widely adopted \emph{rank similarity} (RS) assumption and characterizing it by the relationship it imposes between observed and counterfactual potential outcome distributions. The characterization highlights the restrictiveness of RS, motivating a weaker identifying condition. Under this alternative, we construct identifying bounds on the distributional treatment effects of interest through a linear semi-infinite programming (SILP) formulation. Our identification strategy also clarifies how richer exogenous instrument variation, such as multi-valued or multiple instruments, can further tighten these bounds. Finally, exploiting the SILP's saddle-point structure and Karush-Kuhn-Tucker (KKT) conditions, we establish large-sample properties for the empirical SILP: consistency and asymptotic distribution results for the estimated bounds and associated solutions.