David Van Dijcke, University of Michigan

"Regression Discontinuity Design with Distribution-Valued Outcomes"

Abstract

This article introduces Regression Discontinuity Design (RDD) with DistributionValued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise where treatment is assigned at the level of an aggregate unit (e.g. firms, school districts) but the outcome of interest is an entire within-unit distribution (e.g. employee wages, student test scores). Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. Under a mild continuity assumption on the average of the random quantile functions, the jump in this average at the cutoff identifies a “local average quantile treatment effect”. To estimate this target, I develop a functional local polynomial estimator based on local Fréchet regression. The estimator simultaneously fits the whole quantile curve, requires only one bandwidth, avoids quantile crossing, and—through optimal-transport geometry—yields a meaningful “average distribution”. Exploiting the fact that the Fréchet fit is a projection of pointwise local polynomials onto the space of quantile functions, I establish asymptotic normality, construct robust, bias-corrected uniform confidence bands via a multiplier bootstrap, and propose a data-driven bandwidth selection procedure tailored to distribution-valued outcomes. Simulations confirm the accuracy and rapid convergence of this estimator and demonstrate that standard quantile RDD methods are biased and inconsistent in this setting. I illustrate R3D by estimating the effect of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality–efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

Contact person: Jesper Riis-Vestergaard Sørensen