Regression Discontinuity Design with Distribution-Valued Outcomes

This article introduces a regression discontinuity design for distribution-valued outcomes (R3D), extending the standard RDD framework to settings where outcomes are entire distributions rather than single values. This arises when treatment is assigned at the group level (e.g., firms, schools) but the objects of interest are within-group distributions (e.g., employee wages, student test scores). Standard RDDs are not designed for this two-level structure, as they assume scalar outcomes observed at the same level as treatment assignment. To address this, I develop a novel approach based on random distributions and show that, under a mild continuity condition on the average quantile function, the jump at the cutoff identifies a local average quantile treatment effect. To estimate it, I propose a distribution-valued local polynomial estimator, which fits the full quantile curve with a single bandwidth, avoids quantile crossing, and yields a meaningful “average distribution”. I derive uniform asymptotic normality, valid multiplier bootstrap confidence bands, and a data-driven bandwidth selection method. Simulations demonstrate strong performance and reveal that standard quantile RDD is biased and inconsistent in this setting. An application to U.S. gubernatorial close elections (1984–2010) uncovers an equality–efficiency trade-off under Democratic control, driven by income reductions at the top of the distribution.