research

This paper proposes distributional tests of treatment effect heterogeneity in regression discontinuity designs. While existing tests compare average effects across subgroups defined by observed covariates, our tests instead detect heterogeneity anywhere in the outcome distribution and require no choice of covariates, so they also capture heterogeneity driven by unobservables. The tests ask whether the treated outcome distribution at the cutoff is a constant shift of the control distribution. We compare the two distributions after aligning them by the estimated shift and obtain critical values by permutation inference. When the shift is known, the permutation test is exact in finite samples; when it must be estimated, a Durbin problem arises and the test is no longer valid. We provide two remedies: a Khmaladze martingale transformation that removes the drift induced by estimation, and a prepivoted permutation test that reproduces that drift in the reference distribution. Both are asymptotically valid. In Monte Carlo simulations, the two tests hold their nominal size not only in large samples but also in small ones, and they have power against latent heterogeneity. We apply the tests to the effect of Islamic municipal rule on female education in Turkey and find evidence of heterogeneity within more religious municipalities that a comparison of average effects would miss.