Testing for Heterogeneous Treatment Effects in Regression Discontinuity Designs

We propose a nonparametric test for unobserved treatment effect heterogeneity in regression discontinuity designs. Under the null of no unobserved heterogeneity, a transformed outcome that imputes treated potential outcomes for untreated units must have a continuous conditional distribution at the cutoff. We convert this implication into an integrated conditional-moment restriction using characteristic functions, thereby allowing the conditional local average treatment effect to be an unrestricted function of covariates. We derive the asymptotic distribution of the test statistics via a $U$-process and establish the validity of a multiplier bootstrap procedure for calculating critical values. Monte Carlo experiments show well-controlled size and increasing power. Two empirical applications illustrate how the test distinguishes between heterogeneity explained by observables and that explained by unobserved factors.