  Numerous empirical studies employ regression discontinuity designs with
multiple cutoffs and heterogeneous treatments. A common practice is to
normalize all the cutoffs to zero and estimate one effect. This procedure
identifies the average treatment effect (ATE) on the observed distribution of
individuals local to existing cutoffs. However, researchers often want to make
inferences on more meaningful ATEs, computed over general counterfactual
distributions of individuals, rather than simply the observed distribution of
individuals local to existing cutoffs. This paper proposes a consistent and
asymptotically normal estimator for such ATEs when heterogeneity follows a
non-parametric function of cutoff characteristics in the sharp case. The
proposed estimator converges at the minimax optimal rate of root-n for a
specific choice of tuning parameters. Identification in the fuzzy case, with
multiple cutoffs, is impossible unless heterogeneity follows a
finite-dimensional function of cutoff characteristics. Under parametric
heterogeneity, this paper proposes an ATE estimator for the fuzzy case that
optimally combines observations to maximize its precision.
