  We study identification and estimation in the Regression Discontinuity Design
(RDD) with a multivalued treatment variable. We also allow for the inclusion of
covariates. We show that without additional information, treatment effects are
not identified. We give necessary and sufficient conditions that lead to
identification of LATEs as well as of weighted averages of the conditional
LATEs. We show that if the first stage discontinuities of the multiple
treatments conditional on covariates are linearly independent, then it is
possible to identify multivariate weighted averages of the treatment effects
with convenient identifiable weights. If, moreover, treatment effects do not
vary with some covariates or a flexible parametric structure can be assumed, it
is possible to identify (in fact, over-identify) all the treatment effects. The
over-identification can be used to test these assumptions. We propose a simple
estimator, which can be programmed in packaged software as a Two-Stage Least
Squares regression, and packaged standard errors and tests can also be used.
Finally, we implement our approach to identify the effects of different types
of insurance coverage on health care utilization, as in Card, Dobkin and
Maestas (2008).
