  The causal inference model proposed by Lee (2008) for the regression
discontinuity design (RDD) relies on assumptions that imply the continuity of
the density of the assignment (running) variable. The test for this implication
is commonly referred to as the manipulation test and is regularly reported in
applied research to strengthen the design's validity. The multidimensional RDD
(MRDD) extends the RDD to contexts where treatment assignment depends on
several running variables. This paper introduces a manipulation test for the
MRDD. First, it develops a theoretical model for causal inference with the
MRDD, used to derive a testable implication on the conditional marginal
densities of the running variables. Then, it constructs the test for the
implication based on a quadratic form of a vector of statistics separately
computed for each marginal density. Finally, the proposed test is compared with
alternative procedures commonly employed in applied research.
