  This paper studies the validity of nonparametric tests used in the regression
discontinuity design. The null hypothesis of interest is that the average
treatment effect at the threshold in the so-called sharp design equals a
pre-specified value. We first show that, under assumptions used in the majority
of the literature, for \emph{any} test the power against any alternative is
bounded above by its size. This result implies that, under these assumptions,
any test with nontrivial power will exhibit size distortions. We next provide a
sufficient strengthening of the standard assumptions under which we show that a
novel test in the literature can control limiting size.
