Approximate permutation tests and induced order statistics in the regression discontinuity design

This paper proposes an asymptotically valid permutation test for a testable implication of the identification assumption in the regression discontinuity design (RDD). Here, by testable implication, we mean the requirement that the distribution of observed baseline covariates should not change discontinuously at the threshold of the so-called running variable. This contrasts to the common practice of testing the weaker implication of continuity of the means of the covariates at the threshold. When testing our null hypothesis using observations that are "close" to the threshold, the standard requirement for the finite sample validity of a permutation does not necessarily hold. We therefore propose an asymptotic framework where there is a fixed number of closest observations to the threshold with the sample size going to infinity, and propose a permutation test based on the so-called induced order statistics that controls the limiting rejection probability under the null hypothesis. In a simulation study, we find that the new test controls size remarkably well in most designs. Finally, we use our test to evaluate the validity of the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.