Approximate Permutation Tests and Induced Order Statistics in the Regression Discontinuity Design*

In the regression discontinuity design (RDD), it is common practice to assess the credibility of the design by testing whether the means of baseline covariates do not change at the cut-off (or threshold) of the running variable. This practice is partly motivated by the stronger implication derived by Lee (2008), who showed that under certain conditions the distribution of baseline covariates in the RDD must be continuous at the cut-off. We propose a permutation test based on the so-called induced ordered statistics for the null hypothesis of continuity of the distribution of baseline covariates at the cut-off; and introduce a novel asymptotic framework to analyse its properties. The asymptotic framework is intended to approximate a small sample phenomenon: even though the total number $n$ of observations may be large, the number of effective observations local to the cut-off is often small. Thus, while traditional asymptotics in RDD require a growing number of observations local to the cut-off as $n\to \infty$, our framework keeps the number $q$ of observations local to the cut-off fixed as $n\to \infty$. The new test is easy to implement, asymptotically valid under weak conditions, exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity, and has favourable power properties relative to tests based on means. In a simulation study, we find that the new test controls size remarkably well across designs. We then use our test to evaluate the plausibility of the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.