When can we ignore measurement error in the running variable?

Abstract

In many applications of regression discontinuity designs, the running variable used to assign treatment is only observed with error. We show that, provided the observed running variable (i) correctly classifies treatment assignment and (ii) affects the conditional means of potential outcomes smoothly, ignoring the measurement error nonetheless yields an estimate with a causal interpretation: the average treatment effect for units whose observed running variable equals the cutoff. Possibly after doughnut trimming, these assumptions accommodate a variety of settings where support of the measurement error is not too wide. An empirical application illustrates the results for both sharp and fuzzy designs.