  In many applications of regression discontinuity designs, the running
variable used by the administrator to assign treatment is only observed with
error. We show that, provided the observed running variable (i) correctly
classifies the treatment assignment, and (ii) affects the conditional means of
the 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 to the cutoff. We show that,
possibly after doughnut trimming, these assumptions accommodate a variety of
settings where support of the measurement error is not too wide. We propose to
conduct inference using bias-aware methods, which remain valid even when
discreteness or irregular support in the observed running variable may lead to
partial identification. We illustrate the results for both sharp and fuzzy
designs in an empirical application.
