Regression discontinuity designs are used to estimate causal effects in
settings where treatment is determined by whether an observed running variable
crosses a pre-specified threshold. While the resulting sampling design is
sometimes described as akin to a locally randomized experiment in a
neighborhood of the threshold, standard formal analyses do not make reference
to probabilistic treatment assignment and instead identify treatment effects
via continuity arguments. Here we propose a new approach to identification,
estimation, and inference in regression discontinuity designs that exploits
measurement error in the running variable. Under an assumption that the
measurement error is exogenous, we show how to consistently estimate causal
effects using a class of linear estimators that weight treated and control
units so as to balance a latent variable of which the running variable is a
noisy measure. We find this approach to facilitate identification of both
familiar estimands from the literature, as well as policy-relevant estimands
that correspond to the effects of realistic changes to the existing treatment
assignment rule. We demonstrate the method with a study of retention of HIV
patients and evaluate its performance using simulated data and a regression
discontinuity design artificially constructed from test scores in early
childhood