Before revision:  Under an assumption that the measurement error is exogenous, we show how to consistently estimate causal effects using a class  linear estimators that weight treated and control units so as to balance a latent variable of which the running variable is a noisy measure.
After revision:  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.