We study regression discontinuity designs with the use of additional covariates for estimation of the average treatment effect. We provide a detailed proof of asymptotic normality of the covariate-adjusted estimator under minimal assumptions, which may serve as an accessible text to the mathematics behind regression discontinuity with covariates. In addition, this proof carries at least three benefits. First of all, it allows to draw straightforward consequences concerning the impact of the covariates on the bias and variance of the estimator. In fact, we can provide conditions under which the influence of the covariates on the bias vanishes. Moreover, we show that the variance in the covariate-adjusted case is never worse than in the case of the baseline estimator under a very general invertibility condition. Finally, our approach does not require the existence of potential outcomes, allowing for a sensitivity analysis in case confounding cannot be ruled out, e.g., by a manipulated forcing variable.
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