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We study regression discontinuity designs with the use of additionalcovariates for estimation of the average treatment effect. We provide adetailed proof of asymptotic normality of the covariate-adjusted estimatorunder minimal assumptions, which may serve as an accessible text to themathematics behind regression discontinuity with covariates. In addition, thisproof carries at least three benefits. First of all, it allows to drawstraightforward consequences concerning the impact of the covariates on thebias and variance of the estimator. In fact, we can provide conditions underwhich the influence of the covariates on the bias vanishes. Moreover, we showthat the variance in the covariate-adjusted case is never worse than in thecase 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.

Estimating Average Treatment Effects in Regression Discontinuity Designs with Covariates under Minimal Assumptions