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.

翻译：我们研究了在断点回归设计中引入额外协变量以估计平均处理效应的问题。我们在最小假设下提供了协变量调整估计量渐近正态性的详细证明，该证明可作为理解含协变量断点回归背后数学知识的入门文本。此外，该证明至少具有三项优势。首先，它能够直接推导出协变量对估计量偏差与方差的影响。事实上，我们可以给出使协变量对偏差的影响消失的条件。其次，我们证明在非常一般的可逆条件下，协变量调整情形下的方差绝不会比基准估计量情形更差。最后，我们的方法不需要潜在结果的存在性，从而允许在无法排除混杂因素（例如通过操纵强制变量引发的混杂）时进行敏感性分析。