Inference in Regression Discontinuity Designs with High-Dimensional Covariates

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We study regression discontinuity designs in which many covariates, possibly much more than the number of observations, are available. In such settings, it is not feasible to feed all covariates in a standard procedure for RD problems with covariates. Therefore we provide a two-step algorithm which first selects the set of covariates to be used through a localized Lasso-type procedure, and then, in a second step, estimates the treatment effect by including the selected covariates into the usual local linear estimator. We show that the inclusion of covariates improves the asymptotic variance of the estimator provided that only a moderate number of covariates (in a certain sense) is included. It is natural to ask for the optimal choice of covariates to include. Our results show that, under an approximate sparsity condition, our model selection based on the Lasso is able to find a covariate set which yields the optimal reduction in variance. However, we also show that it is not necessary to find a specific covariate set with high probability. We provide an in-depth analysis of the algorithm's theoretical properties, showing that the resulting estimator is asymptotically normal, with asymptotic bias and variance that are conceptually similar to those obtained in low-dimensional settings. Bandwidth selection and inference can be carried out using standard methods. We also provide simulations and an empirical application.