  This paper studies the case of possibly high-dimensional covariates in the
regression discontinuity design (RDD) analysis. In particular, we propose
estimation and inference methods for the RDD models with covariate selection
which perform stably regardless of the number of covariates. The proposed
methods combine the local approach using kernel weights with
$\ell_{1}$-penalization to handle high-dimensional covariates. We provide
theoretical and numerical results which illustrate the usefulness of the
proposed methods. Theoretically, we present risk and coverage properties for
our point estimation and inference methods, respectively. Under certain special
case, the proposed estimator becomes more efficient than the conventional
covariate adjusted estimator at the cost of an additional sparsity condition.
Numerically, our simulation experiments and empirical example show the robust
behaviors of the proposed methods to the number of covariates in terms of bias
and variance for point estimation and coverage probability and interval length
for inference.
