  We study regression discontinuity designs in which many predetermined
covariates, possibly much more than the number of observations, can be used to
increase the precision of treatment effect estimates. We consider a two-step
estimator which first selects a small number of "important" covariates through
a localized Lasso-type procedure, and then, in a second step, estimates the
treatment effect by including the selected covariates linearly into the usual
local linear estimator. We provide an in-depth analysis of the algorithm's
theoretical properties, showing that, under an approximate sparsity condition,
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.
