  We estimate the linear coefficient in a partially linear model with
confounding variables. We rely on double machine learning (DML) and extend it
with an additional regularization and selection scheme. We allow for more
general dependence structures among the model variables than what has been
investigated previously, and we prove that this DML estimator remains
asymptotically Gaussian and converges at the parametric rate. The DML estimator
has a two-stage least squares interpretation and may produce overly wide
confidence intervals. To address this issue, we propose the
regularization-selection regsDML method that leads to narrower confidence
intervals. It is fully data driven and optimizes an estimated asymptotic mean
squared error of the coefficient estimate. Empirical examples demonstrate our
methodological and theoretical developments. Software code for our regsDML
method will be made available in the R-package dmlalg.
