  We consider inference on a scalar regression coefficient under a constraint
on the magnitude of the control coefficients. A class of estimators based on a
regularized propensity score regression is shown to exactly solve a tradeoff
between worst-case bias and variance. We derive confidence intervals (CIs)
based on these estimators that are bias-aware: they account for the possible
bias of the estimator. Under homoskedastic Gaussian errors, these estimators
and CIs are near-optimal in finite samples for MSE and CI length. We also
provide conditions for asymptotic validity of the CI with unknown and possibly
heteroskedastic error distribution, and derive novel optimal rates of
convergence under high-dimensional asymptotics that allow the number of
regressors to increase more quickly than the number of observations. Extensive
simulations and an empirical application illustrate the performance of our
methods.
