  The increasing popularity of regression discontinuity methods for causal
inference in observational studies has led to a proliferation of different
estimating strategies, most of which involve first fitting non-parametric
regression models on both sides of a treatment assignment boundary and then
reporting plug-in estimates for the effect of interest. In applications,
however, it is often difficult to tune the non-parametric regressions in a way
that is well calibrated for the specific target of inference; for example, the
model with the best global in-sample fit may provide poor estimates of the
discontinuity parameter. In this paper, we propose an alternative method for
estimation and statistical inference in regression discontinuity designs that
uses numerical convex optimization to directly obtain the finite-sample-minimax
linear estimator for the regression discontinuity parameter, subject to bounds
on the second derivative of the conditional response function. Given a bound on
the second derivative, our proposed method is fully data-driven, and provides
uniform confidence intervals for the regression discontinuity parameter with
both discrete and continuous running variables. The method also naturally
extends to the case of multiple running variables.
