It has become standard practice to use local linear regressions in regression discontinuity designs.
This paper highlights that the same theoretical arguments used to justify local linear regression suggest
that alternative local polynomials could be preferred. We show in simulations that the local linear estimator
is often dominated by alternative polynomial specifications. Additionally, we provide guidance on the
selection of the polynomial order. The Monte Carlo evidence shows that the order-selection procedure
(which is also readily adapted to fuzzy regression discontinuity and regression kink designs) performs
well, particularly with large sample sizes typically found in empirical applications.