Regression discontinuity designs, white noise models, and minimax

This paper studies an approximation of sharp regression discontinuity (RD) designs with normal homoskedastic errors by ideal Gaussian white noise models with discontinuous drifts. After establishing an asymptotic equivalence of the two models, we first provide a simple derivation of a lower bound on the minimax optimal convergence rate for estimating the discontinuity size in a derivative (of an arbitrary order) of the regression function. We show that the optimal convergence rate is attained by the local polynomial estimator. We also establish an analogous correspondence between the RD designs with an unknown threshold and a convolution white noise model, and derive the minimax optimal rate for estimating the discontinuity location in a derivative of the regression function.