A Score Based Approach to Wild Bootstrap Inference

We propose a generalization of the wild bootstrap of Wu (1986) and Liu (1988) based upon perturbing the scores of M-estimators. This "score bootstrap" procedure avoids recomputing the estimator in each bootstrap iteration, making it substantially less costly to compute than the conventional nonparametric bootstrap, particularly in complex nonlinear models. Despite this computational advantage, in the linear model, the score bootstrap studentized test statistic is equivalent to that of the conventional wild bootstrap up to order `O_p(n^(-1))`. We establish the consistency of the procedure for Wald and Lagrange Multiplier type tests and tests of moment restrictions for a wide class of M-estimators under clustering and potential misspecification. In an extensive series of Monte Carlo experiments we find that the performance of the score bootstrap is comparable to competing approaches despite its computational savings.