Nearest neighbor matching: M-out-of-N bootstrapping without bias correction vs. the naive bootstrap

Abstract

It is well known that the limiting variance of nearest neighbor matching estimators cannot be consistently estimated by a naive Efron-type bootstrap as the conditional variance of the bootstrap estimator does not generally converge to the correct limit in expectation. In essence this is caused by the fact that the bootstrap sample contains ties with positive probability even when the sample size becomes large. This negative result was originally derived in a simple setting by Abadie and Imbens (ECONOMETRICA, pp. 235–267, 76(6), 2008). A proof of concept for a direct M-out-of-N bootstrap on the data is provided in this setting. It is proven that in this setting the conditional variance of a direct M-out-of-N-type bootstrap estimator without bias-correction does converge to the correct limit in expectation. The key to the proof lies in the fact that asymptotically with probability one there are no ties in the bootstrap sample. The potential of the direct M-out-of-N-type bootstrap is investigated in simulations.