A Causal Bootstrap

The bootstrap, introduced by The Jackknife, the Bootstrap and Other Resampling Plans [(1982), SIAM], has become a very popular method for estimating variances and constructing confidence intervals. A key insight is that one can approximate the properties of estimators by using the empirical distribution function of the sample as an approximation for the true distribution function. This approach views the uncertainty in the estimator as coming exclusively from sampling uncertainty. We argue that for causal estimands the uncertainty arises entirely, or partially, from a different source, corresponding to the stochastic nature of the treatment received. We develop a bootstrap procedure for inference regarding the average treatment effect that accounts for this uncertainty, and compare its properties to that of the classical bootstrap. We consider completely randomized and observational designs as well as designs with imperfect compliance.