We study inference on the average causal effect when the propensity score can take on values arbitrarily close to zero. It is known that no regular root-n consistent estimators exist and standard estimators may fail to be asymptotically normal, suggesting valid inference is difficult. We show that statistical inference need not be difficult: even ordinary Wald inference may remain valid in this challenging setting if small enough (estimated) propensity scores are suitably clipped. The key is to clip at a threshold decaying slowly enough to obtain asymptotic normality but not so slowly that bias dominates. Unlike prior work with seemingly similar results, our results pertain to the standard average causal effect and hold uniformly over a nonparametric model.
Work in progress.