  Inverse Probability Weighting (IPW) is widely used in empirical work in
economics and other disciplines. As Gaussian approximations perform poorly in
the presence of "small denominators," trimming is routinely employed as a
regularization strategy. However, ad hoc trimming of the observations renders
usual inference procedures invalid for the target estimand, even in large
samples. In this paper, we first show that the IPW estimator can have different
(Gaussian or non-Gaussian) asymptotic distributions, depending on how "close to
zero" the probability weights are and on how large the trimming threshold is.
As a remedy, we propose an inference procedure that is robust not only to small
probability weights entering the IPW estimator but also to a wide range of
trimming threshold choices, by adapting to these different asymptotic
distributions. This robustness is achieved by employing resampling techniques
and by correcting a non-negligible trimming bias. We also propose an
easy-to-implement method for choosing the trimming threshold by minimizing an
empirical analogue of the asymptotic mean squared error. In addition, we show
that our inference procedure remains valid with the use of a data-driven
trimming threshold. We illustrate our method by revisiting a dataset from the
National Supported Work program.
