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In this paper, I try to tame "Basu's elephants" (data with extreme selectionon observables). I propose new practical large-sample and finite-sample methodsfor estimating and inferring heterogeneous causal effects (underunconfoundedness) in the empirically relevant context of limited overlap. Idevelop a general principle called "Stable Probability Weighting" (SPW) thatcan be used as an alternative to the widely used Inverse Probability Weighting(IPW) technique, which relies on strong overlap. I show that IPW (or itsaugmented version), when valid, is a special case of the more general SPW (orits doubly robust version), which adjusts for the extremeness of theconditional probabilities of the treatment states. The SPW principle can beimplemented using several existing large-sample parametric, semiparametric, andnonparametric procedures for conditional moment models. In addition, I providenew finite-sample results that apply when unconfoundedness is plausible withinfine strata. Since IPW estimation relies on the problematic reciprocal of theestimated propensity score, I develop a "Finite-Sample Stable ProbabilityWeighting" (FPW) set-estimator that is unbiased in a sense. I also propose newfinite-sample inference methods for testing a general class of weak nullhypotheses. The associated computationally convenient methods, which can beused to construct valid confidence sets and to bound the finite-sampleconfidence distribution, are of independent interest. My large-sample andfinite-sample frameworks extend to the setting of multivalued treatments.

Stable Probability Weighting: Large-Sample and Finite-Sample Estimation  and Inference Methods for Heterogeneous Causal Effects of Multivalued  Treatments Under Limited Overlap