Estimation and Inference of Extremal Quantile Treatment Effects for Heavy-Tailed Distributions

AbstractCausal inference for extreme events has many potential applications in fields such as climate science, medicine, and economics. We study the extremal quantile treatment effect of a binary treatment on a continuous, heavy-tailed outcome. Existing methods are limited to the case where the quantile of interest is within the range of the observations. For applications in risk assessment, however, the most relevant cases relate to extremal quantiles that go beyond the data range. We introduce an estimator of the extremal quantile treatment effect that relies on asymptotic tail approximation, and use a new causal Hill estimator for the extreme value indices of potential outcome distributions. We establish asymptotic normality of the estimators and propose a consistent variance estimator to achieve valid statistical inference. We illustrate the performance of our method in simulation studies, and apply it to a real dataset to estimate the extremal quantile treatment effect of college education on wage. Supplementary materials for this article are available online.Keywords: Asymptotic normalityCausalityCausal Hill estimatorExtrapolationExtreme value theoryPropensity score Supplementary MaterialsThe supplementary material contains the details of the estimated propensity score using sieve method, regularity assumptions for sieve estimation and the central limit theorem, examples satisfying Assumption 6, details of variance estimation, supplementary material for simulations and the real application, and technical proofs.Disclosure StatementThe authors report there are no competing interests to declare.Additional informationFundingSebastian Engelke gratefully acknowledges the Eccellenza grant of the Swiss National Science Foundation. Jinzhou Li gratefully acknowledges support by the SNSF Grant P500PT-210978.