Estimation and Inference for Extreme Continuous Treatment Effects*

Abstract

This paper studies estimation and inference for the treatment effect in deep tails of the potential outcome distributions corresponding to a continuously valued treatment, namely the extreme continuous treatment effect. We consider two measures for the tail characteristics: the quantile function and the tail mean function defined as the conditional mean beyond a quantile level. Then, for a quantile level close to 1, we define the extreme quantile treatment effect (EQTE) and extreme average treatment effect (EATE), which are, respectively, the ratios of the quantile and tail mean at different treatment statuses. We propose estimators for the EQTE and EATE based on tail approximations from the extreme value theory. Our limiting theory is for the EQTE and EATE processes indexed by a set of quantile levels and pairs of different treatment statuses. It facilitates uniform inference for the EQTE and EATE over multiple tail levels and multiple treatment effects. Simulations suggest that our method works well in finite samples, and an empirical study illustrates its practical merits.