Identification and Auto-Debiased Machine Learning for Outcome-Conditioned Average Structural Derivatives

This study proposes a new class of heterogeneous causal quantities, referred to as <i>outcome-conditioned</i> average structural derivatives (OASDs), in a general nonseparable model. An OASD is the average partial effect of a marginal change in a continuous treatment on individuals located on different parts of an outcome distribution, irrespective of individuals’ characteristics. We show that OASDs extend the unconditional quantile partial effects (UQPE) proposed by Firpo, Fortin, and Lemieux to that conditional on a set of outcome values by effectively integrating the UQPE. Exploiting such relationship brings about two merits. First, unlike UQPE that is generally not n-estimable, OASD is shown to be n-estimable. Second, our estimator achieves semiparametric efficiency bound which is a new result in the literature. We propose a novel, automatic, debiased machine-learning estimator for an OASD, and present asymptotic statistical guarantees for it. The estimator is proven to be n-consistent, asymptotically normal, and semi-parametrically efficient. We also prove the validity of the bootstrap procedure for uniform inference for the OASD process. We apply the method to Imbens, Rubin, and Sacerdote’s lottery data.