Localized Debiased Machine Learning: Efficient Estimation of Quantile Treatment Effects, Conditional Value at Risk, and Beyond

Abstract: We consider the efficient estimation of a low-dimensional parameter in the presence of very high-dimensional nuisances that may depend on the parameter of interest. An important example is the quantile treatment effect (QTE) in causal inference, where the efficient estimation equation involves as a nuisance the conditional cumulative distribution evaluated at the quantile to be estimated. Debiased machine learning (DML) is a data-splitting approach to address the need to estimate nuisances using flexible machine learning methods that may not satisfy strong metric entropy conditions, but applying it to problems with estimand-dependent nuisances would require estimating too many nuisances to be practical. For the QTE estimation, DML requires we learn the whole conditional cumulative distribution function, which may be challenging in practice and stands in contrast to only needing to estimate just two regression functions as in the efficient estimation of average treatment effects. Instead, we propose localized debiased machine learning (LDML), a new three-way data-splitting approach that avoids this burdensome step and needs only estimate the nuisances at a single initial bad guess for the parameters. In particular, under a Frechet-derivative orthogonality condition, we show the oracle estimation equation is asymptotically equivalent to one where the nuisance is evaluated at the true parameter value and we provide a strategy to target this alternative formulation. In the case of QTE estimation, this involves only learning two binary regression models, for which many standard, time-tested machine learning methods exist. We prove that under certain lax rate conditions, our estimator has the same favorable asymptotic behavior as the infeasible oracle estimator that solves the estimating equation with the true nuisance functions.