Root-n consistent semiparametric learning with high-dimensional nuisance functions under minimal sparsity

TL;DR

This paper introduces a Double-Calibration method that achieves root-n consistent treatment effect estimation in high-dimensional settings with minimal sparsity assumptions on nuisance functions, even under misspecification.

Contribution

It proposes a novel Double-Calibration strategy that relaxes sparsity requirements for high-dimensional nuisance parameters in treatment effect estimation.

Findings

Achieves $1/\sqrt{n}$-rate under minimal sparsity conditions.

Allows one nuisance parameter to be arbitrarily complex or misspecified.

Applicable to various semiparametric estimation problems.

Abstract

Treatment effect estimation under unconfoundedness is a fundamental task in causal inference. In response to the challenge of analyzing high-dimensional datasets collected in substantive fields such as epidemiology, genetics, economics, and social sciences, various methods for treatment effect estimation with high-dimensional nuisance parameters (the outcome regression and the propensity score) have been developed in recent years. However, it is still unclear what is the necessary and sufficient sparsity condition on the nuisance parameters such that we can estimate the treatment effect at $1 / \sqrt{n}$-rate. In this paper, we propose a new Double-Calibration strategy that corrects the estimation bias of the nuisance parameter estimates computed by regularized high-dimensional techniques and demonstrate that the corresponding Doubly-Calibrated estimator achieves $1 / \sqrt{n}$-rate as long as one of the nuisance parameters is sparse with sparsity below $\sqrt{n} / \log p$, where $p$ denotes the ambient dimension of the covariates, whereas the other nuisance parameter can be arbitrarily complex and completely misspecified. The Double-Calibration strategy can also be applied to settings other than treatment effect estimation, e.g. regression coefficient estimation in the presence of a diverging number of controls in a semiparametric partially linear model, and local average treatment effect estimation with instrumental variables.