A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond

Abstract

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse propensity weighting (AIPW) with cross-fitting has emerged as a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new CLT for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the precross-fit estimates is non-negligible even on the √ n scale. In fact, these crosscovariances turn out to be negative in our setting. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools-approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions. * Author names sorted in alphabetical order.

Augmented Doubly Robust Post-Imputation Inference for Proteomic data

“…Other related works. In high-dimensional nuisance parameter estimation, Jiang et al (2022) and Yadlowsky (2022) derived consistency results for estimated conditional treatment effect with sparsity or distributional assumptions. Double machine learning, proposed by Chernozhukov et al (2018), provides a framework for building an efficient estimator of lowdimensional parameters, with nuisance functions estimated using a high-dimensional black-box model.…”