  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 probability weighting
(AIPW) with cross-fitting has emerged 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 central limit theorem 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
pre-cross-fit estimators is non-negligible even on the root-n scale. 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.
