  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, many 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 for the treatment effect to be $\sqrt{n}$-estimable. 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 diverging
number of controls in a semiparametric partially linear model.
