When is the estimated propensity score better? High-dimensional analysis and bias correction

Anecdotally, using an estimated propensity score is superior to the true propensity score in estimating the average treatment effect based on observational data. However, this claim comes with several qualifications: it holds only if propensity score model is correctly specified and the number of covariates $d$ is small relative to the sample size $n$. We revisit this phenomenon by studying the inverse propensity score weighting (IPW) estimator based on a logistic model with a diverging number of covariates. We first show that the IPW estimator based on the estimated propensity score is consistent and asymptotically normal with smaller variance than the oracle IPW estimator (using the true propensity score) if and only if $n \gtrsim d^2$. We then propose a debiased IPW estimator that achieves the same guarantees in the regime $n \gtrsim d^{3/2}$. Our proofs rely on a novel non-asymptotic decomposition of the IPW error along with careful control of the higher order terms.