When is the estimated propensity score better? Highdimensional analysis and bias correction
Abstract
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 nonasymptotic decomposition of the IPW error along with careful control of the higher order terms.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.17102
 arXiv:
 arXiv:2303.17102
 Bibcode:
 2023arXiv230317102S
 Keywords:

 Statistics  Methodology
 EPrint:
 Fangzhou Su and Wenlong Mou contributed equally to this work