  Doubly robust (DR) estimation is a crucial technique in causal inference and
missing data problems. We propose a novel Propensity score Augmentved Doubly
robust (PAD) estimator to enhance the commonly used DR estimator for average
treatment effect on the treated (ATT), or equivalently, the mean of the outcome
under covariate shift. Our proposed estimator attains a lower asymptotic
variance than the conventional DR estimator when the propensity score (PS)
model is misspecified and the outcome regression (OR) model is correct while
maintaining the double robustness property that it is valid when either the PS
or OR model is correct. These are realized by introducing some properly
calibrated adjustment covariates to linearly augment the PS model and solving a
restricted weighted least square (RWLS) problem to minimize the variance of the
augmented estimator. Both the asymptotic analysis and simulation studies
demonstrate that PAD can significantly reduce the estimation variance compared
to the standard DR estimator when the PS model is wrong and the OR is correct,
and maintain close performance to DR when the PS model is correct. We further
applied our method to study the effects of eligibility for 401(k) plan on the
improvement of net total financial assets using data from the Survey of Income
and Program Participation of 1991.
