Covariate Balancing and the Equivalence of Weighting and Doubly Robust Estimators of Average Treatment Effects

How should researchers adjust for covariates? We show that if the propensity score is estimated using a specific covariate balancing approach, inverse probability weighting (IPW), augmented inverse probability weighting (AIPW), and inverse probability weighted regression adjustment (IPWRA) estimators are numerically equivalent for the average treatment effect (ATE), and likewise for the average treatment effect on the treated (ATT). The resulting weights are inherently normalized, making normalized and unnormalized IPW and AIPW identical. We discuss implications for instrumental variables and difference-in-differences estimators and illustrate with two applications how these numerical equivalences simplify analysis and interpretation.