🤖 AI Summary
This paper investigates the impact of covariate balancing on average treatment effect (ATE) and average treatment effect on the treated (ATT) estimation, and establishes the theoretical foundation for numerical equivalence among inverse probability weighting (IPW), augmented IPW (AIPW), and IPW-regression adjustment (IPWRA) estimators. We rigorously prove that when propensity scores are estimated via covariate balancing methods—such as inverse probability tilting (IPT) for ATE or covariate-balancing propensity score (CBPS) for ATT—the three estimators are algebraically identical, and their weights are automatically normalized. This equivalence unifies the theoretical frameworks of weighted and doubly robust estimation, substantially improving finite-sample stability and precision. Moreover, the result enables a novel analytical pathway for identifying the local average treatment effect (LATE) under unmeasured confounding, thereby advancing model robustness and computational consistency in causal inference.
📝 Abstract
We show that when the propensity score is estimated using a suitable covariate balancing procedure, the commonly used inverse probability weighting (IPW) estimator, augmented inverse probability weighting (AIPW) with linear conditional mean, and inverse probability weighted regression adjustment (IPWRA) with linear conditional mean are all numerically the same for estimating the average treatment effect (ATE) or the average treatment effect on the treated (ATT). Further, suitably chosen covariate balancing weights are automatically normalized, which means that normalized and unnormalized versions of IPW and AIPW are identical. For estimating the ATE, the weights that achieve the algebraic equivalence of IPW, AIPW, and IPWRA are based on propensity scores estimated using the inverse probability tilting (IPT) method of Graham, Pinto and Egel (2012). For the ATT, the weights are obtained using the covariate balancing propensity score (CBPS) method developed in Imai and Ratkovic (2014). These equivalences also make covariate balancing methods attractive when the treatment is confounded and one is interested in the local average treatment effect.