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.

翻译：我们证明，当使用合适的协变量平衡程序估计倾向得分时，用于估计平均处理效应（ATE）或处理组平均处理效应（ATT）的常用逆概率加权（IPW）估计量、带有线性条件均值的增强逆概率加权（AIPW）估计量以及带有线性条件均值的逆概率加权回归调整（IPWRA）估计量在数值上均等价。此外，适当选择的协变量平衡权重会自动标准化，这意味着IPW和AIPW的标准化与非标准化版本完全相同。对于ATE的估计，实现IPW、AIPW和IPWRA代数等价性的权重基于使用Graham、Pinto和Egel（2012）提出的逆概率倾斜（IPT）方法估计的倾向得分。对于ATT，这些权重则通过Imai和Ratkovic（2014）开发的协变量平衡倾向得分（CBPS）方法获得。这些等价性也使得协变量平衡方法在处理变量存在混杂且研究者关注局部平均处理效应时具有吸引力。