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For binary treatment, outcomes $Y_i$, and estimated propensity score $\hat{e}(X_i)$, the inverse-probability-weighted estimator isIt is consistent under conditional exchangeability, consistency of potential outcomes, positivity in causal inference, and a consistent propensity-score estimator.

The overlap-weighted average treatment effect averages the conditional average treatment effect with weight proportional to $e(X)\{1-e(X)\}$, where $e(X)$ is the propensity score. It emphasizes covariate strata in which both treatments are plausible.

Let $e(x)=\mathbb{P}(A=1\mid X=x)$ and let $\hat{e}$ be a consistent estimate. The inverse-probability-weighted estimator of the average treatment effect isUnder the exchangeability, consistency, and positivity conditions in part ii, and consistent estimation of the propensity score, its probability limit isso it consistently estimates the average treatment effect. It does not require the additive outcome-regression model used to interpret the ordinary-least-squares coefficient.

Estimate the propensity score $\pi$ and the untreated outcome regression $\mu$, preferably with cross-fitting when flexible methods are used, and setWith $n_1={\sum }_i{A}_i$, the resulting one-step estimator of the ATT isEquivalently,This is an augmented inverse-probability-weighted estimator; it is consistent when either the propensity model or the untreated outcome model is correct, subject to the usual regularity and positivity conditions.