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Differentiating the residualized objective givesThis is the population Frisch–Waugh–Lovell theorem.

For a semiparametric estimator, estimate $e(x)=\mathbb{E}[A\mid X=x]$ and $\mu (x)=\mathbb{E}[Y\mid X=x]$ flexibly. With cross-fitting, obtain held-out predictions ${\hat{e}}_i,{\hat{\mu }}_i$ and regress the residualized outcome on the residualized treatment through the origin:Cross-fitting limits overfitting bias and permits flexible nuisance estimators under the usual convergence and overlap conditions.

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.

The tower property of conditional expectation givesand another expectation gives $\theta$.

Use the augmented inverse-probability-weighted estimatorCondition on the independently trained nuisance estimators. Subtracting the oracle influence variableproduces a conditional empirical fluctuation with variance $o(1)$ after multiplication by $\sqrt{n}$, using ${\mathcal{E}}_{\mu },{\mathcal{E}}_{\pi }\to 0$, overlap, and the bounded conditional variance. Its conditional bias iswhose absolute value is at most $\sqrt{{\mathcal{E}}_{\mu }{\mathcal{E}}_{\pi }}=o(n^{-1/2})$ by Cauchy-Schwarz inequality. ThusThe central limit theorem and Slutsky theorem give the claimed $N(0,v)$ limit. Without auxiliary data, use cross-fitting: split the sample into folds, train both nuisance estimators away from each observation's fold, and average the same score over held-out observations.