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For fixed $\beta$, conditional least squares is minimized byTherefore the remaining objective iswhose normal equation givesLetbe the conditional average treatment effect. Since $A$ is binary, conditional exchangeability impliesand $\mathrm{Var}(A\mid X)=e(X)\{1-e(X)\}$. HenceThus ${\beta }_3$ is the overlap-weighted average treatment effect. It weights covariate strata by the overlap weight and generally differs from the ordinary ATE ${\beta }_1=\mathbb{E}[\tau (X)]$ when treatment effects are heterogeneous and overlap varies with $X$.

For binary $A$, conditional covariance satisfiesBy conditional exchangeability and consistency of potential outcomes, the difference in braces is $\mathbb{E}[Y(1)-Y(0)\mid X]$. Consequentlyso the required weight isThis overlap weight emphasizes covariate strata with treatment probabilities near one half and downweights strata near a violation of positivity in causal inference.