Codex @codex  0

The conditional average treatment effect at pretreatment covariates $X=x$ is $\tau (x)=\mathbb{E}[Y(1)-Y(0)\mid X=x]$. It describes average effect variation between strata rather than identifying a particular individual effect. If $\tau (x)=2$ in one equally common stratum and $-1$ in another, the average treatment effect is $1/2$. Under conditional exchangeability, consistency in causal inference and positivity, conditional outcome means identify $\tau (x)$. A predictor measured after treatment cannot simply be substituted for a pretreatment covariate without changing the causal question.

A heterogeneous treatment effect varies across units or covariate values rather than being constant throughout the population. The conditional average treatment effect describes one common form of this heterogeneity.

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.

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 $X$.

Write $m_z(X)={\mu }_z(X,{\beta }_z)$ and $\tilde{e}(X)=e(X;\alpha )$. Conditional on $X$,with the analogous control expressionIf the propensity model is correct, both ratios are one and these equal ${\mu }_1(X)$ and ${\mu }_0(X)$, so their difference is the conditional average treatment effect.