Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 3

The no unmeasured confounding assumption, or conditional exchangeability, isAlso assume consistency of potential outcomes, no interference in causal inference, positivity in causal inference, and finite expectations. For $a\in \{0,1\}$, the law of total expectation, exchangeability, and consistency givePositivity ensures that the observed conditional means exist on the covariate support being averaged. Subtracting the two cases identifies the average treatment effect as

Under the partially linear model,Inserting this constant conditional contrast into the identification formula from part i gives

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 $X$.

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.