Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 4 / b

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.

If both outcome models are correct, each residual term in the preceding conditional expectations has mean zero regardless of the working propensity model. Thus ${\tilde{\mu }}_z^{\mathrm{dr}}(X)={\mu }_z(X)$ for $z=0,1$, and their difference is again $\tau (X)$. This proves double robustness.