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

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.