Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 4 / b

In the full sample, covariate adjustment moves the estimated coefficient from $0.73$ to $0.32$, a large change consistent with substantial measured confounding. In the Catholic-middle-school subsample the estimates remain between $0.48$ and $0.60$, supporting the claim that restriction has already improved comparability. The cost is visible in the standard errors, which rise from about $0.08$--$0.09$ to $0.13$--$0.15$ despite similar coefficient magnitudes.

A sufficient causal condition is conditional exchangeabilitytogether with consistency of potential outcomes and positivity in causal inference. For the ordinary-least-squares coefficient itself to equal one common causal effect, also require the correctly specified additive conditional-mean modelThen $\beta$ is the homogeneous treatment effect, and the adjusted coefficient consistently estimates both conditional effects and the average treatment effect in the analyzed population.

Let $e(x)=\mathbb{P}(A=1\mid X=x)$ and let $\hat{e}$ be a consistent estimate. The inverse-probability-weighted estimator of the average treatment effect isUnder the exchangeability, consistency, and positivity conditions in part ii, and consistent estimation of the propensity score, its probability limit isso it consistently estimates the average treatment effect. It does not require the additive outcome-regression model used to interpret the ordinary-least-squares coefficient.