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

Restricting to pupils who already attended a Catholic middle school improves covariate balance: the treatment-group differences in family income, urban residence, and prior mathematics score are all much smaller. This makes severe extrapolation and measured confounding less prominent.

The restriction reduces the sample from $11,839$ to $1,006$, so estimates are less precise. It also changes the target population to Catholic-middle-school pupils, reducing external validity for all United States pupils.

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.

In the bivariate probit model for endogenous treatment, $U$ affects treatment through its threshold equation and $V$ affects the outcome through its threshold equation. When $\rho \ne 0$, the two disturbances are dependent, so treatment status carries information about the latent outcome disturbance even after conditioning on $X$. Consequentlyin general, and the no unmeasured confounding assumption fails.

Fix $\rho$. For an observation with covariates $x$, putand let $\Phi$ be the standard-normal distribution function. The four conditional cell probabilities arewhere the first index is $A$ and the second is $Y$.

Define $({\hat{\alpha }}_{\rho },{\hat{\beta }}_{\rho },{\hat{\gamma }}_{\rho })$ as any maximizer of the log likelihoodThen ${\hat{\beta }}_{\rho }$ is the requested estimator for the fixed sensitivity value $\rho$.

The correlation $\rho$ measures dependence between two normalized latent disturbances; it is not a scale-free measure of the strength of one physical confounder. Different latent-variable constructions can induce the same $\rho$ while producing different treatment-outcome confounding, and the same omitted cause can produce different $\rho$ after changing thresholds or disturbance scales. Moreover, re-estimating $(\alpha ,\beta ,\gamma )$ at each value of $\rho$ changes the entire latent model, so the fitted models do not represent one fixed data-generating mechanism with only its confounder strength varied. At $\rho =\pm 1$ the bivariate normal distribution is singular as well. Varying $\rho$ is a model-based sensitivity analysis, but interpreting the interval $[-1,1]$ as an ordered range of strengths of a single unmeasured confounder is therefore logically unjustified.