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

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.