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 $\rho\ne0$, the two disturbances are dependent, so treatment status carries information about the latent outcome disturbance even after conditioning on $X$. Consequently
$$
Y(a)\not\!\perp A\mid X
$$
in general, and the <no unmeasured confounding assumption> fails.

Solution

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

A bivariate probit model for a binary treatment and outcome uses latent equations
$$
A=\mathbf1_{\{X^T\alpha+U>0\}},\qquad
Y=\mathbf1_{\{A\beta+X^T\gamma+V>0\}},
$$
where $(U,V)$ has a <bivariate normal distribution>. Its correlation parameter represents dependence between the two latent disturbances; when nonzero, treatment is endogenous in the outcome equation.

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