Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 13K / b

The OLS normal equation with an intercept gives $\hat{\beta }=S_{XY}/S_{XX}$. By the law of large numbers it tends to $\mathrm{Cov}(X,Y)/\mathrm{Var}X$. Here $X=Z+U+{ϵ}_X$ and $Y=X+U+{ϵ}_Y$, so the limit is $(3+1)/3=4/3$, showing upward confounding bias.