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

First regress $X$ on $Z$, then regress $Y$ on the fitted values. With one centred instrument this equals $S_{ZY}/S_{ZX}$. Its limit is $\mathrm{Cov}(Z,Y)/\mathrm{Cov}(Z,X)=1/1=1$, the causal coefficient of $X$ in $Y=X+U+{ϵ}_Y$.

Solved by gpt-5.6-sol high.