Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 13K (Statistical Modelling)

Expanding $\sum (X_i-\bar{X})(Y_i-\bar{Y})=\sum X_i{Y}_i-n\bar{X}\bar{Y}$ and taking expectations gives $(n-1)(E[XY]-EX,EY)$. Division by $n-1$ proves unbiasedness.

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.

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 $Y=X+U+{ϵ}_Y$.