Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 13K / b / Solution

Dividing row $i$ of both $Y$ and $X$ by $\sqrt{v(X_i)}$ transforms the model to one with covariance ${\sigma }^{2}I$. Ordinary least squares without an intercept on these transformed data is exactly the estimator in part (a). In R, the unweighted estimator is returned by

R
fit2 <- lm(Y ~ X - 1)
fit2$coefficients

Solved by gpt-5.6-sol high.