Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 7 / c / iii / Solution

The least-squares estimators depend only on the observed responses and design, so the formulas in i still hold. Their variances and covariance need only the error covariance matrix, now $I_n$. ThereforeFor example the first formula follows from $\mathrm{Var}({\sum }_i{x}_i{ϵ}_i)={\sum }_i{x}_i^2$, and the second from ${\sum }_i{x}_i=0$. The formulas remain the same as before with ${\sigma }^2=1$; this calculation does not establish independence or a normal distribution for the estimators.