Past exam of the options I course of the University of Oxford / 2012 / O / O2 / a / ii / Solution

Independence and common error variance give $\mathrm{Var}(ϵ)={\sigma }^2{I}_n$. For the fixed matrix $A=(X^{T}X{)}^{-1}{X}^T$, the covariance transformation rule givesThus the covariance is determined by the error variance and the inverse Gram matrix of the design columns.