This is a question from a past paper. The answer given in the mark scheme is that the minimal assumption is just $\textrm{Var}(\epsilon)=\sigma^2I_n$. I'm struggling to understand why this is the case, based off the standard derivation of the estimator. I would have assumed that the assumptions were $\mathbb{E}(\epsilon)=0$ so that the sum of squares is given by $S(\beta)=(y-Z\beta)^T(y-Z\beta)$ and that $Z$ is of full rank so that $(Z^TZ)^{-1}$ exists.
I am aware that the assumption of equal variance and no correlation is required for the Gauss-Markov theorem to hold, but I can't see its relevance in this case. Could someone please cast some light on this issue?
