Take $X_i = 1$ for all $i$ ; your question boils down to this one: the answer given here is that the independence of the sample mean $\hat \beta$ and $S^2$ occurs only when the error is normal.

Take $X_i = 1$ for all $i$ ; your question boils down to this one: the answer given here is that the independence of the sample mean $\hat \beta$ and $S^2$ occurs only when the error is normal.

Take $X_i = 1$ for all $i$ ; your question boils down to this one: the answer given here is that the independence of the sample mean $\hat \beta$ and $S^2$ occurs only when the error is normal.

Take $X_i = 1$ for all $i$ ; your question boils down to this one: the answer given here is that the independence of the sample mean $\hat \beta$ and $S^2$ occurs only when the error is normal.