Why is the assumption of a normally distributed residual relevant to a linear regression model? [duplicate]

Question

Isn't it just the assumptions of autocorrelation, unbiasedness and homoscedasicty that are relevant to proving the efficiency and the unbiasedness of OLS estimators?

How does the normality in the distribution of the residuals play in here?

Answer 1

You're correct that the assumption of normality is not required to prove unbiasedness. Without the assumption of normality you can also prove efficiency in the class of linear, unbiased estimators via the Gauss-Markov theorem.

If the errors are normally distributed, you can also establish that the least-squares estimators coincide with the maximum likelihood estimators. This lets you talk about things like the asymptotic efficiency of the MLEs in terms of the Cramér-Rao lower bound.

From this you can establish that the OLS estimators are asymptotically best in the class of regular estimators - estimators whose distributions "are not affected by small changes in the parameter", according to Larry Wasserman.

So, the normality assumption is not required but nets you some stronger results.

Answer 2

The usual small-sample inference -- confidence intervals, prediction intervals, hypothesis tests - rely on normality. You can of course make different parametric assumptions.

While Gauss-Markov gives you BLUE, the problem is if you're far enough from normality, all linear estimators may be bad, so choosing the best among them may be nearly useless.