# Normally distributed errors - Why not use the observed residual histogram?

In the context of the classical linear regression model (with all standard assumptions), we know that when the error term is normally distributed, least squares is minimum variance among all linear unbiased estimators and we know the exact statistical distribution of the t statistics.

However, if the residual histogram suggests that the error term is something other than normal, then the t statistics will not have the claimed t distribution (nor will the F statistic be F distributed). Surely, it does not make sense to assume normality if the residuals suggest otherwise. Instead, if we want to know the theoretical distribution of the t statistic, why not assume that the errors are distributed in population based on what we observe from the residual histogram?

• The short answer to that is that testing the significance of the parameter estimate(s) implicitly assumes that the residuals are normally distributed, because that's how the test statistic was determined. Sure, you can test the residuals to see if the normal error assumption is reasonable, but that's not the same thing--that's assumption checking, not model selection. – heropup Jan 22 '14 at 1:46
• You can bootstrap from the residuals to get a significance value. – Jeremy Miles Jan 22 '14 at 2:16
• What hypothesis are you testing there? – Glen_b Jan 22 '14 at 2:22
• @ Glen_b - This is a theoretical question – Christian Jan 22 '14 at 3:01
• @ heropu - Yes, generally we assume that the errors are normally distributed since we need this assumption for exact inference. However, what are the implications if we construct signficance tests based on a bootstrap distribution using the observed distribution of residuals? – Christian Jan 22 '14 at 3:03