# How to understand the normality assumption of the errors?

We know one of the Gauss-Markov conditions is $$\mathrm{Cor}[e_i, e_j]=0$$ for $$i \neq j.$$

Also, we assume the errors are normally distributed, i.e., $$e_i \sim \mathcal{N}(0, \sigma^2).$$

My teacher said that the normality assumption implies that the errors are independent since they are uncorrelated.

However, what I know is only multivariate normal distribution with zero correlation can imply independent. I do not know why we can draw the conclusion that $$e_i$$ are independent.

Thanks!

• Saying the errors are individually (i.e. marginally normal) is insufficient. They have to be jointly normal for the implication to hold, as you say. This is covered in a one or two previous posts on site; I'll try to dig one up. Oct 6, 2019 at 7:13
• For the bivariate case a counterexample is mentioned in the answer here: stackoverflow.com/c/moderators/questions/1316/1584#1584 ... that's not the post I was looking for though Oct 6, 2019 at 7:29

In defense of your teacher, it is not so unusual to write something like $$\epsilon\sim N(0,\sigma^2I)$$ to indicate joint normality of the errors, so it is reasonable to think that you teacher left implicit the joint error normality.