# Assumptions of linear regression

One particular assumption is that $e_{i} \sim N(0,\sigma^{2})$ but I was wondering why we use a QQ plot to test this assumption since we have $n$ iid distributions to look at. When we use a QQ plot, aren't we looking for the normality of all the residuals as opposed to one particular $e_{i}$? I am confused about this assumption. Does the QQ plot somehow imply that each $e_{i}$ is normally distributed?

• You seem to be confounding errors and residuals together there. The assumptions apply to the errors, but the diagnostics are done on the available estimates of them (the residuals). You're interested in checking whether the entire set of errors are consistent with being sample from $N(0,\sigma^2)$ – Glen_b Aug 12 '13 at 13:29
• The issue with errors versus residuals aside, I think you also need to ask yourself why does the QQ plot tell us that the residuals follow a normal distribution. Think about what it means to be on the straight line in the plot. – user25658 Aug 12 '13 at 13:41

First, note that if we observe realizations out of a normally distributed random variable, then indeed the data/realizations should "look" normal with sufficient $n$. If the error terms are iid, it does not matter if we think of it as one random variable spewing out data, or several random events happening iid after each other, as the normal distribution (and therefore our error term) is completely determined by the moments and assumptions we made.
Does the QQ plot imply that each $e_i$ is indeed normal? No. But it doesn't invalidate this assumption. More importantly: We can not observe $e_i$ anyway. We observe the residuals - the difference between what we can explain systematically by our model and what is left in the data.
Since our residuals are defined as: $\hat{e_i} = y_i - \hat{y_i}$, the appearance of normality in the residuals suggests that everything out of our model is random influence. That is, no systematic influence remains in the data that can not be attributed to our data.
If $\hat{y_i}$ is our fitted model, then we can also say that $\hat{e_i}$ is kinda our estimate of the error term. If that doesn't turn out to be normal, something must be wrong in our model!