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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?

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    $\begingroup$ 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 $N(0,\sigma^2)$ $\endgroup$ – Glen_b Aug 12 '13 at 13:29
  • $\begingroup$ 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. $\endgroup$ – user25658 Aug 12 '13 at 13:41
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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.

If the quantile quantile plot of our regression model for our residuals shows a normal distribution of the residuals, then we can be confident that the OLS assumptions, not only about the error term but in general, are met by our model.

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!
This is important for OLS, especially for things like Missing Variable Bias (has nothing to do with the error term, as you see), which would in most cases make your results invalid (which can not be saved by asymptotics!). If our error term assumptions are correct AND our model has all the right parameters included, then our residuals will - on average - represent the error terms. If we look at the distribution of all those residuals and it turns out to be normal, we can assume our model to be correct. (On the other hand if the error terms weren't normally distributed, there would be the chance of our model being so wrong that our residuals would show normality because of our model - but this event is rather theoretical.)

So yes: It's important to understand that residuals only correspond to the error terms if our model is correct. And because of that, a QQ-Plot can never logically VALIDATE our model - since the error terms are not observed.
It can, however, invalidate our model and assumptions. And that's why we need to look at it.

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