# Why check normality of raw residuals if raw residuals do not have the same normal distribution?

It is known that raw residuals $\hat{\varepsilon_i}=y_i-\hat{y_i}$ do not have the same standard error, therefore raw residuals do not have the same distribution.

Then plotting the raw residual against the quantiles of the same (normal) distribution would not make sense.

However, it is traditionally taught to do an eyeball verification of the normal plot of the raw residual for checking the normality assumption -- even in courses for students with major in Statistics!

So why is normal plot of raw residual still taught as the standard procedure?
I mean, ok, it would work as an easy, quick approximation, but not taught as the standard procedure taught and, worse, taught as the only procedure.

On the other hand, it is known now that, if $\varepsilon_i\sim\text{N}(0,\sigma^2)$ i.i.d., then externally studentized residuals $t_{i(i)}$ (a.k.a. jackknifed residuals) $$t_{i(i)} = \frac {\hat{\varepsilon_i}} { \sqrt{ \left( \frac1{n-k-1}\sum_{j\ne i}\hat{\varepsilon_i}^2 \right) \left( 1-\frac1n -\frac {(x_i-\bar{x})^2} {\sum_{j=1}^{n}(x_j-\bar{x})^2} \right) } }$$ have Student's $t$ distribution with $n-k-1$ degrees of freedom in a linear regression with $k$ independent variables.

So, as side questions, why not plotting jackknifed residuals against $t_{n-k-1}$ quantiles for checking normality assumption? Shouldn't it be the standard procedure instead of the analysis of normal plot of raw residuals?

• A pp or qq plot are just used to see potential departures from normality. They are not confirmatory tests. They are descriptive. The KS and Shapiro-Wilk tests for example are confirmatory tests. In linear regression models it is the error term that is assumed to be normal and hence to confirm the assumption the residuals should be approximately normal. – Michael Chernick May 12 '17 at 2:40
• Nonetheless, residuals related to lower and upper values of $x$ are more prone to wider variation than normally expected, which leads systematically to departures from diagonal in both lower and upper tails of residuals in normal plot, even for perfectly normal error. – Marcelo Ventura May 12 '17 at 4:55
• I suppose that sometimes that may be true. But as I said these plots are descriptive and should not be used to draw conclusions. – Michael Chernick May 12 '17 at 5:00
• And I was originally intrigued about why wouldn't the residual of choice especifically for descriptives be the jackknifed ones instead of the raw ones. – Marcelo Ventura May 12 '17 at 5:04
• Some people do use studentized residuals. – Michael Chernick May 12 '17 at 5:08

## 1 Answer

I suspect the reason is because while the studentize residuals are all from the same distribution, they are NOT i.i.d since they have some dependency between them. I've (Sadly) yet to find a normality test for residuals that accounts for that dependancy.