What tests do I use to confirm that residuals are normally distributed? I have some data which looks from plotting a graph of residuals vs time almost normal but I want to be sure. How can I test for normality of error residuals?
 A: The Shapiro-Wilk test is one possibility.
Shapiro-Wilk test
This test is implemented in almost all statistical software packages.  The null hypothesis is the residuals are normally distributed, thus a small p-value indicates you should reject the null and conclude the residuals are not normally distributed.
Note that if your sample size is large you will almost always reject, so visualization of the residuals is more important.
A: *

*No test will tell you your residuals are normally distributed. In fact, you can reliably bet that they are not.

*Hypothesis tests are not generally a good idea as checks on your assumptions. The effect of non-normality on your inference is not generally a function of sample size*, but the result of a significance test is. A small deviation from normality will be obvious at a large sample size even though the answer to the question of actual interest ('to what extent did this impact my inference?') may be 'hardly at all'. Correspondingly, a large deviation from normality at a small sample size may not approach significance.
* (added in edit) -- actually that's much too weak a statement. The impact of non-normality actually decreases with sample size pretty much any time the CLT and Slutsky's theorem is going to hold, while the ability to reject normality (and presumably avoid normal-theory procedures) increases with sample size ... so just when you're most able to identify non-normality tends to be when it doesn't matter$^\dagger$ anyway... and the test is no help when it actually matters, in small samples.
$\dagger$ well, at least as far as significance level goes. Power can still be an issue though if we are considering large samples as here, that may be less of an issue
as well.

*What comes closer to measuring effect size is some diagnostic (either a display or a statistic) that measures degree of non-normality in some way. A Q-Q plot is an obvious display, and a Q-Q plot from the same population at one sample size and at a different sample size are at least both noisy estimates of the same curve - showing roughly the same 'non-normality'; it should at least approximately be monotonically related to the desired answer to  the question of interest.
If you must use a test, Shapiro-Wilk is probably about as good as anything else (the Chen-Shapiro test is typically a bit better on alternatives of common interest, but harder to find implementations of) -- but it's answering a question you already know the answer to; every time you fail to reject, it's giving an answer you can be sure is wrong.
A: From wikipedia: 
Tests of univariate normality include D'Agostino's K-squared test, the Jarque–Bera test, the Anderson–Darling test, the Cramér–von Mises criterion, the Lilliefors test for normality (itself an adaptation of the Kolmogorov–Smirnov test), the Shapiro–Wilk test, the Pearson's chi-squared test, and the Shapiro–Francia test. A 2011 paper from The Journal of Statistical Modeling and Analytics [1] concludes that Shapiro-Wilk has the best power for a given significance, followed closely by Anderson-Darling when comparing the Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors, and Anderson-Darling tests.
