In statistics (in particular time series analysis, like ARCH/GARCH models) I've noticed that residual diagnostics usually look at autocorrelation of residuals and squared residuals. Why both? What does autocorrelation of the squared residuals tell us that is different to just the regular residuals?
Apparently there are cases when residuals are not auto-correlated, but the square residuals are, and this observation goes back to at least the Engle's 1982 ARCH paper. See
(even if you don't have full access to JSTOR, the first page should be enlightening.)
EDIT The real point is this. Consider a day like August 4, 2011 at the markets. At the end of the day, it was not clear if the next day would be up or down, but it was clear that it would be volatile. So, the square (or the absolute value) of the changes are correlated, while it is not clear that the actual changes are themselves...