Correlated Residuals - The Big Bad Wolf of Statistics? In a probability and statistics class, we are often told about the requirement of residuals being uncorrelated when fitting a regression model.
We are often told that the reason for this requirement is that the absence of this usually suggests low quality data.
But mathematically, is there a mathematical reason as to why residuals need to be uncorrelated? What happens if the residuals are in fact correlated?
Can someone please comment on this?

Correlated errors are not the big bad wolf --- they will not blow your house down
Strictly speaking, it is the error terms that are assumed to be uncorrelated in a standard regression model.  The residuals are actually slightly correlated owing to their mutual dependence on the estimated parameters in the model.  If the error terms in a regression analysis are correlated then it is possible to build this into the model, which is essentially what is done in a linear mixed model.  Moreover, it is simple to test the assumption that the errors are uncorrelated using diagnostic tests; usually this would involve looking at correlation or auto-correlation estimates for the residuals.
If you don't account for correlated errors in a regression model (i.e., if they are actually correlated but you fit a model that assumes they are uncorrelated) then this will mean that your estimated variance for the coefficient estimators and residuals will sytematically under or over-estimate the true values.  For example, if the error terms are positively correlated (which is the more common case) and you treat them as uncorrelated, you will tend to underestimate the true variance in the coefficient estimators in the model.
Correlated error terms are certainly not the big bad wolf of statistics --- they will not blow your house down and there are much nastier problems that can be encountered.  The main thing that keeps them in check is the fact that diagnostic tests in regression can generally estimate residual correlation well, so they can be used to check the plausibility of underlying assumptions about the correlation of the error terms.  If you find positive correlation in your error terms you can model this using a linear mixed model, where the additional random effects term in the model induces positive correlation in the overall error term and thereby generalises to a case that allows such correlation.
