Why is the residual independence assumption important in OLS regression and how may it be violated? I haven't found a clear answer to this question yet. What parts of the equation does it affect exactly, and how can b weights still be unbiased? Does it bias your results? I know the answer to the last question is yes, but how and in what ways? People say time series designs and clusters violate this assumption, but what other ways can it be violated?
 A: Independence of the residuals or error term from predictors is a core assumption of all regression modeling regardless of the method used to estimate the model, whether it be OLS, maximum likelihood, FIML, whatever. Jeffrey Wooldridge in Econometric Analysis of Cross Section and Panel Data has the clearest explanation for why this is "bad" for models beginning on page 50, 

An explanatory variable is said to be endogenous if it is correlated
  with u (the errors or disturbance)...This usually arises in one of
  three ways: Omitted variables...these appear when we would like to
  control for one or more additional variables but...they cannot be
  included in the model...Measurement error...or Simultaneity which
  arises when at least one of the explanatory variables is determined
  simultaneously along with y.

None of these are desirable properties for any model. Regrettably, many of the workarounds and "solutions," e.g., 2SLS in time series, etc., don't do that great of a job of solving these problems either. 
