Question about the assumption that residuals are uncorrelated with the predictors I'm studying the review for one of my graduate classes, Correlation and Regression Analysis, and this is one of the questions:

A friend says that you don't have to worry about the assumption that the residuals are uncorrelated with each predictor variable:  each time she has conducted a regression analysis, the residuals always turn out to be uncorrelated with the predictors by mathematical necessity.  Explain for your friend (a) what is really meant by this assumption, (b) why your friend came to the wrong conclusion trying to detect violations of the assumption in the manner described, and (b) the consequences of violating the assumption.

I'd really appreciate some help with it.
 A: I think the author of this question is confused about the distinction between errors (unobservable quantities that in the model shift observed values from their conditional expectation) and residuals (sample quantities that are calculable, and which are distinct from the errors, though related to them). 
"Residuals are uncorrelated with the predictors" is not an assumption for regression. If they've told you that it is, they're wrong, and there's no real way to handwave this error away. It's wrong at a pretty fundamental level.
There is an assumption about the errors (which are not the residuals) being uncorrelated with the predictors.
As the hypothetical friend in the problem suggests, the residuals are uncorrelated with the predictors by construction and there's no issue there. The friend's conclusion is correct and the premise of the question is false.
There's no difficulty that is not caused by the author of the question apparently conflating errors with residuals. 
I'd be very interested to see what the materials you have no doubt already been given on this subject have to say on this issue. 
