Regressors that are correlated with varying coefficients

Michael Papenfus writes, regarding this article on multilevel modeling,

I [Papenfus] am currently working on trying to better understand the assumptions underlying the random effects (both varying intercepts and varying slopes) in hierarchical models. My question is: are there any hierarchical modeling techniques which allow one to include regressors which are correlated with the random effects or is this situation an example of what these models cannot do?

My short answer is that, when an individual-level predictor x is correlated with the group coefficients (I prefer to avoid the term “random effects”), you can include the group-level average of x as a group-level predictor. See here for Joe’s entry on this topic, along with a link to our paper on the subject. (We also briefly discuss this issue in Section 21.7 of our new book.)