I saw that Andrew Gelman had discussed this issue in one of his books, and a relevant excerpt is presented here. He also very briefly discusses this issue in his blog post here.

In the book excerpt Gelman gives a nice example in which a correlation between individual-level variables and group-level errors causes residual variance to increase when an individual-level predictor is added to the model. However, he doesn't explain whether a correlation of that sort is necessary for including a predictor to increase the residual variance.

Can this phenomenon only occur if there is a correlation between a variable at one level and errors at another level? Are there are conditions that are sufficient and/or necessary?



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