Usually, when we run regressions or any of its near relatives, we are dealing with the changes in the means. Typically, especially if you are thinking about variances, this is not the question that you want to know about.
It is not unreasonable that, an increase in test scores on average as a function of some variable also increases the variance–i.e. more students’ scores might drop while, at the same time, others gain even more. This is not just a matter of statistical nuance, but a substantial pattern with real consequences. (see the whole debate over inequality, following Piketty’s book) Of course, this involves dispensing altogether with the homoskedasticity assumption–because, if that were true, this would not even be a problem. But, precisely because homoskedasticity is built into the assumptions of OLS regression, nobody can even think about this problem systematically.
There are potentially approaches one can take that leverages off of tests and “remedies” for heteroskedasticity to deal with this, perhaps at a systematic level. That, however, might take a bit more work than not… Has anyone already done any significant work on this?