# Regression - Dealing with Correlated, Zero-Sum Predictors

I'm currently working on a regression problem where a subset of the predictor variables are zero-sum. By zero-sum I don't mean they all sum to zero, I simply mean that increasing one implies a decrease in one, all or some of the others. This fact makes these variables correlated, but it also means that there is a built-in opportunity cost when one variable has a very high value since all the other predictors must be correspondingly lower.

I was just curious to see if anyone has dealt with this kind of predictor variables before and, if so, have any suggestions on how to deal with these types of predictors, especially in terms of variable selection?

• If "zero sum" does not mean zero sum, what distinguishes this from any question about (multi)collinearity, as the signs of correlations are arbitrary in that context? To the extent that a constant-sum constraint is approximated, it is more likely that one of those predictors will be redundant and is better omitted. Modern regression diagnostics should help. Commented Jun 4, 2013 at 19:04
• Is the result of this perfect multicollinearity? What are your goals for this analysis? (Note that variable selection is a very difficult activity, & may often not be necessary.) Commented Jun 4, 2013 at 19:36

1. If the interest is in inference, then by adequately adjusting for the desired stratification variables, we conserve the interpretation of the regression coefficient for our parameter of interest. That is, that "a unit difference in $X$ is associated with a $\beta$ difference in $Y$ holding $U_1$, $U_2$, ... fixed.