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so for my master's thesis, I am examining the influence of union density (% of the workforce in a union) and top marginal tax rates on pre-tax CEO pay. These two independent variables are very highly correlated in my sample. When I include both variables in the regression, union density has a significant negative association with CEO pay, and top marginal tax rates are insignificant. The VIF is very high for both variables. However, it is my understanding that multicollinearity does not bias coefficients, and it is accurately captured in the standard errors (see Stephen Voss' multicollinearity paper). On the other hand, in this case, multicollinearity may suggest union density and top tax rates are causally related and not independent--this would violate a key OLS assumption. Further, union density and top tax rate changes are both associated with left government ideology (a third variable I can include in my analysis), but I expect they aren't related to each other EXCEPT for their relation to left government ideology. So my question is this: if I include left goverment ideology as a control variable, and union density is still significantly associated with CEO pay, and tax rates are not, will my coefficients be unbiased? Can I then ignore multicollinearity? Another question: do I even need to control for government ideology, if union density and tax rates are only related to each other because of their association with government ideology? If I only include union density and tax rates in the regression are my coefficients unbiased? Previous research mostly suggests top tax rates are not associated with CEO pay changes, but a few authors find they are (and in opposite directions). Thanks very much in advance for any insight.

Here is an excerpt from Voss' multicollinearity paper that has me concerned: At some point the idea of “variable packages” becomes hazy. Two variables may not capture the exact same underlying concept, but researchers nevertheless may be aware that they are causally related to each other in the larger population, such that multicollinearity in the sample is no accident. This sort of multicollinearity, although theoretically meaningful, nonetheless can pose an obstacle to the analyst who wishes to distinguish two or more concepts statistically; it can hinder an analysis based on fine theoretical distinctions. The appearance of such multicollinearity may be helpful, because it offers a warning that concepts may not be as theoretically distinct as a modeler initially assumed, but it still risks leaving the analyst with regrettably hesitant causal conclusions. The analyst would only be able to generalize about the overlapping variables as a package, even if a project’s needs demand otherwise.

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The problem with colinearity is not so much bias in the parameter estimates as huge variance in those estimates and extreme sensitivity to small changes in the input data. David Belsley, in his two books on the subject, proposes using condition indexes rather than VIF to diagnose collinearity; there's also my dissertation on the subject: Multicollinearity diagnostics for multiple regression: A Monte Carlo study.. Belsley gives examples where changes in the 4th significant digit of the input data reverses the signs of the parameter estimates. If you have R, a good way to check this for your example is the perturbpackage.

If your goal is explanation of the phenomena in question, which it seems to be, then you cannot ignore collinearity. Your conclusions may be the opposite of what a very similar data set would give.

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