Test for multicollinearity with binary and continuous independent variables

Question

I have a question concerning multicollinearity: I have several independent variables. Some are binary and some continuous. The dependent variable is binary. Can I use the Pearson correlations to test for multicollinearity?

Answer 1

Correlations are not good direct approximations of collinearity. The issue is that they only consider the covariance between two variables and not the entire model. The usual go-to is the variance inflation factor (VIF), defined as:

$$ \operatorname{VIF}(\hat{\beta_j}) = \frac{1}{1-R^2_j} $$

where $R^2$ is simply the typical $R^2$ obtained by regressing the $j$th predictor on the remaining predictors. By its naming, this approximates how much the variance is inflated by including a given predictor, the variance being the standard errors of the coefficients (which typically explode with things like perfect collinearity). I note that this collinearity issue may or may not matter depending on the context in this recent answer.

Answer 2

Shawn is right that correlations are not a good approximation to collinearity (although they are often used for that); he also gives the reason for that. He is also right that VIF is the usual method. But I prefer condition indices, as does David Belsley, one of the top experts on collinearity. One big reason I like them is that, with their allied proportion of variance statistic, they tell you where the problem is. VIF gives you single variables and the VIFs. But collinearity, by its nature, involves at least two variables.

For much more on this, see Belsley's books: Conditioning Diagnostics and Regression Diagnostics (coauthored with Kuh and Welsch), or my dissertation: Multicollinearity Diagnostics for Multiple Regression: A Monte Carlo Study.