# If we have a binary variable in our linear regression, the VIF for its coefficient estimate uses the $R^2$ of a linear probability model. What gives?

The variance inflation factor (VIF) in an ordinary least squares linear regression coefficient is calculated using the $$R^2$$ of a linear model that uses the other features to predict the feature to which the coefficient corresponds.

If the feature for which we want to calculate the variance inflation factor is binary, then such a model is a linear probability model, which has several known problems, among them being that such a model can make predictions that are literally impossible (e.g., predicted probability above $$1$$).

I suppose the math of the working with the feature covariances leads to such a model, but it seems bizarre that such a characteristic (VIF) would relate to a model that can make impossible predictions. Is there any resolution beyond a somewhat unsatisfying, "Yep, that's just what happens when you work with the feature covariance matrix"?

• On the one hand, you could do a logistic regression and use a pseudo-$R^2$ to calculate the VIF of a binary predictor. On the other hand, the VIF if very much a rule of thumb, so I don't think this change would make a major difference as to where you would actually detect collinearity. On the third hand, you could use Belsley's collinearity diagnostics instead of the VIF. Mar 17 at 6:26