I was trying to determine the biasing constant in ridge regression when I came across a phenomenon that seems quite puzzling, to me at least. I let the GCV criterion choose a constant for me and then I got the Variance Inflation Factors of the new model by computing
$$ \left( \mathbf{R_{XX}} +c\mathbf{I} \right)^{-1} \mathbf{R_{XX}} \left( \mathbf{R_{XX}} +c\mathbf{I} \right)^{-1} $$
and extracting the diagonal elements of this matrix. What I found puzzling was the fact that these VIFs were very close to zero. It seems to me that that would require negative $R^2$s, no? I know that this can happen occasionally, for example in Regression Through the Origin, but I cannot quite justify it in this context.
I am wondering then, what does a VIF close to zero mean? Then, would my choice of this constant be acceptable or should I look for another solution that keeps the VIFs close to 1, as they ought to be in the absence of multicollinearity?