# What is the definition of the generalized variance inflation factors (GVIF)?

I am currently working on a statistical project at my school, in short, it is about finding the "best" linear regression model to explain the price of houses in a giving community.

The model we have created contains some factor variables with more than 2 degrees of freedom, and thus when using the car package in R, to measure the multicollinearity we get a table that gives the GVIF and the $\text{GVIF}^{1/(2\cdot df)}$.

As far as I can tell from the documentation of the VIF function, the GVIF adjust for the dimensions when dealing with these degrees of freedom, and from this question (and answer) Which variance inflation factor should I be using: $\text{GVIF}$ or $\text{GVIF}^{1/(2\cdot\text{df})}$? it seems that there is no "correct way" to set a threshold for the GVIF.

Something that made me quite confused is the mention of that VIF and GVIF is the same when the degree of freedom is 1, which would make sense since the VIF function only prints the VIF in R, but then I saw this definition of the GVIF (http://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/Collinearity): $$\text{GVIF} = \text{VIF}^{\frac{1}{2\cdot df}},$$ which then boils down to that the GVIF squared is equal to the VIF. And then I read in the previously mentioned thread which made me even more confused:

"If we then simply apply the same standard rules of thumb for GVIFˆ(1/(2*Df)) values as recommended in the literature for the VIF, we simply need to square GVIFˆ(1/(2*Df))."

Now my question is: Is there a precise way to define (and interpret) the GVIF such that the connection between the VIF and GVIF becomes clear, and thus the interpretation becomes more clear?

I hope my question makes sense. Thank you.