Classical VIF coefficient calculated using aux regression as $VIF_i = \frac{1}{1 - R^2_i}$. But this formula derived from classical linear model assumptions (spherical matrix of errors) under which $Var(\hat{\beta}_{OLS}) = \sigma^2 (X'X)^{-1}$. My question is any generalisation of VIF formula exist that based on robust representation of coefficient variance $Var(\hat{\beta}_{OLS}) = (X'X)^{-1} X' \Sigma X (X'X)^{-1}$ that accomodates both, for example, White and Newey-West estimators?
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$\begingroup$ What (type of) model are you investigating? A normal linear regression? $\endgroup$– Michael MCommented Dec 17, 2023 at 16:22
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$\begingroup$ @MichaelM Yes, classical linear regression with possible violations of conditional homoscedasticity and (or) no serial correlation in errors assumptions. $\endgroup$– kissmemiauCommented Dec 17, 2023 at 16:24
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2$\begingroup$ See this page. There is a general form of VIF that can be calculated for any model with a variance-covariance matrix of coefficient estimates. The “classical” version you show happens to be what you get from that approach in OLS. $\endgroup$– EdMCommented Dec 17, 2023 at 16:31
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$\begingroup$ @EdM Duplicate? $\endgroup$– DaveCommented Dec 20, 2023 at 12:12
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$\begingroup$ @Dave this is a borderline case for duplication. Linked page was specifically about logistic regression, although the same generalized VIF can apply to the situation here, linear regression with a robust coefficient covariance matrix. $\endgroup$– EdMCommented Dec 20, 2023 at 14:37
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John Fox and Georges Monette, in "Generalized collinearity diagnostics," Journal of the American Statistical Association 87: 178-183 (1992) described a generalized variance inflation factor that can be calculated for models with a variance-covariance matrix of coefficient estimates, including generalized linear models. It also can handle multi-level categorical predictors and continuous predictors fitted with polynomials or splines sensibly. In the limiting case of unweighted ordinary least squares regression this gives the same result as the "classical VIF."
The vif() function of the R car package implements that, with an outline of the default code shown on this page. As the variance-covariance matrix outside of simple linear regression models depends on both outcomes and predictor values, the generalized VIF in the most general case also depends on both outcomes and predictor values.
In your application, the "classical VIF" will estimate the multiple correlation of one predictor with the others. The "generalized VIF" will tell how much those correlations matter in terms of being able to distinguish one predictor's association with outcome from that of the others. My sense is that the latter is of more general interest.
