Detect multicollinearity in maximum likelihood scenarios

Question

I'm estimating a binary logit discrete choice model with BIOGEME and want to check for multicollinearity of my predictors. BIOGEME uses maximum likelihood estimation (MLE) and not ordinary least square (OLS) method.

In OLS regression models, one can use $VIF = \frac{1}{1-R^2}$ to assess for multicollinearity.

My questions are:

1. Can I use VIF in the context of MLE as well?
2. In case, that yes: McFaddens $R^2$ is defined exactly the same as $\rho^2 = 1- \frac{L_1}{L_0}$, where $L_1$ is the estimated model and $L_0$ the nullmodel. Can I then derive: $VIF = \frac{1}{1-\rho^2}$ ?
3. If not: are there any other measures for multicollinearity in MLE models?

Obviously, I check for correlations in between the independent variables before the modelling. However, I'd like to have an indicator for my final model as well.

Answer 1

There needs to be a distinction made between predictor multicollinearity and VIF.

The variance-inflation factor (VIF) represents relationships among the coefficients of the independent variables, as captured in the coefficient variance-covariance matrix. The equation given above for unweighted ordinary least squares (OLS) takes advantage of the independence of the entries of that matrix from the dependent-variable values, outside of the estimate of overall variance. There is a direct relationship between the predictor multicollinearity calculated from that equation and the covariances among the estimated coefficients.

For models fit by maximum likelihood (ML), the relationships between the predictors and the dependent variable can influence the variance-covariance matrix, unlike with OLS. In an ML model the OLS formula for VIF will similarly show multicollinearity among predictors, but the relationship between the predictor multicollinearity and the coefficient covariances won't necessarily be so direct.

You might want to consider a different approach, rather than VIF, for evaluating predictor multicollinearity per se. The generalized VIF helps if your independent variables include categorical factors, and for ML models it will provide an estimate based directly on the variance-covariance matrix. See this page.