Multicollinearity and Interaction Effects I know similar questions were asked before. However, none of the existing answers help me with my problem:
I have a gls model with y=B0+B1X1+B2X2+B3X1X2+e. The VIF value for X1 and the interaction effect X1X2 are 413.67 and 414.08 respectively. The VIF value for X2 is only 1.044729. Furthermore, the resulting coefficients are very high for X1 and the ineteraction effect X1X2 but not statistically signficant.
It is written on many webistes that one does not have to consider multicollinearity between a variable and its interaction effect. But could it here lead to a lower p-value? Furhtermore, it seems strange that the VIF values for X1 and the interaction effect are so close to each other.
Help is much appreciated. Thanks in advance.
 A: I think that you might be getting misled by the p-values of the individual coefficients for X1 and the interaction term. Multicollinearity often isn't a big problem, particularly for predictive models. It just can make it hard to get precise estimates of individual coefficients.
When two predictors are highly correlated, the standard errors of their individual coefficients can be very large--that's the problem with multicollinearity. There will, however, typically be a compensating negative correlation between their coefficient estimates. There's a simple example here. That's not typically reported in standard model reports, but the coefficient variance-covariance matrix is an important component of model results.
If you do a test that evaluates the overall association of X1 with outcome, like a Wald "chunk" test on all coefficients involving it or a likelihood-ratio test between your model and one that completely omits X1 and its interactions, then you could still get a highly significant result. Similarly, for predictions from a model, the covariances among coefficient estimates can help correct for the high variances of the individual coefficient estimates and lead to reliable predictions.
Furthermore, with an interaction term, the individual coefficient for X1 is arbitrary: it depends on the centering of the X2 variable with which it interacts. When a predictor is involved in an interaction, it thus can be misleading to try to interpret the p-value of its individual coefficient.
