# Linear regression: is near multicollinearity really a problem ?

I have the following model for continuous variables $$Y = \beta +\beta_1 X1 + \beta_2 X2 + \beta_3 X3 + \beta_4 X4 + \beta_5 X5 + \epsilon$$

Everything works out very well, the model passes all kinds of tests, respects all assumptions but one: $X1$ and $X2$ are highly correlated (>0.9) so maybe we have multicollinearity. They have huge VIF (around 20). When I remove $X2$ from the model, the VIF of each variable is under 4, and I obtain similar results but slightly worse. In every kind of criteria.

So if I follow theory, I should remove $X2$. But practical tests shows that the first model is slightly better (just a tiny bit). What should I do ? Why are the models behaving like this in opposition to what theory says ?

• What do you mean by "better"? What do you want to do with the model- predict y values given x values? Do inference on the cofficients $\beta$? Commented Apr 13, 2018 at 14:40
• Do you only care about the predictive power of your model? Or do you want to interpret the coefficient values and the effect of each predictor? Commented Apr 13, 2018 at 14:42
• What I mean by better is that: - Confidence interval for $\beta$'s are smaller - The coefficient R is higher - AIC, BIC are higher - Errors are smaller - MSE in k-cross validation is smaller Commented Apr 13, 2018 at 14:43
• I am more interested in the predictive power of the model. Commented Apr 13, 2018 at 14:45

1. There is no assumptions of uncorrelated $X$s, hence it does not violate any theoretical consideration.
2. However, high correlation (surely huge VIF as $20$) is important practical consideration regarding model's stability.
4. However, if you are interested in dealing with the multicolinearity with a little more sophisticated way than just dropping out $X_2$, then you can:
(i) Perform principle components regression on the subsets of $k$ best PCs. https://en.wikipedia.org/wiki/Principal_component_regression