I was reading the following thread: link when I came across this discussion regarding dropping regressors that demonstrate multicollinearity from a linear regression model:
But what if you have multicollinearity and removing a variable reduces it? (This isn't the case in the original question, but often is in other data). Isn't the resulting model often superior in all sorts of ways (reduce variance of estimators, signs of coefficients more likely to reflect underlying theory, etc)? If you still use the correct (original model) degrees of freedom. – Peter Ellis Feb 13 '12 at 23:08
It is still better to include both variables. The only price you pay is the increased standard error in estimating one of the variable's effects adjusted for the other one. Joint tests of the two collinear variables are very powerful as then they combine forces rather than compete against one another. Also if you want to delete a variable, the data are incapable of telling you which one to delete. – Frank Harrell
I am curious about the part in bold. What kind of statistical test becomes more powerful when using two or more multi-collinear variables? The textbooks I've been using to study linear regression seem to agree that we should seek to remove it in almost all cases. Thank you!