The problem I encounter is the following:
Imagine a (perfect) inverted U-shaped relation between an independent variable and a dependent variable. When you look at the curve estimation there is indeed a perfect inverted U-shaped relation.
If you conduct a regression analysis you have the normal independent variable and the squared term of the independent variable. In such a case, usually the normal independent variable gives a positive beta and the independent variable squared gives a negative beta. If these are both significant, this indicates that there is an inverted U-shaped relationship. However, it is well-known that in this way you have a very high VIF value (multicolliniearity), because there is an almost perfectly positive correlation between the normal independent variable and independent variable squared (because you square it of course).
The solution for reducing the multicolliniearity is to mean center the independent variable, and after then taking the squared term of the mean centered independent variable. If i perform a regression now (with both of the two predictors mean centered) I do not have problems anymore with multicollinearity, but I have two negative beta's. How is this possible? How can i have two negative beta's while i have a perfect inverted U shaped relationship according to the curve estimation and data? Is it still possible to prove the inverted U shaped relation with two negative beta's?