One of the most common causes of multicollinearity is when predictor variables are multiplied to create an interaction term or a quadratic or higher order terms (X squared, X cubed, etc.).
Why does this happen? When all the X values are positive, higher values produce high products and lower values produce low products. So the product variable is highly correlated with the component variable. I will do a very simple example to clarify. (Actually, if they are all on a negative scale, the same thing would happen, but the correlation would be negative).
In a small sample, say you have the following values of a predictor variable X, sorted in ascending order:
2, 4, 4, 5, 6, 7, 7, 8, 8, 8
It is clear to you that the relationship between X and Y is not linear, but curved, so you add a quadratic term, X squared (X2), to the model. The values of X squared are:
4, 16, 16, 25, 49, 49, 64, 64, 64
The correlation between X and X2 is .987–almost perfect.
To remedy this, you simply center X at its mean. The mean of X is 5.9. So to center X, I simply create a new variable XCen=X-5.9.
These are the values of XCen:
-3.90, -1.90, -1.90, -.90, .10, 1.10, 1.10, 2.10, 2.10, 2.10
Now, the values of XCen squared are:
15.21, 3.61, 3.61, .81, .01, 1.21, 1.21, 4.41, 4.41, 4.41
The correlation between XCen and XCen2 is -.54–still not 0, but much more managable. Definitely low enough to not cause severe multicollinearity. This works because the low end of the scale now has large absolute values, so its square becomes large.
The scatterplot between XCen and XCen2 is:
If the values of X had been less skewed, this would be a perfectly balanced parabola, and the correlation would be 0.
Tonight is my free teletraining on Multicollinearity, where we will talk more about it. Register to join me tonight or to get the recording after the call.