Can I include a quadratic regression in a linear model?

Question

I have several independent variables & one dependent variable for a regression model.

One of these IVs have a much better curve fit (with the DV) as quadratic regression.

I was thinking about transforming the IV (by squaring it) and then adding both the IV and its squared value in the multiple linear regression model.

However, I've heard that if I do that, I can only keep either one or the other (i.e. either the IV or its squared values) in the regression model because they're co-dependent.

What would be the correct way to go about this? Can I add quadratic regression to a multiple linear model at all?

Answer 1

It's totally fine to include both the linear ($X$) and quadratic ($X^2$) terms in the model, and I recommend you do so. Whoever you were talking to is right that you can run into problems with multicollinearity when adding polynomial terms, though, if you don't center your predictor before making the higher order polynomial terms. If you center $X$ before making $X^2$, then that problem will be greatly reduced and chances are you'll have two relatively independent variables in your model. This is even mentioned in the wikipedia article on multicollinearity:

Mean-center the predictor variables. Generating polynomial terms (i.e., for $x_1, x_1^2, x_1^3$, etc.) or interaction terms (i.e., $x_1*x_2$, etc.) can cause some multicollinearity if the variable in question has a limited range (e.g., [2,4]). Mean-centering will eliminate this special kind of multicollinearity.

You can always check the level of multicollinearity for your predictors by getting the tolerance or variance inflation factors for your predictors after you estimate the model.