# Interaction between a predictor and its quadratic form?

Does it ever make sense to form an interaction between a predictor and its quadratic form?

Suppose I have the model:

$y = \beta_{0} + \beta_{1} x + \beta_{2} x^2$

Is there ever an instance that adding an interaction term between the $x$ and $x^2$ terms makes sense?

$y = \beta_{0} + \beta_{1} x + \beta_{2} x^2 + \beta_{3} x \cdot x^2$

This CV post suggests that a quadratic term is very similar to an interaction term, but the question/answers are in the context of interactions between two different variables.

I'm wondering if an interaction between a variable and it's own quadratic makes sense.

I ask because doing so in a generalized least squares (gls) model improves the model (lower AIC and lower RMSE). I'm not sure if this is valid. For context, I'm creating a predictive model so I'm really after optimizing model performance.

• The interaction term between x and $x^2$ is $x^3$ . So you are just creating a cubic polynomial regression rather than a quadratic polynomial regression. In general one can create degree n polynomial regressions. Will adding the cubic term, or any other degree term, increase the predictive power of the model ? That is a question that CV or train/test will answer.
– meh
Commented Oct 18, 2016 at 18:19
• I would not say it never makes sense, but higher order terms will be more difficult to reliably estimate and increase the chances of over-fitting. So I do not think a decrease in error over the training data is necessarily a good reason to add another term. (Cross validation may be more reliable here than AIC.) Commented Oct 18, 2016 at 18:20
• @aginensky You could write that as an answer. Commented Oct 18, 2016 at 19:18
• I'd suggest to edit the question and replace "its quadratic form" by "its square", since this would make it clear you're not asking about something rather more general Commented Oct 18, 2016 at 22:29

The interaction term between x and $x^2$ is $x^3$ . So you are just creating a cubic polynomial regression rather than a quadratic polynomial regression. In general one can create degree n polynomial regressions. Will adding the cubic term, or any other degree term, increase the predictive power of the model ? That is a question that CV or train/test will answer.