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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.

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theforestecologist
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    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 Oct 18 '16 at 18:19
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    I would not say it *never* makes sense, but higher order terms will be more difficult to [reliably estimate](http://stats.stackexchange.com/questions/233414/why-are-there-large-coefficents-for-higher-order-polynomial/) 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.) – GeoMatt22 Oct 18 '16 at 18:20
  • @aginensky You could write that as an answer. – Firebug Oct 18 '16 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](https://en.wikipedia.org/wiki/Quadratic_form) – Silverfish Oct 18 '16 at 22:29

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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
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