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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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    $\begingroup$ 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. $\endgroup$ – aginensky Oct 18 '16 at 18:19
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    $\begingroup$ 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.) $\endgroup$ – GeoMatt22 Oct 18 '16 at 18:20
  • $\begingroup$ @aginensky You could write that as an answer. $\endgroup$ – Firebug Oct 18 '16 at 19:18
  • $\begingroup$ 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 $\endgroup$ – 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.

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