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.