# Inclusion of Interaction terms when higher order polynomial effects are present

Suppose that there are 2 models,

1. $$y$$ ~ $$x_1+x_2+x_2^2+x_1:x_2+x_1:x_2^2$$
2. $$y$$ ~ $$x_1+x_2+x_2^2+x_1:x_2^2$$

For both models, their adjusted $$R^2$$ values are the same and BIC values are similar with the 2nd model having a slightly lower BIC. However, the interaction term, $$x_1:x_2$$ in model 1 is insignificant.

Not quite sure which model I should adopt as a result. More specifically, do I need to include the interaction term with all present polynomial orders for $$X_2$$? Are there any mathematical considerations in doing so?

• At stats.stackexchange.com/a/408855/919 I discuss a few of the considerations related to using polynomials in multiple variables for regressors. See the section titled "polynomials in multiple variables."
– whuber
Commented Oct 6, 2022 at 14:54

You can think about the $$x_1:x_2^2$$ term as an interaction between $$x_1:x_2$$ and $$x_2$$. In that context consider:
with respect to your observation that "the interaction term, $$x_1:x_2$$ in model 1 is insignificant."
For a choice between model 1 and model 2, that's mostly a function of how you want to apply your model and your tradeoff between accuracy and parsimony. If model 1 isn't overfit, then it would be fine. If you do choose model 1, however, maintain its "insignificant" $$x_1:x_2$$ term. See: