Suppose that there are 2 models,
- $y$ ~ $x_1+x_2+x_2^2+x_1:x_2+x_1:x_2^2$
- $y$ ~ $x_1+x_2+x_2^2+x_1:x_2^2$
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?