Quadratic moderator with a quadratic independent variable My model has two moderators and two independent variables. All four have a quadratic relationship with the dependent variable.
Y = f(x1, x1^2, x2, x2^2, m1, m1^2, m2, m2^2)

I need to include the interaction term. But unsure what would be the right way if I want to just test for linear interaction [i.e., the level shift and not the curvature change]. Can I just take the interaction between the linear term of the independent variables and the moderators, i.e., x1m1, x1m2, x2m1, x2m2?
I found links like these but couldn't find the solution to my question - How to model interaction effects when both the independent variable and the moderating variable have quadratic main effects? and https://www.researchgate.net/publication/257584254_Moderation_in_Management_Research_What_Why_When_and_How
Kindly advice. Thanks.

Let me attempt to answer your question. Generally the way one specifies a model is informed by a number of factors, including the theory underpinning the studies, interpretability, etc.
So you can include as many interaction terms as possible so long as theory suggests there are some technical interactions between or among those variables. But as you do that do not also forget that your model may be difficult to interpret, and the issues with modeling such as collinearity and endogeneity may become predominant.
To summarize, yes, you can include as many interaction terms as theory supports, interpretable, and improves the model overall. But too many interaction terms may also increase the test error of your model, especially if some of the interaction terms are unnecessary and reduce the generalization of your model.
