Interpreting two-way interaction in the presence of quadratic interaction

In an OLS regression with continuous variables Y, X1, X2:

• the interaction term X1*X2 is not significant in model 1,
• the interaction term X1*X2 becomes significant when the quadratic interaction term X1*X2^2 is added in model 2.

Am I correct that it seems "wrong" or problematic to conclude a significant interaction X1*X2 on the basis of model 2 (similarly to it being problematic to interpret the main effects X1 and X2 in a model containing an interaction X1*X2)?

In principle there is nothing wrong with such a model. Say $x_2$ is age ranging from 0 till 20 years. So our interaction effect measures how the effect of $x_1$ changes as a child grows older. Going from 0 to 1 year is something very different than going from 10 to 11 years, so it is reasonable to expect some non-linearity in both the main effect but also in the interaction effect.

You have two questions, which I will translate into a more "statistical language":

1. How can I test the null hypothesis that there is no interaction effect?
2. How can I interpret the resulting parameters?

As to question 1: For that you would need to test the hypothesis that the coefficients for $x_1\times x_2$ and $x_1 \times x_2^2$ are both simultaneously 0. You can do that with an F-test.

As to question 2: You can in principle interpret $x_1\times x_2$. It is the change in effect of $x_1$ for a unit change in $x_2$ when $x_2$ is 0. In our example that could make sense: it is the interaction effect for babies. However, if we were studying elderly people, then this would be a gross extrapolation (assuming you haven't centered $x_2$ to a reasonable value withing the range of the data, e.g. the mean).

In practice I would prefer to interpret the interaction effects through a graph. That way you not only see the interaction effect for babies but for the entire age range. We added the square term because we thought that the interaction effect is not the same for each age, so it makes sense to look at that.

I final note: I would make the main effect at least as flexible as the interaction effect. So if you add an interaction with age and age squared, than in the main effects I would also add at least age and age squared. Otherwise the square term in the interaction effect may capture non-linearities in the main effect.