How to specify a non-linear interaction term in a linear model Say I have 2 continuous variables (x & y) in a logit model. I want to test the following hypotheses in the model:

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*For low values of x, increasing the value of y will increase the value of the response variable

*For high values of x, increasing the value of y will decrease the value of the response variable

Is there a way to specify this using interaction terms? I have only used interaction terms in which y has a linear impact on the effect of x on the response variable.

It seems to me that you want:

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*A two-way interaction, $XZ$, between $X$ and $Z$. This will estimate the expected change in the response for a 1 unit change in $X$, when $Z$ also changes by 1 unit.


*If you want this two-way interaction also to also vary with $X$ then you can interact the two-way interaction with $X$, which will be $X^2Z$. This 3-way interaction will estimate the expected change in the response for a 1 unit change in $X$ when the interaction $XZ$ also changes by 1 unit.
