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Is it acceptable to have a quadratic interaction term in OLS regression? Does this violate the assumption of OLS regression as predictors should have a linear relationship?

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Is it acceptable to have a quadratic interaction term in OLS regression?

Statistically speaking this is not specifically a problem, though in some situations such a term may seem odd unless other terms are also in the model. (You might also run into issues with interpretation or multicollinearity.)

If you mean "is it acceptable to my colleagues/audience/supervisor?" ... that depends on your colleagues (etc).

In some situations there may be quite good reasons to avoid such a model.

Does this violate the assumption of OLS regression as predictors should have a linear relationship?

It does not violate the assumptions of linear regression (OLS is an estimator, not a model, please don't conflate the model with the method you use to estimate its parameters).

You may be slightly confused about what the linearity refers to. It is not required that $E(Y|x)$ be linear in any particular predictor; it will be linear in the set of columns in the $X$-matrix ("design matrix"). In that case, it will still be linear regression (indeed "linear" in linear regression really refers to linearity in the parameters, but that also means linearity in the columns of $X$).

You can have linear regressions with powers of $x$, or terms like $\log(x)$ or $x\log(x)$ or $\sin(ax)$ and $\cos(ax)$ (with specified $a$), or the set of basis functions in a natural cubic spline fit, or many other things besides. See some of the examples in the links here (some of the later examples are not linear regression, but it should be clear in the context; the linear ones are under item 1. and transformations of $x$ in item 2.).

It's also possible for such things to occur in interaction terms but naturally the more complicated the model the harder it may be to interpret (let alone issues like multicollinearity).

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  • $\begingroup$ Note that the $a$ in those $\sin(ax)$ and $\cos(ax)$ have to be specified (so pick $\sin(\pi x)$ or $\cos(2x)$ or whatever). If they are estimated, then the regression is not linear. $\endgroup$
    – Dave
    Commented Nov 12, 2022 at 3:27
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    $\begingroup$ Yes, I should clarify. $\:$ also ... @User1865345 Sorry about the rejected edit; that was not intentional -- we were just both editing at the same time and my save led to an auto-reject. I will incorporate those edits. $\endgroup$
    – Glen_b
    Commented Nov 12, 2022 at 3:39
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    $\begingroup$ Not a problem @Glen_b. +1. $\endgroup$ Commented Nov 12, 2022 at 3:40
  • $\begingroup$ @Glen_b Thank you! Regarding the multicollinearity issue, I only have one unique independent variable, I am adding interaction terms among my control variables. Does the multicollinearity matter in this case? $\endgroup$
    – Laiy
    Commented Nov 12, 2022 at 14:51
  • $\begingroup$ It might. There's no way to tell from the information you have given. $\endgroup$
    – Glen_b
    Commented Nov 13, 2022 at 1:27

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