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I am finishing up an econometrics assignment and this problem has me stuck.

I have estimated a regression equation for ln hourly wages on a gender dummy variable, several race dummy variables, a quadratic form of experience, and education.

ln(hourly wages) = β0 + β1female + β2black + β3american_indian + β4asian + β5other + β6education + β7experience + βexperience^2

I now want to re-estimate my model allowing the relationship between experience and wages to differ by sex.

I know I need to include the βexperience*female but do I also need to include βexperience^2*female?

I apologize if my notation is a bit off. Any help/explanation is greatly appreciated.

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Yes. The relationship between experience and wages is described by the function β7*experience + β*experience^2. To allow the relationship to vary by gender, you should interact the function with a gender dummy:

(β7*experience + β*experience^2)*female = β7*experience * female + β*experience^2*female.

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You can do, & @standard_error has shown how to, but you don't necessarily need to; in fact it's quite common to consider only linear interactions, to spare degrees of freedom, when modelling a curvilinear relationship through polynomials. See e.g. Venables (1998), "Exegeses on linear models", S-Plus Users' Conference, Washington DC. The difference is in how you want to allow the relationship between wages and experience to differ by sex: without interaction the slopes are the same at any experience level; with linear interaction the slopes can differ at any experience level, but the rate of change in slope is constrained to be the same; with quadratic interaction the slopes can have different rates of change. (Geometrically, a linear interaction allows sex to shift the axis of symmetry of the parabola representing the relationship but not to stretch it.) As always, the limit to the overall complexity of models you can usefully fit depends on how much data you have; the usual approaches to model selection and hypothesis testing apply.

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