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In my textbook I often see a quadratic term in the regression. For example let's say I have the regression model: $\log(\mathrm{wage})=\beta_0+\beta_1\mathrm{educ}+\beta_2\mathrm{exp}+\beta_3\mathrm{exp}^2+\varepsilon$. Why is it so common to put the squared term in there? Here's a quote from my book:

It often makes sense to add quadratic terms of any significant variables to a model.

Why is that? And if this is true, then if I'm studying endogeneity and a quadratic variable isn't in there is it likely to be that case the the squared term is endogenous if it's omitted?

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    $\begingroup$ Well, for starters, frequently relationships are not linear. Unless I had a priori reason to, I'd usually only do it if the residual analysis suggested the linear model was clearly inadequate. On the other hand if I had no a priori sense of what form the relationship might take I'd fit something more general from the outset. $\endgroup$ – Glen_b Nov 26 '12 at 1:10
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Often the relationship between y and x is nonlinear. There are a variety of solutions. One solution is to add polynomial terms and the first one to look at is usually $x^2$. But you should first look at a scatterplot of x and y; you should also look at the residuals from the linear model without the quadratic term. But it turns out that many relationships are pretty well fit by $y \sim b_0 + b_1x + b_2x^2$ (plus any other x variables, of course).

It is also possible to add cubic, quartic and even higher order terms, but such models quickly become hard to interpret. Another possibility is to look at a spline regression.

I don't really understand your last bit about endogeneity. A variable can be endogenous or exogenous to a model, but that doesn't seem to relate to this ... at least, not in a way that's clear to me.

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  • $\begingroup$ I can see why you would be confused. I appologize for that, but a friend explained it to me. Thanks for you explanation. $\endgroup$ – Kyle Nov 26 '12 at 4:54
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    $\begingroup$ Adding arbitrary polynomials to "account for nonlinearity" is almost as arbitrary as asserting linearity. Best to use a nonparametric smoothing regression in the absence of explicit theory regarding functional relationship, right? And if a parametric estimate is needed, following that with an algorithmically determined fractional polynomial, or appropriate custom nonlinear least squares, or something similar second, yes? A quadratic or cubic function generally does a poor job of representing linear threshhold or saturation effects, for example. $\endgroup$ – Alexis Dec 10 '18 at 0:59

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