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In the textbook, Introductory Econometrics (Wooldridge), the concept of specifying log/log, level/log ... , is explained as a tool to understand the relationships of the regerssors either in relative or absolute terms. Using log transformation is also explained as having desirable properties like capturing non-linear relationships in a linear regression, reducing the influence of outliers and so forth.

Question: My question mostly refers to the point on log capturing non-linear relationships. If log transforming captures non-linear relationships, why are there examples in the textbook and in the real world where we have both log and squared or cubed terms? something like:

log(y) = c + log(x1) + log(x1^2) + log(xk) + u

Wooldridge also points out ambiguity in what log squared terms mean. It could be as above, or the one below (where we log first, then square the whole term):

 log(y) = c + log(x1) + log(x1)^2 + log(xk) + u

In either event, is it fair to say that log transforming alone does not adequately capture non-linear components, and thus we include squared/cubed terms? Or, perhaps using log is mainly for theory (relative/absolute inference) and including squared/cubed terms still is needed?

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    $\begingroup$ $\log(x_1^2) = 2\log(x_1)$ so as written this does not make a lot of sense - are you sure this formulation is what appears in the textbook (and in "the real world")? $\endgroup$ – Juho Kokkala May 12 '18 at 10:54
  • $\begingroup$ @JuhoKokkala Is almost the same, I am inept at latex encoding, I'll try to fix though. $\endgroup$ – Arash Howaida May 12 '18 at 10:55
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Logs certainly don't capture all nonlinear relationships. Sometimes a polynomial term is better, sometimes you really need something more flexible such as a spline.

And sometimes we want logs of variables because it makes substantive sense. We might, conceivably, even want a spline of a log term.

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  • $\begingroup$ Time permitting, I would love to see an example specification using logs and splines together. $\endgroup$ – Arash Howaida May 12 '18 at 14:25
  • $\begingroup$ I don't have an example but, suppose income was one of the IVs and it was related in nonmonotonic ways to the DV. Then that would be a good candidate. $\endgroup$ – Peter Flom - Reinstate Monica May 12 '18 at 15:33

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