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I was taught that if I include a regressor in an interaction, then I should include it separately, too. That is, if I regress $y$ on $x_1 \times x_2$, then I should also include $x_1$ and $x_2$ separately. Although I have never seen this spelled out in a text, this makes sense to me -- if I include the "cross partial," then I should include the first order effect, too.

However, in published economics papers (in good journals) I see a lot of $$y_i = \alpha + \beta x_{1,i} + \gamma x_{1,i} \times x_{2,i} + \epsilon_i$$ when it seems that $$y_i = \alpha + \beta x_{1,i} + \gamma x_{1,i} \times x_{2,i} + \delta x_{2,i} + \epsilon_i$$ is more correct.

Is it OK to omit the non-interacted $x_2$? Is there a good reference on this? Thanks!

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    $\begingroup$ Would you mind posting a few references to some of the economics papers that you have in mind? $\endgroup$ – Jake Westfall Aug 1 '12 at 5:43
  • $\begingroup$ I highly doubt you'll get a definite answer, mostly rules of thumb. $\endgroup$ – probabilityislogic Aug 1 '12 at 7:41
  • $\begingroup$ It really depends on the area of application; if, for example, you're trying to estimate the weight of a tree, based on its height and circumference at the base, clearly the "interaction" (product) of height and circumference will be important, and, given that's in the model, neither the height nor the circumference by itself will contribute much. $\endgroup$ – jbowman Aug 1 '12 at 18:23
  • $\begingroup$ @Jake -- Sorry for the delay (getting lost in studying for my comprehensive exams :) ). The paper that motivated me was "The Political Economy of the Subprime Mortgage Credit Expansion," but I see it fairly frequently. +jbowman's comment and +JDav's answer clears it up. I guess they've tried the model with the regressor separately and the model fits best without it (and there's no need to explicitly address it). Thanks! Here's a link to the paper. papers.ssrn.com/sol3/papers.cfm?abstract_id=1623004 $\endgroup$ – Richard Herron Aug 3 '12 at 1:38
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In textbooks, many techniques are presented so that they could be applied in general contexts e.g. let's assume you have a relationship $y=h(x_1,x_2)$ where $h$ is unknown and non-linear. Nevertheless it may be possible to approximate $y$ linearly through a Taylor expansion of a certain order (let's say 2) evaluated at $x=0$.

$y\approx a + bx_1 + cx_2 +dx_1x_2 + b_2x_1^2 +c_2x_2^2 $

where the model parameters stand for the first and second partial derivatives evaluated at $x=0$

I prefer this approach as it nests two modeling issues that are taught in applied econometrics: Interactions and non-linear relationships. The second issue is very common in wage equations where experience squared determines the observed wages, but you will not necessarily find interactions of experience with other variables.

To conclude, the Taylor approach or the one that you found in your textbook are exploratory approaches in order to find a reasonable estimating equation to start with. Once confronted with data, many of these terms (a,b,c, etc.) may become non-significant or irrelevant.

The papers that you mention probably analyzed several estimating equations and reported the relevant ones. Or alternatively it may be a well known issue that $x_2$ by itself is irrelevant to explaining $y$.

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  • $\begingroup$ According to your assumption of the relationship, two subjects with the same x_! and x_2 should have the same y. But in the statistical world, it is not true. $\endgroup$ – user158565 Apr 25 '17 at 4:54

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