Include interaction regressors separately? 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!
 A: 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 $y$.
