Including the interaction but not the main effects in a model
I'm studying logistic regression now. And I have a question:
Suppose I have a logistic regression model as below:
$\beta_0$ is the "constant" coeffficient (I wonder if that is a correct name), $\beta_1$ is the coefficient for $x_1$, $\beta_2$ for $x_2$, and $\beta_3$ is the coefficient for the interaction term $x_1 x_2$.
During the variable selection processing, is it appropriate to include $x_2$ as an interaction term only, but not the main effect? I think this imply that $x_2$ itself has no effect on the outcome, but has effect when interacting with $x_1$. I wonder this kind of partial variable selection is an appropriate way to do modelling.