Consider the following model for a regression : $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2+ \beta_3 x_1 x_2 + u$.
Suppose $x_1$ is exogenous and $x_2$ is endogenous. Endogeneity precludes me from analyzing the effect of $x_2$ on $y$. However, I still want to interpret the effect of a change of $x_1$ on $y$. The fact that $x_2$ is endogenous wouldn't bother me too much normally, but here, because of the interaction, the effect of $x_1$ is affected by the value of $x_2$. Not only is $x_2$ potentially correlated with the error term, but it is likely correlated with $x_1$ ($x_1$ and $x_2$ are not independent variables). Can I still interpret the marginal effect ($\beta_1 + \beta_3 x_2$) as the effect of $x_1$ on $y$, holding all else constant?
In my problem $x_2$ is endogenous because it is jointly determined with $y$. I study international agreements and these variables are characteristics which are decided at the same time by the parties.