If I want to run a 2 stage least squares (2SLS) regression with:
Relationship of interest: $Y = \alpha + \beta X + \varepsilon $, where $X$ is the endogenous explanatory variable of interest.
If I have an instrument $Z$ where I can safely assume that it is both relevant and fulfils the exclusion restriction, can I interact it with other variables in the first stage, that do not fulfil the exclusion restriction, as long as I control for these other variables in the second stage?
So I am thinking
Estimating the first stage: $X = \gamma + \delta_1 Z + \delta_2 W_1 + \delta_3 W_2 + \delta_4 Z*W_1 + \delta_5 Z* W_2 + \epsilon$, where the exclusion restriction only holds for $Z$ but not for the $W$s, to get $\hat X$
Second stage: $Y = \omega + \eta_1 \hat X + \eta_2 W_1 + \eta_3 W_2 + e $
So does this procedure allow me to kind of side-step the exclusion restriction? If so, is there any paper / book / article that talks about that?