Interacting covariates with the instrument in the first stage

Question

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?

Answer 1

Short answer: No.

Your model is $Y=\alpha + \beta X + \varepsilon$. Even when $X$ is exogenous, if you regress $Y$ on $X$, $W_1$ and $W_2$, then the OLS estimator is inconsistent (for $\beta$) unless $W_1$ and $W_2$ do not affect $Y$ on average (after controlling for $X$) or $X$ is uncorrelated with $W_1$ and $W_2$. When $X$ is endogenous, there is no reason why your strategy should work.

Now suppose that your model is $Y=\alpha + \delta X + \gamma_1 W_1 + \gamma_2 W_2 + \varepsilon$, instead. (This is not your model; $\delta \ne \beta$.) You are saying $X$, $W_1$ and $W_2$ are endogenous, while $Z$ is exogenous. You need instruments for $X$, $W_1$ and $W_2$, but you don't. The parameters are not identified.