# Interacting covariates with the instrument in the first stage

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?

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.
• I already accepted this answer, but just realized that I actually do not completely follow. Would $\eta_1$ not be interpretable as causal under the conditional mean independence assumption? And why would regressing Y on an exogenous X (and endogenous Ws) yield an inconsistent estimator for $\beta$? From my understanding it would only be a problem if the Ws where not included and are correlated with X and Y.
• I think it's a matter of what the model is. Let us accept it that the relationship of interest is $Y = \alpha + \beta X + \varepsilon$. If X is exogenous but W is included in the RHS, the estimator is inconsistent (in the sense that $\eta_1$ is not equal to $\beta$) in general (unless X and W are uncorrelated) - the opposite case of omitted important variables. Feb 22 at 1:23