Leaving out other endogenous variables when having one instrument

Question

Let's suppose we have a regression $E(Y|X)$ with two known endogenous variables $X_1$ and $X_2$ and one instruments $Z_1$.

My regression is: $y=\alpha+b_1 X_1+b_2 X_2 + \Gamma X_{exo}+u$

$Z_1$ seems to be a very good instrument for $X_1$; informative and valid.

My understanding is that when running a regression with both endogenous variable included in the model and instrumenting just for $X_1$, it will still produce a bias of the coefficient $b_1$. In other words, you need to instrument for all other endogenous variables to have an unbiased estimate of $b_1$.

So what is the best strategy here to proceed? Would it be possible to instead entirely leave out $X_2$ from the regression and merely instrument for $X_1$. That is, we could run instead:

$y=\alpha+b_1 X_1+ \Gamma X_{exo}+u$

while instrumenting for $X_1$ with $Z_1$. Would not that correct the omitted variable bias altogether and the problem solved?

Answer 1

It will correct the omitted variable bias only in the case that $E[Z_1|X_2]=0$. In the case of an informative instrument, you will have an unbiased estimate of the coefficient $b1$.

Edit: Then, as $X_2$ is endogenous it affects the other coefficients in $X_{exo}$--no matter if you leave it in our take it out. However, personally I wold leave $X_2$ in the regression if there is any chance it potentially could strengthen the validity of your instrument as a necessary control. This has to be decided on a case by case basis.