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I have three variables: $y$, $x_1$ and $z$.

I'm interested in finding the effect of $x_1$ on $y$. Unfortunately $x_1$ is endogenous so I use an instrument z.

I know that $z$ is correlated with $x_1$. The problem is, $z$ is correlated with another variable ($x_2$) that arguably has an effect on $y$. $z$ is unlikely to have a causal effect on $x_2$, but $x_2$ may still be endogenous for other reasons (e.g. reverse causality, etc.).

Will the following procedure give me a consistent estimate of beta-1: two stage least squares with $y$ regressed on $x_1$ and $x_2$, with $z$ as an instrument for $x_1$?

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Why is this just not an endogeneity issue due to omitting $x_2$ and that can be solved like you said: Include the omitted variable $x_2$ and you just happen to have an IV $z$ in hand that's correlated with (both) regressors? –  Michael Apr 7 at 15:10

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