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Can you use two instrumental variables $z_1$ and $z_2$ at the same time for $x_1$ and $x_2$, in the following regression model

$y=a+bx_1+cx_2+dX_{other}+e$, where

$y$ is the explained variable,

$x_1$ and $x_2$ are two variables of interest that you want to study but are endogenous,

$X_{other}$ are other exogenous control variables,

$e$ is the error term.

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  • $\begingroup$ Yes. In that case, technically, the instrumental variables are $z_1$, $z_2$ and $X_{other}$. The exogeneity requirement is clear. The relevance condition is more complicated when there are multiple (in your case, two) endogenous regressors. See help ivregress_postestimation (estat firststage) of Stata. $\endgroup$ – chan1142 Nov 17 '17 at 2:21
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If you have endogeneity between a dependent variable and error term the use of Instrument variables are the way to go.

as long as $\mathbf{COV}(x_1,z_1)\ne0,\mathbf{COV}(x_2,z_2)\ne0,\mathbf{COV}(z_2,e)=0$ and $\mathbf{COV}(z_1,e)=0$ you can do so.

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  • $\begingroup$ Isn't it also required that $z_1,z_2$ are uncorrelated with $X_{other}$? $\endgroup$ – Adam Bailey Nov 16 '17 at 22:46
  • $\begingroup$ @AdamBailey the issue is perfect multicollinearity. as long as the correlation between all combinations of $z_1,z_2$ and $X_{other}$ are not 1 or -1, its good. en.wikipedia.org/wiki/Multicollinearity#Definition $\endgroup$ – EconJohn Nov 17 '17 at 1:16
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    $\begingroup$ @EconJohn I think you meant to say COV(z2,e)=0 whereas you said COV(z1,e)=0 twice. Also more generally you could allow for Cov(x1, z2)≠0 and Cov(x2, z1)≠0. The point is as long as there are as many instruments that are informative as there are variables that are endogenous (ie the rank condition) $\endgroup$ – Andrew M Nov 17 '17 at 2:03
  • $\begingroup$ @AndrewM you are correct. edited $\endgroup$ – EconJohn Nov 17 '17 at 3:25
  • $\begingroup$ Thanks @EconJohn for your time. what is your comments on "Models with multiple endogenous variables are indeed hard to identify and the results can be hard to interpret. So we don’t usually like to see them – for one thing it’s not clear why you’re tackling two causal questions at the same time; one is hard enough." from mostlyharmlesseconometrics.com/2010/02/… $\endgroup$ – Max Shang Nov 17 '17 at 14:25

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