Can you use more than one instrumental variables in an Econometric model?

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.

• 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. – chan1142 Nov 17 '17 at 2:21

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.
• Isn't it also required that $z_1,z_2$ are uncorrelated with $X_{other}$? – Adam Bailey Nov 16 '17 at 22:46
• @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 – EconJohn Nov 17 '17 at 1:16