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

Question

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, Commented Nov 17, 2017 at 2:21

EconJohn · Accepted Answer · 2017-11-17 03:24:45Z

4

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.

answered Nov 16, 2017 at 18:56

EconJohn♦

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$\begingroup$ Isn't it also required that $z_1,z_2$ are uncorrelated with $X_{other}$? $\endgroup$
– Adam Bailey
Commented Nov 16, 2017 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 ♦
Commented Nov 17, 2017 at 1:16
2

$\begingroup$ @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
Commented Nov 17, 2017 at 2:03
$\begingroup$ @AndrewM you are correct. edited $\endgroup$
– EconJohn ♦
Commented Nov 17, 2017 at 3:25
1

$\begingroup$ 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
Commented Nov 17, 2017 at 14:25

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