<p>In general, if $c_1$ is endogenous, you need instruments for $c_1$ as well.</p>

<p>Example: Even when $x_1$ is exogenous, if $c_1$ is endogenous and $x_1$ and $c_1$ are correlated, then OLS (which is the IV estimator using $x_1$ as IV for $x_1$) is generally inconsistent.</p>

<p>Exceptions exist. For example, when exactly identified, if $(Ezc') (Ecc')^{-1} (Ecu)=0$, then your IV estimator is consistent, I guess. (Can derive this using standard technique involving the law of large numbers.)</p>

CC BY-SA 3.0
2016-12-12T17:25:38.273Z
2016-12-12T17:25:38.273Z
chan1142
<p>Suppose we want to estimate
\begin{equation} y = \alpha_0 + x_1 \beta + c'\gamma + \varepsilon
\end{equation}
where $\beta$ is the variable of interest and $c$ is a vector of controls. We suspect $E[x\varepsilon]\neq0$ but we have an instrument $z$ such that $E[z\varepsilon]=0.$ My question is - as long as IV is valid - should we care that there could be an endogenous control $c_1$ such that $E[c_1\varepsilon]\neq0$? Why or why not? </p>

<p>My hunch is that it doesn't because as long as IV is valid (it satisfies inclusion-exclusion principle) the correlation between controls and unobservables should not lead to the inconsistency of the estimate of $\beta$.</p>

<econometrics>
General question about IV
certainlyUncertain
# General question about IV

(Asked by: username_0 on 2016-12-12)

Suppose we want to estimate
\begin{equation} y = \alpha\_0 + x\_1 \beta + c'\gamma + \varepsilon
\end{equation}
where $\beta$ is the variable of interest and $c$ is a vector of controls. We suspect $E[x\varepsilon]\neq0$ but we have an instrument $z$ such that $E[z\varepsilon]=0.$ My question is - as long as IV is valid - should we care that there could be an endogenous control $c\_1$ such that $E[c\_1\varepsilon]\neq0$? Why or why not?

My hunch is that it doesn't because as long as IV is valid (it satisfies inclusion-exclusion principle) the correlation between controls and unobservables should not lead to the inconsistency of the estimate of $\beta$.

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2016-12-12. Score: 3 | Accepted answer

In general, if $c\_1$ is endogenous, you need instruments for $c\_1$ as well.

Example: Even when $x\_1$ is exogenous, if $c\_1$ is endogenous and $x\_1$ and $c\_1$ are correlated, then OLS (which is the IV estimator using $x\_1$ as IV for $x\_1$) is generally inconsistent.

Exceptions exist. For example, when exactly identified, if $(Ezc') (Ecc')^{-1} (Ecu)=0$, then your IV estimator is consistent, I guess. (Can derive this using standard technique involving the law of large numbers.)


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