<p>I'm going to assume that by &quot;conditions for instrumental variables to work&quot; you mean &quot;instrumental variables is consistent.&quot; However, there are other properties to consider, like performance in small samples, etc.
<span class="math-container">$
\newcommand{\Cov}{\text{Cov}}
$</span></p>
<p>In this simple case, the IV estimator in sample of size <span class="math-container">$n$</span> is
<span class="math-container">$$
\hat \gamma_2 = \frac{\sum_{i=1}^n (Z_i - \bar Z)(Y_i - \bar Y)}{\sum_{i=1}^n (Z_i - \bar Z)(D_i - \bar D)}.
$$</span>
For the instrumental variables estimator to be consistent, we only need
<span class="math-container">$$
\text{Cov}(Z, \epsilon_2) = 0 \quad \text{ (instrument exogeneity)} \tag{1}
$$</span>
and
<span class="math-container">$$
\text{Cov}(Z, D) \neq 0. \quad \text{ (instrument relevance)} \tag{2}
$$</span>
As long as these conditions are satisfied, it doesn't matter if <span class="math-container">$Z$</span> is correlated with <span class="math-container">$\epsilon_1$</span>. The IV estimator will be consistent. To see this, let's analyze the model in terms of populations,
<span class="math-container">\begin{align}
\Cov(Z,Y) &amp;= \gamma_2 \Cov(Z, D) + \Cov(Z, \epsilon_2),
\end{align}</span>
and solve
<span class="math-container">$$
\gamma_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)} - \frac{\Cov(Z, \epsilon_2)}{\Cov(Z, D)}.
$$</span>
If conditions (1) and (2) hold, then <span class="math-container">$\frac{\Cov(Z, \epsilon_2)}{\Cov(Z, D)} = 0$</span> and
the IV estimate<br />
<span class="math-container">$$
\gamma_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)}
$$</span>
is well-defined. The estimate is consistent since the probability limit is the population analog,
<span class="math-container">$$
\hat \gamma_2 \overset{p}{\rightarrow} \frac{\Cov(Z, Y)}{\Cov(Z, D)} = \gamma_2.
$$</span></p>
<p>Note that weak instruments become a problem as soon as we relax the assumption that
<span class="math-container">$\text{Cov}(Z, \epsilon_2) = 0$</span> exactly, as you might want to do when analyzing the properties of the estimator in small samples.</p>
<p>So, are there any consequences if <span class="math-container">$Z$</span> is correlated with <span class="math-container">$\epsilon_1$</span>? Well, it means that a regression of <span class="math-container">$D$</span> onto <span class="math-container">$Z$</span> will give a biased and inconsistent estimate of <span class="math-container">$\gamma_1$</span>. However, the usual assumption underlying IV is that you don't really care about estimating <span class="math-container">$\gamma_1$</span>. You want to consistently estimate <span class="math-container">$\gamma_2$</span>, which you can do as long as assumptions (1) and (2) hold, as shown above.</p>

CC BY-SA 4.0
2021-03-27T21:14:13.420Z
2021-04-01T02:06:20.570Z
2021-04-01T02:06:20.570Z
jmbejara
<p>Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and <span class="math-container">$cov(Z,\epsilon_1)\neq 0$</span> would violate them (see A Guide to Modern Econometrics by Verbeek).</p>
<p>Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have <span class="math-container">$F$</span>-statistics above <span class="math-container">$10$</span> (as a rule of thumb). Also the instrument should have an effect on <span class="math-container">$Y$</span> only through <span class="math-container">$D$</span> - the so called exclusion restriction (see Mostly Harmless Econometrics by Angrist and Pischke).</p>

CC BY-SA 4.0
2020-09-27T22:35:49.243Z
relevance is equal to weak instrument. relevance mean the  the correlation between Z and D must be significantly larger than 0. exogeneity condition is similar to exclusion restriction. they are just different names for the same thing
FARRAF
CC BY-SA 4.0
2020-09-27T22:36:48.233Z
@FARRAF no it’s not equal. Even with non-zero correlation, and individual betas being significant using t test, the F-statistics can be <10
1muflon1
CC BY-SA 4.0
2020-09-27T22:12:09.527Z
2020-09-27T23:40:13.840Z
2020-09-27T23:40:13.840Z
1muflon1
<p>We have this model:</p>
<p><span class="math-container">$$D=\alpha_1+\gamma_1 Z + \epsilon_1$$</span></p>
<p><span class="math-container">$$Y=\alpha_2+\gamma_2 D + \epsilon_2$$</span></p>
<p>Notations are as usual: Y is the outcome, D is the treatment, Z is the instrument</p>
<p>Two conditions for instrument variable to work in the literature: relevance (there is a correlation between Z and D) and exogeneity (no correlation between Z and error term <span class="math-container">$\epsilon_2$</span>)</p>
<p>However, it seems no one cares about endogeneity in the first stage regression, which is the correlation between Z and <span class="math-container">$\epsilon_1$</span>. Why is that the case? I have never seen any applied micro paper addressing this.</p>

<instrumental-variable><endogeneity>
is endogeneity a problem in the first-stage regression in a two-stage least square regression?
FARRAF
# is endogeneity a problem in the first-stage regression in a two-stage least square regression?

(Asked by: username_0 on 2020-09-27)

We have this model:

$$D=\alpha\_1+\gamma\_1 Z + \epsilon\_1$$

$$Y=\alpha\_2+\gamma\_2 D + \epsilon\_2$$

Notations are as usual: Y is the outcome, D is the treatment, Z is the instrument

Two conditions for instrument variable to work in the literature: relevance (there is a correlation between Z and D) and exogeneity (no correlation between Z and error term $\epsilon\_2$)

However, it seems no one cares about endogeneity in the first stage regression, which is the correlation between Z and $\epsilon\_1$. Why is that the case? I have never seen any applied micro paper addressing this.

**Tags:** instrumental-variable, endogeneity

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## 2 answers

### Answer by username_1 on 2021-03-27. Score: 1

I'm going to assume that by "conditions for instrumental variables to work" you mean "instrumental variables is consistent." However, there are other properties to consider, like performance in small samples, etc.
$
\newcommand{\Cov}{\text{Cov}}
$

In this simple case, the IV estimator in sample of size $n$ is
$$
\hat \gamma\_2 = \frac{\sum\_{i=1}^n (Z\_i - \bar Z)(Y\_i - \bar Y)}{\sum\_{i=1}^n (Z\_i - \bar Z)(D\_i - \bar D)}.
$$
For the instrumental variables estimator to be consistent, we only need
$$
\text{Cov}(Z, \epsilon\_2) = 0 \quad \text{ (instrument exogeneity)} \tag{1}
$$
and
$$
\text{Cov}(Z, D) \neq 0. \quad \text{ (instrument relevance)} \tag{2}
$$
As long as these conditions are satisfied, it doesn't matter if $Z$ is correlated with $\epsilon\_1$. The IV estimator will be consistent. To see this, let's analyze the model in terms of populations,
\begin{align}
\Cov(Z,Y) &= \gamma\_2 \Cov(Z, D) + \Cov(Z, \epsilon\_2),
\end{align}
and solve
$$
\gamma\_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)} - \frac{\Cov(Z, \epsilon\_2)}{\Cov(Z, D)}.
$$
If conditions (1) and (2) hold, then $\frac{\Cov(Z, \epsilon\_2)}{\Cov(Z, D)} = 0$ and
the IV estimate  
$$
\gamma\_2 = \frac{\Cov(Z, Y)}{\Cov(Z, D)}
$$
is well-defined. The estimate is consistent since the probability limit is the population analog,
$$
\hat \gamma\_2 \overset{p}{\rightarrow} \frac{\Cov(Z, Y)}{\Cov(Z, D)} = \gamma\_2.
$$

Note that weak instruments become a problem as soon as we relax the assumption that
$\text{Cov}(Z, \epsilon\_2) = 0$ exactly, as you might want to do when analyzing the properties of the estimator in small samples.

So, are there any consequences if $Z$ is correlated with $\epsilon\_1$? Well, it means that a regression of $D$ onto $Z$ will give a biased and inconsistent estimate of $\gamma\_1$. However, the usual assumption underlying IV is that you don't really care about estimating $\gamma\_1$. You want to consistently estimate $\gamma\_2$, which you can do as long as assumptions (1) and (2) hold, as shown above.


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### Answer by username_2 on 2020-09-27. Score: 1

Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and $cov(Z,\epsilon\_1)\neq 0$ would violate them (see A Guide to Modern Econometrics by Verbeek).

Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have $F$-statistics above $10$ (as a rule of thumb). Also the instrument should have an effect on $Y$ only through $D$ - the so called exclusion restriction (see Mostly Harmless Econometrics by Angrist and Pischke).

> **Comments:**
> **username_0**: relevance is equal to weak instrument. relevance mean the  the correlation between Z and D must be significantly larger than 0. exogeneity condition is similar to exclusion restriction. they are just different names for the same thing  
> **username_2**: @FARRAF no it’s not equal. Even with non-zero correlation, and individual betas being significant using t test, the F-statistics can be <10  

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