<p>Generally, following Verbeek's guide to modern econometrics there are three main attributes a good instrument should have:</p>
<ol>
<li><p>It should be relevant. The instrument should be able to explain your the variable being instrumented and through it also the explanatory variable. For example, in two stages least squares (TSLS) which is a model from the instrumental variable (IV) family the estimator converges to  <span class="math-container">$\beta_2 = \frac{cov(z_i, y_i)}{
cov(z_i, x_i)} $</span> where <span class="math-container">$y$</span> is the dependent variable, <span class="math-container">$x$</span> the independent variable that is endogenous and <span class="math-container">$z$</span> the instrument. If there is no covariance between <span class="math-container">$z_i$</span> and <span class="math-container">$x_i$</span> then the estimator is simply not defined.</p>
</li>
<li><p>Instrument should not be 'weak'. Generally it is not  enough that simply <span class="math-container">$cov(z,x)\neq 0$</span>, instrument also has to be significant and generally the first stage of regression has to be significant itself. This can be tested by looking at <span class="math-container">$F$</span>-statistics. A rule of thumb  for TSLS is that instrument is weak if the first stage <span class="math-container">$F$</span>-statistics is less than <span class="math-container">$10$</span> (see mostly harmless econometrics by Angrist and Pischke).</p>
</li>
<li><p>Instrument should satisfy exclusion assumption. It should have effect on <span class="math-container">$y$</span> only through <span class="math-container">$x$</span> not in other ways. The instrument <span class="math-container">$z$</span> also has to be independent.</p>
</li>
</ol>
<p>Moreover, more on a practical note if you are using a quadratic model the program will ask you also to provide an instrument for both the variable itself and  the quadratic term. As mentioned by Michael in his +1 comment you should check if the above assumptions still hold for the quadratic term as well.</p>

CC BY-SA 4.0
2020-08-13T01:06:53.807Z
"...you can use square of the original instrument for this purpose"---exogeneity of $z$ need not imply exogeneity of $z^2$, i.e. $E[z \epsilon] = 0$ does not imply $E[z^2 \epsilon] = 0$, although it wouldn't be surprising if parts of the empirical literature act as if this were the case, as you say.
Michael
CC BY-SA 4.0
2020-08-13T01:09:20.073Z
@Michael thanks for this insight... indeed parts of the literature act that way
1muflon1
CC BY-SA 4.0
2020-08-12T10:18:31.547Z
2020-08-13T01:12:09.223Z
2020-08-13T01:12:09.223Z
1muflon1
<p>I have a quadratic regression model and one of the variables is endogenous. How do I choose an instrument to deal with this issue?</p>

<econometrics>
How do I choose an appropriate Instrument to solve the endogeneity issue?
Esther Blaese
# How do I choose an appropriate Instrument to solve the endogeneity issue?

(Asked by: username_0 on 2020-08-12)

I have a quadratic regression model and one of the variables is endogenous. How do I choose an instrument to deal with this issue?

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2020-08-12. Score: 2 | Accepted answer

Generally, following Verbeek's guide to modern econometrics there are three main attributes a good instrument should have:

1. It should be relevant. The instrument should be able to explain your the variable being instrumented and through it also the explanatory variable. For example, in two stages least squares (TSLS) which is a model from the instrumental variable (IV) family the estimator converges to $\beta\_2 = \frac{cov(z\_i, y\_i)}{
   cov(z\_i, x\_i)} $ where $y$ is the dependent variable, $x$ the independent variable that is endogenous and $z$ the instrument. If there is no covariance between $z\_i$ and $x\_i$ then the estimator is simply not defined.
2. Instrument should not be 'weak'. Generally it is not enough that simply $cov(z,x)\neq 0$, instrument also has to be significant and generally the first stage of regression has to be significant itself. This can be tested by looking at $F$-statistics. A rule of thumb for TSLS is that instrument is weak if the first stage $F$-statistics is less than $10$ (see mostly harmless econometrics by Angrist and Pischke).
3. Instrument should satisfy exclusion assumption. It should have effect on $y$ only through $x$ not in other ways. The instrument $z$ also has to be independent.

Moreover, more on a practical note if you are using a quadratic model the program will ask you also to provide an instrument for both the variable itself and the quadratic term. As mentioned by Michael in his +1 comment you should check if the above assumptions still hold for the quadratic term as well.

> **Comments:**
> **username_2**: "...you can use square of the original instrument for this purpose"---exogeneity of $z$ need not imply exogeneity of $z^2$, i.e. $E[z \epsilon] = 0$ does not imply $E[z^2 \epsilon] = 0$, although it wouldn't be surprising if parts of the empirical literature act as if this were the case, as you say.  
> **username_1**: @Michael thanks for this insight... indeed parts of the literature act that way  

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