<p>Wooldridge's wording is possibly confusing, but I believe it is correct.</p>
<p>Wooldridge is saying that if <span class="math-container">$E(v_i|x_i)=0$</span> then <span class="math-container">$E(x_i\otimes v_i)=0$</span>.</p>
<p>Wooldridge is not saying that if <span class="math-container">$E(x_i\otimes v_i)=0$</span> then <span class="math-container">$E(v_i|x_i)=0$</span>.</p>

CC BY-SA 4.0
2022-07-08T06:33:10.010Z
2022-07-08T06:33:10.010Z
Michael Gmeiner
<p>Consider the following regression model using panel data: <span class="math-container">$$y_{it}=x_{it} + u_{it}$$</span></p>
<p>According to Wooldridge (2010), Chapter 7.4, the first assumption required for Generalized Least Squares (GLS) in a panel data setting can be stated as:</p>
<p>Assumption SGLS.1:    <span class="math-container">$$E(X_i \otimes u_i) = 0    $$</span>
Where <span class="math-container">$X_i$</span> is a matrix containing all <span class="math-container">$T$</span> observations of the dependent variables for individual <span class="math-container">$i$</span> and and <span class="math-container">$u_i$</span> is a vector containing all <span class="math-container">$T$</span> idiosyncratic error terms for individual <span class="math-container">$i$</span>.</p>
<p>I am confused as to whether this assumption implies strict exogenetiy or not. As far as I understand, given what the kronecker product means, SGLS.1 only implies uncorrelatedness of x with all elements of u, which does not necesarily imply zero conditional mean on all elements of u, which is needed for strict exogeneity. However, in a later section (Chapter 10.3), Wooldridge introduces the assumptions for Random Effects (RE) estimation by stating the assumption
<span class="math-container">$$E(v_i | x_i)=0  , $$</span> where <span class="math-container">$v$</span> is the composite error <span class="math-container">$v_i = c_i + u_i$</span> of a model with unobserved heterogeneity <span class="math-container">$c_i$</span>:    <span class="math-container">$$y_{it}=x_{it} \beta + v_{it}$$</span> He refers to this assumption as satisfying <em>&quot;the exogeneity assumption SGLS.1 (see Chapter 7)&quot;</em>.</p>
<p>How comes he can make this reference if SGLS.1 does not necessarily imply strict exogeneity?
I would appreciate some orientation here. Thanks in advance!</p>

CC BY-SA 4.0
2022-07-08T05:00:20.513Z
Welcome. It's just a matter of how your "strict exogeneity" is defined. If it's defined in terms of covariances, $E(X_i\otimes u_i)=0$ is strict exogeneity; if it's defined in terms of conditional means, not. SGLS.1 is written in terms of covariances. Zero conditional mean implies zero covariance, not vice versa.
chan1142
<econometrics><paneldata>
Is the exogeneity condition required for GLS with panel data a form of strict exogeneity?
meryasks
# Is the exogeneity condition required for GLS with panel data a form of strict exogeneity?

(Asked by: username_0 on 2022-07-07)

Consider the following regression model using panel data: $$y\_{it}=x\_{it} + u\_{it}$$

According to Wooldridge (2010), Chapter 7.4, the first assumption required for Generalized Least Squares (GLS) in a panel data setting can be stated as:

Assumption SGLS.1: $$E(X\_i \otimes u\_i) = 0 $$
Where $X\_i$ is a matrix containing all $T$ observations of the dependent variables for individual $i$ and and $u\_i$ is a vector containing all $T$ idiosyncratic error terms for individual $i$.

I am confused as to whether this assumption implies strict exogenetiy or not. As far as I understand, given what the kronecker product means, SGLS.1 only implies uncorrelatedness of x with all elements of u, which does not necesarily imply zero conditional mean on all elements of u, which is needed for strict exogeneity. However, in a later section (Chapter 10.3), Wooldridge introduces the assumptions for Random Effects (RE) estimation by stating the assumption
$$E(v\_i | x\_i)=0 , $$ where $v$ is the composite error $v\_i = c\_i + u\_i$ of a model with unobserved heterogeneity $c\_i$: $$y\_{it}=x\_{it} \beta + v\_{it}$$ He refers to this assumption as satisfying *"the exogeneity assumption SGLS.1 (see Chapter 7)"*.

How comes he can make this reference if SGLS.1 does not necessarily imply strict exogeneity?
I would appreciate some orientation here. Thanks in advance!

**Tags:** econometrics, paneldata

> **Comments:**
> **username_1**: Welcome. It's just a matter of how your "strict exogeneity" is defined. If it's defined in terms of covariances, $E(X_i\otimes u_i)=0$ is strict exogeneity; if it's defined in terms of conditional means, not. SGLS.1 is written in terms of covariances. Zero conditional mean implies zero covariance, not vice versa.  

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## 1 answers

### Answer by username_2 on 2022-07-08. Score: 2

Wooldridge's wording is possibly confusing, but I believe it is correct.

Wooldridge is saying that if $E(v\_i|x\_i)=0$ then $E(x\_i\otimes v\_i)=0$.

Wooldridge is not saying that if $E(x\_i\otimes v\_i)=0$ then $E(v\_i|x\_i)=0$.


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