<p>The equation-wise OLS estimators are consistent under the further normalization that $E(u_{i1})=0$ and $E(u_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x_i' u_{i1}) \ne 0$ is irrelevant because $x_i$ is not a regressor in the first equation.</p>

CC BY-SA 3.0
2018-04-17T09:26:55.503Z
2018-04-17T09:26:55.503Z
chan1142
<p>Consider the following regression model:</p>
<blockquote>
<p><span class="math-container">$y_{i1}=\beta_1 +u_{i1}$</span></p>
<p><span class="math-container">$y_{i2}=\beta_{21}+\beta_{22}x_i+u_{i2}$</span>.</p>
</blockquote>
<p>If <span class="math-container">$E(x_i' u_{i1})\neq 0$</span> and <span class="math-container">$E(x_i' u_{i2})=0$</span>, will we get consistent estimators for <span class="math-container">$\beta_1$</span> and <span class="math-container">$\beta_{21}$</span> using GLS?</p>
<p>I think we will not get consistent estimators for both, because the assumption <span class="math-container">$E(x_i\otimes u_i)=0$</span> is violated. Please correct me if I am wrong. Thanks in advance!</p>

CC BY-SA 3.0
2018-04-18T00:18:22.337Z
Your question needs a lot of polishing. First of all, can you state the statistical relationship between $u_{i1}$ and $u_{i2}$? Are they independent, mean independent, orthogonal? Secondly, in order to estimate the parameter of interest using GLS, we need further information about the (conditional) second moment $E(u_{i2}^2\vert x_i)$.
MauOlivares
<econometrics><regression>
Constant Regressor in GLS
Chunjing GU
# Constant Regressor in GLS

(Asked by: username_0 on 2018-04-16)

Consider the following regression model:

> $y\_{i1}=\beta\_1 +u\_{i1}$
>
> $y\_{i2}=\beta\_{21}+\beta\_{22}x\_i+u\_{i2}$.

If $E(x\_i' u\_{i1})\neq 0$ and $E(x\_i' u\_{i2})=0$, will we get consistent estimators for $\beta\_1$ and $\beta\_{21}$ using GLS?

I think we will not get consistent estimators for both, because the assumption $E(x\_i\otimes u\_i)=0$ is violated. Please correct me if I am wrong. Thanks in advance!

**Tags:** econometrics, regression
> **Comments:**
> **username_1**: Your question needs a lot of polishing. First of all, can you state the statistical relationship between $u_{i1}$ and $u_{i2}$? Are they independent, mean independent, orthogonal? Secondly, in order to estimate the parameter of interest using GLS, we need further information about the (conditional) second moment $E(u_{i2}^2\vert x_i)$.  

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## 1 answers

### Answer by username_2 on 2018-04-17. Score: 0 | Accepted answer

The equation-wise OLS estimators are consistent under the further normalization that $E(u\_{i1})=0$ and $E(u\_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x\_i' u\_{i1}) \ne 0$ is irrelevant because $x\_i$ is not a regressor in the first equation.


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