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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!

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  • $\begingroup$ 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 $E(u_{i2}^2\vert x_i)$. $\endgroup$ – MauOlivares Apr 18 '18 at 0:18
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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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