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Suppose you have a linear regression with an endogenous regressor $x$ that can be represented as follows:

$x = z'\delta + \epsilon_1$

$y = \beta x + w'\gamma + \epsilon_2$

where

$\begin{pmatrix}\epsilon_1\\ \epsilon_2\\ \end{pmatrix} \sim N(0,\Sigma)$.

As we have assumed that $x$ is endogenous, we know that the off-diagonals of $\Sigma$ will be $\neq 0$. Is there any meaningful interpretation to the covariances of $\epsilon_1$ and $\epsilon_2$?

For instance, a commonly used error covariance matrix specification gives the off-diagonals as $\sigma_{12} = \rho(\sigma_{11}\sigma_{22})^{0.5}$ where $\rho \in [-1,1]$ is a measure of the correlation of the errors.

Does this measure $\rho$ give me any insight other than the strength of the endogeneity? I.e. does the sign of $\rho$ tell me anything about the bias of $\beta_{OLS}$ or similar?

Thanks!

yrx1702
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  • $Cov(x,y) = \beta \sigma_{11} +\sigma_{12}$, $Var(y) = \beta^2\sigma_{11} +2\beta \sigma_{12} +\sigma_{22}$. No more else I can get. – user158565 Nov 21 '18 at 15:47
  • Related: https://stats.stackexchange.com/questions/187271/what-is-the-result-of-violated-exclusion-restrictions/187290#187290 – Christoph Hanck Nov 22 '18 at 12:56

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