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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!

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