# Interpretation of correlation in endogenous regression model

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!

• $Cov(x,y) = \beta \sigma_{11} +\sigma_{12}$, $Var(y) = \beta^2\sigma_{11} +2\beta \sigma_{12} +\sigma_{22}$. No more else I can get. – user158565 Nov 21 '18 at 15:47
• – Christoph Hanck Nov 22 '18 at 12:56