Suppose we have a linear regression model with stochastic $X$ such that
$$ Y = \beta X + \epsilon,$$
and $X$ and $\epsilon$ follow a bivariate normal distribution with $\mu_x = \mu_{\epsilon} = 0$, unknown variances $\sigma_X^2$ and $\sigma_{\epsilon}^2$, and correlation $\rho = Corr(X, \epsilon)$. Also, let $(X_i, Y_i)$ be iid random samples for all $i \in \mathbb{N}$. How does one find the bias of the least squares estimator $\hat{\beta}$?
It seems to me that we no longer have that $\hat{\beta} = (X^TX)^{-1}X^TY$. First, $\rho = \frac{E[X\epsilon]}{\sigma_x\sigma_{\epsilon}}$, and then the needed assumption doesn't hold
$$E[\epsilon|X] \neq 0 ~~\Rightarrow~~ E[\epsilon|X] = \mu_x + \rho\frac{\sigma_x}{\sigma_{\epsilon}}(Y - \mu_{\epsilon}) = \rho Y\frac{\sigma_{x}}{\sigma_{\epsilon}} = \frac{E[X\epsilon]}{\sigma_{\epsilon}^2}Y.$$
So, I guess I would need to find the new derivation of $\hat{\beta}$ first, but am not sure how to proceed?