Consider a gaussian linear model with random regressors

$X_i \sim N(\mu_X,\sigma_X^2), 1 \leq i \leq n$, i.i.d.

$Y_i | X_i \sim N(\beta_0+\beta_1 X_i,\sigma_\varepsilon^2)$

Are either of the ordinary least squares coefficient estimates $\hat\beta_0$ or $\hat\beta_1$ correlated with the regressors $X_i$?

My intuition says no, because the mean of the coefficient estimates depends only the true coefficients. Increasing $X_i$ increases the mean of $Y_i$ such that the two effects somehow cancel in computing the OLS estimates $\hat\beta$.

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    $\begingroup$ Did you perhaps intend the second "$\sigma_X^2$" to be "$\sigma_Y^2$"? Shall we presume the $Y_i$ are conditionally independent? $\endgroup$ – whuber Oct 20 '19 at 17:29
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    $\begingroup$ Thanks @whuber. I have fixed the variance. And yes, I assume the $Y_i$ are conditionally independent. $\endgroup$ – Chad Oct 20 '19 at 20:44

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