# In linear regression, are the regressors correlated with coefficient estimates?

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 $$\hat\beta$$.

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