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I wonder why the covariance between estimates of slope ($\hat{\alpha}$) and intercept ($\hat{\beta}$) is $-\bar{X}\times Var(\hat{\beta})$.

How can I derive this solution by not using matrix?

  • I add a new picture of my solving steps. I tried to use the rule of variance - which is $Var(x+y)=var(x)+var(y)+cov(x,y)$ - to derive covariance of $\hat{\alpha}$ and $\hat{\beta}$. But as you can see, the final solution I got is something wrong. I can't understand why '$-\sigma^2/2n$' adds to $-\bar{X}\times Var(\hat{\beta})$.

Can you plz help me find the stage in which I made mistake...?