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I wonder why the covariance between estimates of slope(alaph hat) and intercept(beta hat) is -Xbar*Var(beta hat).

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 alpha hat and beta hat. But as you can see, the final solution I got is something wrong. I can't understand why '-sigma^2/2n' adds to -Xbar*Var(beta hat).

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

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