Covariance between estimates of slope and intercept

Asked Apr 16 '19 at 10:53

Active Sep 06 '20 at 16:44

Viewed 4,467 times

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...?

edited Sep 06 '20 at 16:44

develarist

asked Apr 16 '19 at 10:53

Dayeon

2

First show that $\mbox{Cov}(\bar y,\hat\beta)=0$. – Jarle Tufto Apr 16 '19 at 10:56
1

See: https://stats.stackexchange.com/questions/127316/clarification-the-covariance-of-intercept-and-slope-in-simple-linear-regression – kjetil b halvorsen Apr 16 '19 at 11:37
This question has been in various forms on this site. Like https://stats.stackexchange.com/questions/68151/how-to-derive-variance-covariance-matrix-of-coefficients-in-linear-regression?noredirect=1&lq=1, https://stats.stackexchange.com/questions/171125/correlation-between-ols-estimators-for-intercept-and-slope. – StubbornAtom Apr 16 '19 at 12:15
I added a new picture.. plz help me:(:(:( – Dayeon Apr 16 '19 at 12:33
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Please explain us why none of the links can help you? – kjetil b halvorsen Apr 16 '19 at 12:52
Please type your question as text, do not just post a photograph or screenshot (see here). – gung - Reinstate Monica Sep 06 '20 at 16:04
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Found a more elegant way of proving the same.Hope it helps. – Paritosh Sep 06 '20 at 12:42
Please type your answer as text, do not just post a photograph or screenshot (see here). – gung - Reinstate Monica Sep 06 '20 at 16:00

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