How could I demonstrate

#1
Supposing that we have a regression model 1572248456373.png that fits the conditions of normal, homoscedasticity and independent residuals, I would like to demonstrate that the variance of the estimators is:

1572248391830.png

Where 1572248409903.png is the Coefficient of Determination and 1572248423579.png

How should I start this?
 
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staassis

Active Member
#4
One approach is deriving the variance of Beta = (Beta_0, Beta_1, ..., Beta_p) in the matrix form, as

sigma^2 * (X' * X)^{-1}, (1)

where X = (1, x) is an N-by-(p+1) matrix. Then you would rewrite formula (1) for the case of simple linear regression (p = 1).
 

staassis

Active Member
#6
I know it is not obvious to you. That is why you have to figure it out. I intentionally did not expose the complete proof, giving you hints instead. Or were you expecting to have somebody do the whole exercise for you?.. You cannot move smoothly between statistical topics without knowing linear algebra. Use linear algebra to prove the result I stated in the previous post.
 
#7
I didn't expect anybody doing the exercise for me. I just want to learn by getting some leash
It would be really helpful if you could detail the previous steps until you get (1)