Unbiasedness of the Variance of the Error Term in an intercept-only Model

#1
Hi folks,

Is there anyone that might help me with the following derivation to show that the estimated variance error term is unbiased? I have managed to do the first and the second step, however I am currently stuck at the third step since I don't know how to replace Yi by µ^.

The model itself is an intercept-only model: Yi = µ + ui

Thank you so much,
Naomi
 

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Dason

Ambassador to the humans
#2
Keep in mind that µ^ (in this case) is just the mean of the Yi terms.

Is that enough of a hint? If not please slow what work you've done so far.
 
#3
Keep in mind that µ^ (in this case) is just the mean of the Yi terms.

Is that enough of a hint? If not please slow what work you've done so far.
Hi Dason,

first of all, thank you so much for your reply and your hint. I'm sorry that I have only managed to answer now.

In step (i.), I have just inserted the expression for µ^ (i.e. 1/n Σ Yi) and then, I have solved the expectation sign to get 1/n Σ E(Yi). Then, I have inserted the expression for Yi (i.e., µ + ui) and solved this expression to get 1/n*n*µ which is µ.

In step (ii.), I have solved the equation backwards which meant that I have written the right hand side as a binomial formula, so that I could see how, -µ got into the equation.

In step (iii.), I have now tried to insert the expression 1/n Σ Yi for µ^. My problem is that I don't know how to get rid of the 1/n and of the Σ to get to the second expression in that equation and to then, get to the final expression (i.e., variance of u divided by n).

Do you think you may help me once again? :)

In any case, thanks a lot and stay safe!