# Thread: Show that s^2 is an unbiased estimator for sigma^2

1. ## Show that s^2 is an unbiased estimator for sigma^2

I am trying to complete a proof showing that s^2 = sum(Yi - Yi hat)^2/(n-2) is unbiased for sigma^2

In other words, show that E[s^2] = sigma^2

I have...

E[s^2] = E [sum(yi - yi hat)^2/(n-2)]

which is equivalent to...

E[s^2(n-2)] = E[sum(yi - yi hat)^2]

From here,

E[sum(yi - yi hat)^2] = Var(Yi - Yi hat) + [E(Yi - Yi hat)]^2

So now I have to show that Var(Yi - Yi hat) + [E(Yi - Yi hat)]^2 = sigma^2(n-2), but I keep getting stuck at this point, especially in terms of taking the variance...should I make a substitution for yi hat? Anything I try doesn't seem to get me where I want!

If anyone can provide me with some direction that would be great, thanks

2. Originally Posted by statgirl11

If anyone can provide me with some direction that would be great, thanks

Someone else asked this question a few days ago. See my reply here:

http://www.talkstats.com/showthread.php?t=9685

3. I looked at your solution, however you have...

Subtracting (2) from (1) gives
(3) (Y_i – Ybar) = Beta1(X_i – Xbar) + (u_i – ubar), and go from there...

however, there is a difference in that you are using Y_i - Ybar, rather than Y_1 - Yhat, which is what I am trying to do...

4. Originally Posted by statgirl11
I looked at your solution, however you have...

however, there is a difference in that you are using Y_i - Ybar, rather than Y_1 - Yhat, which is what I am trying to do...
I think you mean: Y_i - Yhat_i (?).

This is what e_i is in (4) and (5)...i.e. e_i = Y_i - Yhat_i

5. Yes that is what I meant, and I see what you mean now, thank you!

I'm going to try and work through your other post and see if I can figure it out from there.

Thanks a lot

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