Regression Proof

[jdcozza]

I need to prove that in the basic two-variable regression model Yi= alpha + (beta)Xi that the expected value of alpha(hat) (the least squares intercept estimator term) is alpha (aka that alpha hat is unbiased), and also that the variation of alpha(hat) is rsquared(sigma Xi squared)/(N sigma(Xi-X(bar))squared.

I was able to prove for the expected value and variance of beta(hat) but am having a hard time knowing how to get started on the proofs for alphs.

Thank you very much,
Joe

 

[DrMike]

I would suggest finding the expected value of \(\displaystyle \hat{\alpha}+\hat{\beta}\bar{x}\). I would expect a lot of stuff to cancel out while you are doing the calculation, making the working out easier, and then you can use the fact that \(\displaystyle E(\hat{\alpha}+\hat{\beta}\bar{x})=E(\hat{\alpha})+E(\hat{\beta})\bar{x}\)