Beta-hat is an unbiased estimator (OLS estimator is unbiased)

Just a quick post to put the "proof" out there for anyone who is looking for it.

B_hat = [X'X]^-1 X'y

where y = XB + E

B_hat = [X'X]^-1 X'(XB + E)

B_hat = [X'X]^-1 X'XB + [X'X]^-1 X'XBE

The [X'X]^-1 X' measures the linear relationship between X and the value you are multiplying it by. So when you have [X'X]^-1 X'*y it's the linear relationship between X and y. Thus, the [X'X]^-1 X'XB equals (1)B since the linear relationship between X and X will be 1. And [X'X]^-1 X'E is 0 since by assumption it is assumed that the errors are uncorrelated with the explanatory variables.

Therefore, Beta_hat is an unbiased estimator.

 

 

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