03 Finite Sample Properties of OLS Estimators

03 Finite Sample Properties of OLS Estimators - Economics...

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Economics 241B Finite Sample Properties of OLS Estimators We deal in turn with the estimator B and the estimator S 2 . Linear Estimators From B = ( X 0 X ) 1 X 0 Y we see that B is a linear estimator. Unbiased Estimators To verify that the OLS estimators are unbiased, note B = ( X 0 X ) 1 X 0 U: Thus E ( B j X ) = ( X 0 X ) 1 X 0 E ( U j X ) = 0 : Because E ( B j X ) = 0 whenever E ( B j X ) = we have shown that B is conditionally unbiased. Further, by the law of total expectations E [ E ( B j X )] = so that the estimator is (unconditionally) unbiased. Variance of the OLS Estimators of Let A = ( X 0 X ) 1 X 0 , so AA 0 = ( X 0 X ) 1 . Because is not random V ( B j X ) = V ( B j X ) = V ( AU j X ) = AV ( U j X ) A 0 : Yet V ( U j X ) = E ( UU 0 j X ) = ± 2 I n , so V ( B j X ) = ± 2 ( X 0 X ) 1 :
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Gauss-Markov Optimality For the classic regression model, the OLS estimator is the best linear unbiased
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03 Finite Sample Properties of OLS Estimators - Economics...

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