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1 Properties of the Least squares Estimators Theorem The OLS estimator B is unbiased. Furthermore, if limT-oo(X'Q2X/T)is finite, B Proof: E(B)=B+E(X'X)-IX'E=B, which proves unbiasedness Also plim B=B+ lim plim- T→∞0 X' has zero mean and covariance matrix If limT-oo(X'QLX/) is finite, then xnX=0. Hence x= has zero mean and its covariance matrix vanishes asymptotically, which implies plim AT=0, and therefore, plim 6=6 Theorem The covariance matrix of B is o(X'X)XnX(XX) Proof: E(B-B)(-B)=E(XX)1x∈∈X(XX)- o(XX-XQX(XX fote that the covariance matrix of B is no longer equal to 2(X'X)-.It may be either"larger"or"smaller", in the sense that(X'X)IX'QX(XX-1 (XX)-can be either positive semidefinite, negative semidefinite, or neither Theorem e'e/(T-k) is(in general)a biased and inconsistent estimator of1 Properties of the Least Squares Estimators Theorem: The OLS estimator βˆ is unbiased. Furthermore, if limT→∞(X0ΩX/T) is finite, βˆ is consistent. Proof: E(βˆ) = β + E(X0X) −1X0ε = β, which proves unbiasedness. Also plim βˆ = β + lim T→∞  X0X T −1 plim X0ε T . But X0ε T has zero mean and covariance matrix σ 2X0ΩX T 2 . If limT→∞(X0ΩX/T) is finite, then σ 2 T X0ΩX T = 0. Hence X0ε T has zero mean and its covariance matrix vanishes asymptotically, which implies plim X0ε T = 0, and therefore, plim βˆ = β. Theorem: The covariance matrix of βˆ is σ 2 (X0X) −1X0ΩX(X0X) −1 . Proof: E(βˆ − β)(βˆ − β) 0 = E(X0X) −1X0 εε0X(X0X) −1 = σ 2 (X0X) −1X0ΩX(X0X) −1 . Note that the covariance matrix of βˆ is no longer equal to σ 2 (X0X) −1 . It may be either ”larger” or ”smaller”, in the sense that (X0X) −1X0ΩX(X0X) −1 − (X0X) −1 can be either positive semidefinite, negative semidefinite, or neither. Theorem: s 2 = e 0e/(T − k) is (in general) a biased and inconsistent estimator of σ 2 . 2
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