Multiple regression analysis y-Bo+Bx+Bx2+... Bkk+u 23 Asymptotic Properties Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 3. Asymptotic Properties
Consistency o Under the Gauss-Markov assumptionS OLS IS BLUE, but in other cases it wont al ways be possible to find unbiased estimators In those cases, we may settle for estimators that are consistent, meaning as n>o0, the distribution of the estimator collapses to the parameter value Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Consistency Under the Gauss-Markov assumptions OLS is BLUE, but in other cases it won’t always be possible to find unbiased estimators In those cases, we may settle for estimators that are consistent, meaning as n → ∞, the distribution of the estimator collapses to the parameter value
Sampling Distributions as n t ni <n<n 2 Bi Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 Sampling Distributions as n b1 n1 n2 n3 n1 < n2 < n3
Consistency of ols o Under the Gauss-Markov assumptions, the OLS estimator is consistent(and unbiased) o Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness o Will need to take probability limit(plim)to establish consistency Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 Consistency of OLS Under the Gauss-Markov assumptions, the OLS estimator is consistent (and unbiased) Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness Will need to take probability limit (plim) to establish consistency
Proving Consistency B=(x1-元/∑(x1-元 =B+n∑(x1-元x((-) plim B,=B,+Cov(x, u / var(x=B because Cov(xi, u)=0 Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Proving Consistency ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ) ( ) because ( , ) 0 , ˆ plim ˆ 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 = = + = = + − − = − − − − Cov x u Cov x u Var x n x x u n x x x x y x x i i i i i i b b b b b