multreg3

# multreg3 - Economics 20 Prof Anderson 1 Multiple Regression...

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Unformatted text preview: Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = β + β 1 x 1 + β 2 x 2 + . . . β k x k + u 3. Asymptotic Properties 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 Economics 20 - Prof. Anderson 3 Sampling Distributions as n ↑ β 1 n 1 n 2 n 3 n 1 < n 2 < n 3 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 Economics 20 - Prof. Anderson 5 Proving Consistency ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , because , ˆ plim ˆ 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 = = + =-- + =-- = ∑ ∑ ∑ ∑-- u x Cov x Var u x Cov x x n u x x n x x y x x i i i i i i β β β β β Economics 20 - Prof. Anderson 6 A Weaker Assumption For unbiasedness, we assumed a zero conditional mean – E( u|x 1 , x 2 ,…,x k ) = 0 For consistency, we can have the weaker assumption of zero mean and zero correlation – E( u ) = 0 and Cov(...
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multreg3 - Economics 20 Prof Anderson 1 Multiple Regression...

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