By the law of large numbers plimx xn σ xx a positive

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By the Law of Large numbers, plim(X X/n)= Σ XX , a positive definite matrix of full rank plim(X u/n)=0 plim(b)= β + Σ XX -1 0 = β Fall 2008 under Econometrics Prof. Keunkwan Ryu 8
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 9 Assumption MLR.4 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( x j ,u ) = 0, for j = 1, 2, …, k Without this assumption, OLS will be biased and inconsistent!
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 10 Deriving the Inconsistency Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case ( 29 ( 29 1 2 1 2 1 1 2 2 1 1 0 2 2 1 1 0 , where ~ plim and, that so , : You think : model True x Var x x Cov v x u u x y v x x y = + = + = + + = + + + = δ δ β β β β β β β β β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 Asymptotic Bias (cont) So, thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted variable Main difference is that asymptotic bias uses the population variance and covariance, while bias uses the sample counterparts Remember, inconsistency is a large sample problem – it doesn’t go away as add data
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 12 Large Sample Inference Recall that under the CLM assumptions, the sampling distributions are normal, so we could derive t and F distributions for testing This exact normality was due to assuming the population error distribution was normal This assumption of normal errors implied that the distribution of y , given the x ’s, was normal as well
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