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Unformatted text preview: 4/21/2010 1 OVB More Formally  I • Now let’s consider OVB more formally. • Specifically, lets consider the effects of OVB on our estimator β 1 . Recall our equation for β 1 : ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 2 2 1 1 1 ˆ 1 , n n i i i i i i n n i i i i i i i X X Y Y X X Y Y n X X X X n Cov X Y Var X β = = = = = = ≈ ∑ ∑ ∑ ∑ OVB More Formally  II ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 1 1 , , ˆ , , , = , , = , = , = , i i i i i i i i i i i i i i i i i i i i i i i i i i i i i Cov X Y Cov X X u Var X Var X Cov X Cov X X Cov X u Var X Cov X X Cov X u Var X Var X Cov X u Var X Cov X u SD u Corr X u Var X SD X β β β β β β β β β + + ≈ = + + + + + + + = + 4/21/2010 2 OVB More Formally  III • So we have: • What does this equation imply? • 1) If Cov ( X i , u i ) = 0, β 1 u2248 β 1 , i.e. β 1 is an unbiased and consistent estimator of β 1 . (i.e. no OVB) • 2) If Cov ( X i , u i ) > 0, β 1 will generally be greater than β 1 , i.e. β 1 is biased upwards (positive bias). • 3) If Cov ( X i , u i ) < 0, β 1 will generally be less than β 1 , i.e. β 1 is biased downwards (negative bias). ( 29 ( 29 1 1 , ˆ i i i Cov X u Var X β β ≈ + OVB More Formally  IV • So the direction of OVB bias depends directly on the sign of Cov ( X i , u i ). • Often we can come up with a good guess of whether Cov ( X i , u i ) is positive or negative. This can be helpful, as then we know whether our estimate β 1 is biased positively or negatively. • So let’s think about this more. What determines the sign of Cov ( X i , u i )? 4/21/2010 3 OVB More Formally  V • To fix ideas, consider a specific omitted variable V i . Since it is an omitted variable, V i is part of u i . Suppose this relationship is: • Note that in this equation , ω i represents the omitted variables other than V i . Let’s assume that Cov ( X i , ω i ) = 0, i.e. the other omitted variables are not correlated with...
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 Spring '07
 SandraBlack
 Econometrics, vi, Cov

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