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103lect8hand - OVB More Formally - I Now let's consider OVB...

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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 : () 11 1 22 1 ˆ 1 , nn ii i XXYY n XX n Cov X Y Var X β == −− ∑∑ OVB More Formally - II ( ) ( ) ()( )() ) ( ) 01 1 1 1 ,, ˆ , = 0, , = = , = , i i i i i i i i i i i i i Cov X Y Cov X X u Var X Var X C o vX C o X C o vXu 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 ββ ++ ≈= += + OVB More Formally - III • So we have: • What does this equation say? • 1) If Cov ( X i , u i ) = 0, β 1 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). ( ) , ˆ i Cov X u Var X ≈+
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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 )? 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 X i . ii i uV δ ω =+ OVB More Formally - VI • From the above equation, the sign of δ measures whether the omitted variable V i positively or negatively affects u i .
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103lect8hand - OVB More Formally - I Now let's consider OVB...

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