Econ 399 Chapter3d - 3.3 Omitted Variable Bias -When a...

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3.3 Omitted Variable Bias -When a valid variable is excluded, we  UNDERSPECIFY THE MODEL and OLS  estimates are biased -Consider the true population model: (3.40) 2 2 1 1 0 u x x y + + + = β -Assume this satisfies all 4 assumptions and that  we are concerned with x 1 -if we exclude x 2 , our estimation becomes: 1 1 0 ~ ~ x y β+ =
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3.3 Omitted Variable Bias -From (3.23) we know that: (3.43) ~ ˆ ˆ ~ 2 1 1 δ β + = -where Bhats come from regressing y on ALL x’s  and deltatilde comes from regressing x 2  on x 1 -since deltatilde depends on independent  variables, it is considered fixed -we also know from Theorem 3.1 that Bhats are  unbiased estimators, therefore: (3.45) ~ ) ~ ( 2 1 1 + = E
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3.3 Omitted Variable Bias -From this we can calculate Btilde’s bias: (3.46) ~ ˆ ) ~ ( ) ~ ( 2 1 1 1 δ β = - = E Bias -this bias is often called OMITTED VARIABLE  BIAS -From this equation, B 1 tilde is unbiased in two  cases: 1) B 2 =0; x 2  has no impact on y in the true model 2) deltatilde=0
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3.3 Deltatilde=0 -deltatilde is equal to the covariance of x 1  and x 2   over the variance of x 1 , all in the sample -deltatilde is equal to zero only if x 1  and x are  uncorrelated -therefore if they are uncorrelated, B 1 hat is  unbiased -it is also unbiased if we can show that: ) ( ) | ( 2 1 2 x E x x E =
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3.3 Omitted Variable Bias -As B 1 hat’s bias depends on B 2  and deltatilde, the  following table summarizes the possible biases: Corr(x 1 ,x 2 )>0 Corr(x 1 ,x 2 )<0 B 2 hat>0 Positive Bias Negative Bias B 2 hat<0 Negative Bias Positive Bias
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3.3 Omitted Variable Bias
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Econ 399 Chapter3d - 3.3 Omitted Variable Bias -When a...

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