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

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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 -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
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3.3 Omitted Variable Bias -From (3.23) we know that: (3.43) ~ ˆ ˆ ~ 2 1 1 -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 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 2 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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