Including irrelevant variables in a regression model

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Including Irrelevant Variables in a Regression Model What happens if we include variables in our specification that don’t belong? There is no effect on our parameter estimate, and OLS remains unbiased What if we exclude a variable from our specification that does belong? OLS will usually be biased
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 17 Omitted Variable Bias ( 29 ( 29 - - = + + = + + + = 2 1 1 1 1 1 1 1 0 2 2 1 1 0 ~ then , ~ ~ ~ estimate but we , as given is model true the Suppose x x y x x u x y u x x y i i i β β β β β β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 18 Omitted Variable Bias (cont) ( 29 ( 29 ( 29 ( 29 ( 29 i i i i i i i i i i i i i u x x x x x x x u x x x x u x x y - + - + - = + + + - + + + = 1 1 2 1 1 2 2 1 1 1 2 2 1 1 0 1 1 2 2 1 1 0 becomes numerator the so , that so model, true the Recall β β β β β β β β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 19 Omitted Variable Bias (cont) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - + = = - - + - - + = 2 1 1 2 1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 1 ~ have we ns expectatio taking 0, ) E( since ~ x x x x x E u x x u x x x x x x x i i i i i i i i i i β β β β β β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 20 Omitted Variable Bias (cont) ( 29 ( 29 ( 29 ( 29 1 2 1 1 2 1 1 2 1 1 1 1 1 0 2 1 2 ~ ~ so ~ then ~ ~ ~ on of regression he Consider t δ β β β δ δ δ + = - - = + = E x x x x x x x x x i i i
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 21 Summary of Direction of Bias Corr( x 1 , x 2 ) > 0 Corr( x 1 , x 2 ) < 0 β 2 > 0 Positive bias Negative bias β 2 < 0 Negative bias Positive bias
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 22 Omitted Variable Bias Summary Two cases where bias is equal to zero β 2 = 0, that is x 2 doesn’t really belong in model x 1 and x 2 are uncorrelated in the sample If correlation between x 2 , x 1 and x 2 , y is the same direction, bias will be positive If correlation between x 2 , x 1 and x 2 , y is the opposite direction, bias will be negative
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 23 The More General Case Technically, can only sign the bias for the more general case if all of the included x ’s are uncorrelated Typically, then, we work through the bias assuming the x ’s are uncorrelated, as a useful guide even if this assumption is not strictly true
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 24 3.4Variance of the OLS Estimators Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this
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