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day7lm - 2/24/12 1 PADP 8130: Linear Models Specifca(on...

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Unformatted text preview: 2/24/12 1 PADP 8130: Linear Models Specifca(on Angela Fer¡g, Ph.D. Specifca¡on is about deciding which regressors to put in the regression. • OmiHed variable bias • Including irrelevant variables • Specifca¡on tests – OmiHed Variable Test – Chow Test – RESET 2/24/12 2 What is the consequence of omiSng an important regressor? y = X 1 β 1 + X 2 β 2 + ε y = X 1 β 1 + ε b 1 = (X 1 ' X 1 )-1 X 1 ' y = (X 1 ' X 1 )-1 X 1 ' ( X 1 β 1 + X 2 β 2 + ε ) = β 1 + (X 1 ' X 1 )-1 X 1 ' X 2 β 2 + (X 1 ' X 1 )-1 X 1 ' ε E ( b 1 ) = β 1 + (X 1 ' X 1 )-1 X 1 ' X 2 β 2 True model: Es-mated model: Omi2ed Variable Bias! Direc:on of Bias E ( b 1 ) = β 1 + (X 1 ' X 1 )-1 X 1 ' X 2 β 2 If X 1 ' X 2 > 0, then b 1 will be biased upwards. If X 1 ' X 2 < 0, then b 1 will be biased down. 2/24/12 3 What about variance? • The variance is smaller when you omit an important variable. • So, you’ll es:mate the wrong answer really precisely. Var ( b 1 ) = σ 2 ( X 1 ' X 1 ) − 1 < σ 2 ( X ' X ) − 1 because ( X 1 ' X 1 ) − 1 < ( X ' X ) − 1 True model Es:mated model What is the consequence of including irrelevant variables in the regression? y = X 1 β 1 + ε y = X 1 β 1 + X 2 + ε y = X 1 β 1 + X 2 β 2 + ε y = X β + ε b = (X ' X)-1 X ' y = (X ' X)-1 X ' ( X β + ε ) E ( b ) = β = β 1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ True model: Es-mated model: Unbiased! 2/24/12 4 But, what about variance? • The variance is larger when you include an irrelevant variable. • So, you’ll es:mate the right answer imprecisely. Var ( b ) = σ 2 ( X ' X ) − 1 > σ 2 ( X 1 ' X 1 ) − 1 because ( X 1 ' X 1 ) − 1 < ( X ' X ) − 1 Es:mated model True model Specifca:on Tests OmiHed Variable Test ¡ put in a rich set oF...
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This note was uploaded on 03/28/2012 for the course PADP 8130 taught by Professor Fertig during the Spring '12 term at LSU.

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day7lm - 2/24/12 1 PADP 8130: Linear Models Specifca(on...

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