Use proxy variable if not available past exam

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Unformatted text preview: or dependent variable: b2 = = = n ¯ ¯ i =1 (X2,i − X2 )(Yi − Y ) n ¯2 )2 i =1 (X2,i − X n ¯ ¯ ¯ ¯ i =1 (X2,i − X2 )(β2 (X2,i − X2 ) + β3 (X3,i − X3 ) + (ui − u )) n ¯2 )2 i =1 (X2,i − X n n ¯ ¯2 )(X3,i − X3 ) ¯ (X2,i − X2 )(ui − (X2,i − X β2 + β3 i =1 n + i =1 n 2 ¯ ¯2 i =1 (X2,i − X2 ) i =1 (X2,i − X2 ) (3) (4) ¯ u) (5) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences The Bias Result Take expectation: E(b2 ) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) + n ¯ i =1 (X2,i − X2 )(E(ui ) − n ¯2 i =1 (X2,i − X2 ) ¯ E(u )) =0 (6) (7) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences The Bias Result Take expectation: E(b2 ) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) + n ¯ i =1 (X2,i − X2 )(E(ui ) − n ¯2 i =1 (X2,i − X2 ) ¯ E(u )) =0 (6) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) Bias → GENERALLY b2 is biased estimator of β2 . (7) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences The Bias Result Take expectation: E(b2 ) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) + n ¯ i =1 (X2,i − X2 )(E(ui ) − n ¯2 i =1 (X2,i − X2 ) ¯ E(u )) =0 (6) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) Bias → GENERALLY b2 is biased estimator of β2 . Bonus Question When would there not be a bias in b2 even if we omit X3 ? (7) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences The Bias Result Take expectation: E(b2 ) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) + n ¯ i =1 (X2,i − X2 )(E(ui ) − n ¯2 i =1 (X2,i − X2 ) ¯ E(u )) =0 (6) = β2 + β3 n ¯ i =1 (X2,i − X2 )(X3,i − n ¯2 i =1 (X2,i − X2 ) ¯ X3 ) Bias → GENERALLY b2 is biased estimator of β2 . Bonus Question When would there not be a bias in b2 even if we omit X3 ? If X2 and X3 are uncorrelated then omitting X3 does not cause bias in b2 estimator. (pg. 202-203) (7) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences Example for Bias & Intuition Regress books spending on IQ. → But books bought also depends on income. Income is omitted relevant variable! Translation: Y is book spending, X2 is IQ & X3 is income Expect higher income to boost book spending: β3 > 0 Expect higher IQ to boost earnings, so positive correlation between X2 and X3 : n ¯ ¯ i =1 (X2,i − X2 )(X3,i − X3 ) >0 (8) n ¯2 )2 i =1 (X2,i − X b2 will be biased upwards! Just looking at book spending and IQ, we tend to overestimate strength of relation between the two. Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question Consequences OVB Consequences: Tricky Bits Other consequences: Standard errors invalid! Test statistics inva...
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