R 2 from regressing x on z since r 2 1 iv standard

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R 2 from regressing x on z Since R 2 < 1, IV standard errors are larger However, IV is consistent, while OLS is inconsistent, when Cov( x,u ) ≠ 0 The stronger the correlation between z and x , the smaller the IV standard errors
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 8 The Effect of Poor Instruments What if our assumption that Cov( z,u ) = 0 is false? The IV estimator will be inconsistent, too Can compare asymptotic bias in OLS and IV Prefer IV if Corr( z,u )/Corr( z,x ) < Corr( x,u ) x u x u u x Corr x z Corr u z Corr σ σ β β σ σ β β + = + = ) , ( ~ plim : OLS ) , ( ) , ( ˆ plim : IV 1 1 1 1
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 9 IV Estimation in the Multiple Regression Case IV estimation can be extended to the multiple regression case Call the model we are interested in estimating the structural model Our problem is that one or more of the variables are endogenous We need an instrument for each endogenous variable
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 10 Multiple Regression IV (cont) Write the structural model as y 1 = β 0 + β 1 y 2 + β 2 z 1 + u 1 , where y 2 is endogenous and z 1 is exogenous Let z 2 be the instrument, so Cov( z 2 ,u 1 ) = 0 and y 2 = π 0 + π 1 z 1 + π 2 z 2 + v 2 , where π 2 ≠ 0 This reduced form equation regresses the endogenous variable on all exogenous ones
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 [Eg 15.4] Using College Proxy as an IV for Education
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 12 Two Stage Least Squares (2SLS) It’s possible to have multiple instruments Consider our original structural model, and let y 2 = π 0 + π 1 z 1 + π 2 z 2 + π 3 z 3 + v 2 Here we’re assuming that both z 2 and z 3 are valid instruments – they do not appear in the structural model and are uncorrelated with the structural error term, u 1
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 Best Instrument Could use either z 2 or z 3 as an instrument The best instrument is a linear combination of all of the exogenous variables, y 2 * = π 0 + π 1 z 1 + π 2 z 2 +
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