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Econ 104 PS8 Solutions(1)

Econ 104 PS8 Solutions(1) - Econ 104 PS8 Solutions Molin...

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Econ 104 PS8 Solutions Molin Zhong December 17, 2011 1 Problem 1 1.1 Part A The key conditions for the validity of the instrumental variable is that: 1. cov ( z, x ) 6 = 0 2. cov ( z, v ) = 0 where v is the error term We know z is correlated with x. Here, v = β 2 w + u . Because z and w are uncorrelated, z and v are uncorre- lated because z and u are uncorrelated by assumption. Hence, z is still a valid instrument and the 2SLS procedure goes through. 1.2 Part B If z and w are correlated, z is no longer uncorrelated with the error term v so long as β 2 6 = 0. Hence, it does not fulfill the exogeneity condition and z cannot be used as an instrument. Therefore, the 2SLS estimator is no longer consistent. This is because in the second step, the new regressor is a function of z, which is correlated with the error term, so the OLS step will lead to an inconsistent estimate. 2 Problem 2: 12.7 The test of overidentifying restrictions regresses b u T SLS i = δ 0 + δ 1 z 1 i + δ 2 z 2 i + i where b u T SLS i is the residuals from the two stage least squares regression. The J test statistic tests if these δ 1 and δ 2 are jointly equal to 0, which should be the case if the exogeneity condition holds for both of the instrumental variables.
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