Π 3 z 3 we can estimate y 2 by regressing y 2 on z 1

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π 3 z 3 We can estimate y 2 * by regressing y 2 on z 1 , z 2 and z 3 – can call this the first stage If then substitute ŷ 2 for y 2 in the structural model, get same coefficient as IV
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 More on 2SLS While the coefficients are the same, the standard errors from doing 2SLS by hand are incorrect, so let Stata do it for you Method extends to multiple endogenous variables – need to be sure that we have at least as many excluded exogenous variables (instruments) as there are endogenous variables in the structural equation
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 15 Addressing Errors-in-Variables with IV Estimation Remember the classical errors-in-variables problem where we observe x 1 instead of x 1 * Where x 1 = x 1 * + e 1 , and e 1 is uncorrelated with x 1 * and x 2 If there is a z , such that Corr( z,u ) = 0 and Corr( z,x 1 ) ≠ 0, then IV will remove the attenuation bias
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 16 Testing for Endogeneity Since OLS is preferred to IV if we do not have an endogeneity problem, then we’d like to be able to test for endogeneity If we do not have endogeneity, both OLS and IV are consistent Idea of Hausman test is to see if the estimates from OLS and IV are different
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 17 Testing for Endogeneity (cont) While it’s a good idea to see if IV and OLS have different implications, it’s easier to use a regression test for endogeneity If y 2 is endogenous, then v 2 (from the reduced form equation) and u 1 from the structural model will be correlated The test is based on this observation
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 18 Testing for Endogeneity (cont) Save the residuals from the first stage Include the residual in the structural equation (which of course has y 2 in it) If the coefficient on the residual is statistically different from zero, reject the null of exogeneity If multiple endogenous variables, jointly test the residuals from each first stage
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