Econometrics-I-13

# N 2 regress lwage on this prediction of ed and

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n (2) Regress LWAGE on this prediction of ED and everything else. n Standard errors must be adjusted for the predicted ED ™  8/61

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Part 13: Endogeneity OLS ™  9/61
Part 13: Endogeneity The weird results for the coefficient on ED happened because the instruments, MS,FEM,BLK are all dummy variables. There is not enough variation in these variables. ™  10/61

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Part 13: Endogeneity Source of Endogeneity n LWAGE = f( ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ) + n ED = f( MS,FEM,BLK, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ) + u ™  11/61
Part 13: Endogeneity Remove the Endogeneity n LWAGE = f( ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ) + u + n Strategy p Estimate u p Add u to the equation. ED is uncorrelated with  when u is in the equation. ™  12/61

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Part 13: Endogeneity Auxiliary Regression for ED to Obtain Residuals ™  13/61
Part 13: Endogeneity OLS with Residual (Control Function) Added 2SLS ™  14/61

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Part 13: Endogeneity A Warning About Control Function ™  15/61
Part 13: Endogeneity The Problem ™  16/61 1 2 Cov( , ) , K variables Cov( , ) , K variables is OLS regression of y on ( , ) cannot estimate ( , ) consistently. Some other estimator is needed. Additional structure: = + wh = + + = endogenous y X Y X 0 Y 0 Y X Y Y Z V β δ ε ε ε β δ Π ere Cov( , )= . An estimator based on ( , ) may be able to estimate ( , ) consistently. instrumental variable ( Z 0 X IV) Z ε β δ

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Part 13: Endogeneity Instrumental Variables p Framework: y = X + , K variables in X . p There exists a set of K variables, Z such that plim( Z’X/n )  0 but plim( Z’ /n) = 0 The variables in Z are called instrumental variables. p An alternative (to least squares) estimator of is b IV = ( Z’X )-1 Z’y p We consider the following: n Why use this estimator? n What are its properties compared to least squares? p We will also examine an important application ™  17/61
Part 13: Endogeneity IV Estimators Consistent b IV = ( Z’X )-1 Z’y = ( Z’X /n)-1 ( Z’X /n) β + ( Z’X /n)-1 Z’ε /n = β + ( Z’X /n)-1 Z’ε /n  β Asymptotically normal (same approach to proof as for OLS) Inefficient – to be shown. ™  18/61

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Part 13: Endogeneity The General Result By construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not. ™  19/61
Part 13: Endogeneity LS as an IV Estimator The least squares estimator is ( X X )-1 Xy = ( X X )-1i x iyi = + ( X X )-1i x iεi If plim( X’X /n) = Q nonzero plim( X’ε /n) = 0 Under the usual assumptions LS is an IV estimator X is its own instrument. ™  20/61

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Part 13: Endogeneity IV Estimation Why use an IV estimator ? Suppose that X and are not uncorrelated. Then least squares is neither unbiased nor consistent. Recall the proof of consistency of least squares: b = + ( X’X /n)-1( X’ /n). Plim b = requires plim( X’ /n) = 0. If this does not hold, the estimator is inconsistent. ™  21/61
Part 13: Endogeneity A Popular Misconception A popular misconception. If only one variable in X is correlated with , the other coefficients are consistently estimated. False.

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