Neglecting the inconsistency asyvarls this is the

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Neglecting the inconsistency, Asy.Var[LS]    = ( ' ) (This is the variance of LS around its mean, not  ) Asy.Var[2SLS]   Asy.Var[LS] in the matrix sense. Com σ σ X X X X β -1 -1 2 2 2 Z Z pare inverses: ˆ ˆ { Asy.Var[LS]}  - { Asy.Var[2SLS]} (1 / )[ ' ' ] (1/ )[ ' '( ) ]=(1/ )[ ' ] This matrix is nonnegative definite. (Not positive definite as it might have some rows and columns = σ = σ - σ X X - X X X X - X I M X X M X  which are zero.) Implication for "precision" of 2SLS. The problem of "Weak Instruments"
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Part 13: Endogeneity Estimating σ2 ™  31/61 2 2 n 1 i 1 i n Estimating the asymptotic covariance matrix -  a caution about estimating  . ˆ Since the regression is computed by regressing y on  , one might use  ˆ                        (y ) ˆ This is i = σ σ = Σ - 2sls x x'b 2 n 1 i 1 i n nconsistent.  Use                       (y ) ˆ (Degrees of freedom correction is optional. Conventional, but not necessary.) = σ = Σ - 2sls x'b
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Part 13: Endogeneity Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are EXP  = work experience WKS  = weeks worked OCC  = occupation, 1 if blue collar,  IND  = 1 if manufacturing industry SOUTH  = 1 if resides in south SMSA = 1 if resides in a city (SMSA) MS  = 1 if married FEM  = 1 if female UNION  = 1 if wage set by union contract ED  = years of education BLK = 1 if individual is black LWAGE  = log of wage = dependent variable in regressions These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel  Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied  Econometrics, 3, 1988, pp. 149-155.  See Baltagi, page 122 for further analysis.  The data  were downloaded from the website for Baltagi's text.  ™  32/61
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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 ™  33/61
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Part 13: Endogeneity OLS ™  34/61
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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. ™  35/61
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Part 13: Endogeneity Endogeneity Test? (Hausman) Exogenous Endogenous OLS Consistent, Efficient Inconsistent 2SLS Consistent, Inefficient Consistent Base a test on d = b2SLS - bOLS Use a Wald statistic, d’[Var(d)]-1d What to use for the variance matrix? Hausman: V2SLS - VOLS ™  36/61
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Part 13: Endogeneity Hausman Test ™  37/61
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Part 13: Endogeneity Endogeneity Test: Wu p Considerable complication in Hausman test (text, pp. 322-323) p Simplification: Wu test. p Regress y on X and X^ estimated for the endogenous part of X. Then use an ordinary Wald test.
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