Econometrics-I-13

N(2 linearly regress y on these k predictions p this

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Unformatted text preview: n (2) Linearly regress y on these K predictions. p This is two stage least squares ˜˜˜™™™™ ™ 26/61 Part 13: Endogeneity Algebraic Equivalence p Two stage least squares is equivalent to n (1) each variable in X that is also in Z is replaced by itself. n (2) Variables in X that are not in Z are replaced by predictions of that X with all the variables in Z that are not in X . ˜˜˜™™™™ ™ 27/61 Part 13: Endogeneity 2SLS Algebra ˜˜˜˜™™™ ™ 28/61 1 1 ˆ ˆ ˆ ˆ ( ) But, = ( ) and ( ) is idempotent. ˆ ˆ ( )( ) ( ) so ˆ ˆ ( ) = a real IV estimator by the definition. ˆ Note, plim( /n) = -- = = =-1 2SLS-1 Z Z Z Z Z 2SLS X Z(Z'Z) Z'X b X'X X'y Z(Z'Z) Z'X I - M X I - M X'X = X' I - M I - M X = X' I - M X b X'X X'y X' ε-1 ˆ since columns of are linear combinations of the columns of , all of which are uncorrelated with ( ) ] ( ) 2SLS Z Z X Z b X' I - M X X' I - M y ε. =[ Part 13: Endogeneity Asymptotic Covariance Matrix for 2SLS ˜˜˜˜™™™ ™ 29/61 2 1-1 IV IV 2 1-1 2SLS 2SLS General Result for Instrumental Variable Estimation E[( )( ) '| ] ( ) ' ( ) ˆ Specialize for 2SLS, using = ( ) ˆ ˆ ˆ ˆ E[( )( ) '| ] ( ) ' ( ) ---- = σ-- = σ Z b b X, Z Z'X Z Z X'Z Z X = I - M X b b X, Z X'X X X X'X β β β β 2 1-1 2 1 ˆ ˆ ˆ ˆ ˆ ˆ ( ) ' ( ) ˆ ˆ ( )-- = σ = σ X'X X X X'X X'X Part 13: Endogeneity 2SLS Has Larger Variance than LS ˜˜˜˜™™™ ™ 30/61 2-1 2-1 A comparison to OLS ˆ ˆ Asy.Var[2SLS]= ( ' ) 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" 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 Part 13: Endogeneity Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years...
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n(2 Linearly regress y on these K predictions p This is two...

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