Econometrics-I-13

# The problem is smeared over the other coefficients

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The problem is “smeared” over the other coefficients. ™  22/61 1 11 1 21 -1 1 1 1 Suppose only the first variable is correlated with  0 Under the assumptions, plim( /n) =   . Then ... . 0 plim   =  plim( /n) ... ... .    times K q q q ε ε ε ε σ ÷ ÷ ÷ ÷ σ ÷ ÷ ÷ ÷ = σ ÷ ÷ ÷ ÷ ÷ = σ ε X' ε b - X'X β -1  the first column of    Q

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Part 13: Endogeneity Asymptotic Covariance Matrix of bIV ™  23/61 1 IV 1 -1 IV IV 2 1 -1 IV IV ( ) ' ( )( ) ' ( ) ' ' ( ) E[( )( ) ' | ] ( ) ' ( ) - - - - = - - = - - = σ b Z'X Z b b Z'X Z Z X'Z b b X, Z Z'X Z Z X'Z β ε β β εε β β
Part 13: Endogeneity Asymptotic Efficiency Asymptotic efficiency of the IV estimator. The variance is larger than that of LS. (A large sample type of Gauss-Markov result is at work.) (1) It’s a moot point. LS is inconsistent. (2) Mean squared error is uncertain: MSE[estimator| β ]=Variance + square of bias. IV may be better or worse. Depends on the data ™  24/61

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Part 13: Endogeneity Two Stage Least Squares How to use an “excess” of instrumental variables (1) X is K variables. Some (at least one) of the K variables in X are correlated with ε . (2) Z is M > K variables. Some of the variables in Z are also in X , some are not. None of the variables in Z are correlated with ε. (3) Which K variables to use to compute Z’X and Z’y? ™  25/61
Part 13: Endogeneity Choosing the Instruments p Choose K randomly? p Choose the included Xs and the remainder randomly? p Use all of them? How? p A theorem: (Brundy and Jorgenson, ca. 1972) There is a most efficient way to construct the IV estimator from this subset: n (1) For each column (variable) in X , compute the predictions of that variable using all the columns of Z . n (2) Linearly regress y on these K predictions. p This is two stage least squares ™  26/61

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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' 0 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. =[

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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]= ( ' )

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