Econometrics-I-14 - Econometrics I Professor William Greene...

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Part 14: Generalized Regression Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 14: Generalized Regression Econometrics I Part 14 – Generalized                 Regression ™  1/44
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Part 14: Generalized Regression Generalized Regression Model Setting: The classical linear model assumes that E[  ] = Var[ ] = 2 I . That is, observations are uncorrelated and all are drawn from a distribution with the same variance. The generalized regression ( GR ) model allows the variances to differ across observations and allows correlation across observations. ™  2/44
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Part 14: Generalized Regression Implications p The assumption that Var[ ] = 2 I is used to derive the result Var[ b ] = 2( XX ) -1 . If it is not true, then the use of s2( XX ) -1 to estimate Var[ b ] is inappropriate. p The assumption was used to derive most of our test statistics, so they must be revised as well. p Least squares gives each observation a weight of 1/n. But, if the variances are not equal, then some observations are more informative than others. p Least squares is based on simple sums, so the information that one observation might provide about another is never used. ™  3/44
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Part 14: Generalized Regression GR Model p The generalized regression model: y = X + , E[ |X ] = 0 , Var[ |X ] = 2 . Regressors are well behaved. We consider some examples Trace = n. (This is a normalization with no content.) p Leading Cases n Simple heteroscedasticity n Autocorrelation n Panel data and heterogeneity more generally. ™  4/44
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Part 14: Generalized Regression Least Squares p Still unbiased . (Proof did not rely on ) p For consistency , we need the true variance of b , Var[ b|X ] = E[( b - β )( b - β ) ’|X ] = ( X’X ) -1 E[ X’εε’X ] ( X’X ) -1 = 2 ( X’X ) -1 XX ( X’X ) -1 . Divide all 4 terms by n . If the middle one converges to a finite matrix of constants, we have the result, so we need to examine (1/n) XX = (1/n)ij ij x i x j . This will be another assumption of the model. p Asymptotic normality ? Easy for heteroscedasticity case, very difficult for autocorrelation case. ™  5/44
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Part 14: Generalized Regression Robust Covariance Matrix p Robust estimation: Generality p How to estimate Var[ b|X ] = 2 ( X’X ) -1 XX ( X’X ) -1 for the LS b ? p The distinction between estimating 2 an n by n matrix and estimating the KxK matrix 2 XX = 2 ijij x i x j p NOTE…… VVVIR s for modern applied econometrics. n The White estimator n Newey-West. ™  6/44
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Part 14: Generalized Regression The White Estimator ™  7/44 n 1 2 1 i i i i 1 n 2 i 2 i 1 2 i i i 2 1 2 2 2 Est.Var[ ] ( ) e ( ) e Use      ˆ n ne ˆ ˆ             =  ,   =diag( ) note tr( )= n ˆ ˆ ˆ ˆ ˆ Est.Var[ ] n n n n ˆ Does       ˆ n n - - = = - = σ = ϖ ϖ σ σ = ÷ ÷ ÷ ÷ σ - σ ÷ ÷ b X'X x x ' X'X Ω Ω X'X X' X X'X Ω b X' X X' X Ω Ω
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