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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 15. Generalized Regression Model Generalized Regression Model Setting: The classical linear model assumes that E[ εε′ ] = Var[ ε ] = σ 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. Implications The assumption that Var[ ε ] = σ 2 I is used to derive the result Var[ b ] = σ 2 ( X ′ X )1 . If it is not true, then the use of s 2 ( X ′ X )1 to estimate Var[ b ] is inappropriate. The assumption was used to derive most of our test statistics, so they must be revised as well. 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. Least squares is based on simple sums, so the information that one observation might provide about another is never used. GR Model The generalized regression model: y = X β + ε , E[ ε X ] = , Var[ ε X ] = σ 2 Ω . Regressors are well behaved. We consider some examples Trace Ω = n. (This is a normalization with no content.) Leading Cases Simple heteroscedasticity Autocorrelation Panel data and heterogeneity more generally. Least Squares Still unbiased . (Proof did not rely on Ω ) For consistency , we need the true variance of b , Var[ bX ] = E[( b β )( b β ) ’X ] = ( X’X )1 E[ X’ εε ’X ] ( X’X )1 = σ 2 ( X’X )1 X ′ Ω 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 ′ Ω X = (1/n) Σ i Σ j ϖ ij x i x j ′ . This will be another assumption of the model. Asymptotic normality ? Easy for heteroscedasticity case, very difficult for autocorrelation case. Robust Covariance Matrix Robust estimation: How to estimate Var[ bX ] = σ 2 ( X’X )1 X ′ Ω X ( X’X )1 for the LS b ? The distinction between estimating σ 2 Ω an n by n matrix and estimating σ 2 X ′ Ω X =...
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.
 Spring '11
 PP
 Econometrics

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