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Econometrics-I-15 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 15. Generalized Regression Model
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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. 
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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.
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GR Model The generalized regression model:              y  =  X β  +  ε           E[ ε |X ] =  0 , Var[ ε |X ]  =   σ 2       Regressors are well behaved.  We consider some  examples  Trace   = n.  (This is a normalization with  no content.) Leading Cases Simple heteroscedasticity Autocorrelation Panel data and heterogeneity more generally.
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Least Squares Still  unbiased . (Proof did not rely on  ) 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                                      =   σ ( X’X ) -1   X ′ Ω ( 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. 
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Robust Covariance Matrix Robust estimation: How to estimate Var[ b|X ] =   σ ( X’X ) -1   X ′ Ω ( X’X ) -1  for  the LS  b ?   The distinction between estimating                σ 2  an n by n matrix       and estimating               σ X ′ Ω X  =  σ Σ i Σ j ϖ ij   x i   x j NOTE……   VVVIR s for modern applied econometrics.  The White estimator Newey-West.
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