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Econometrics-I-16

# Econometrics-I-16 - 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 16. Applications of the Generalized Regression Model
Two Step Estimation of the Generalized Regression Model     Use the Aitken (Generalized Least Squares -  GLS) estimator with an estimate of  1.     is parameterized by a few estimable  parameters.  Examples, the heteroscedastic  model 2.  Use least squares residuals to estimate the  variance functions 3.  Use the estimated   in GLS - Feasible GLS,  or FGLS

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General Result for Estimation When Is Estimated True GLS uses   [ X ′ Ω   -1  X ] X ′ Ω   -1  y    which  converges in probability to  β . We seek a vector which converges to the same  thing that this does.   Call it FGLS, based on    [ X      -1  X ] X     -1  y

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FGLS Feasible GLS is based on finding an estimator which has the same properties as the true GLS. Example Var[ ε i ] = σ 2 Exp( γ ′ z i ). True GLS would regress y/[ σ Exp((1/2) γ ′ z i )] on the same transformation of x i . With a consistent estimator of [ σ , γ ], say [s, c ], we do the same computation with our estimates. So long as plim [s, c ] = [ σ , γ ], FGLS is as good as true GLS.
FGLS vs. Full GLS VVIR    To achieve full efficiency, we do not need an  efficient estimate of the parameters in  , only a  consistent one.  Why?

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Heteroscedasticity Setting:   The regression disturbances have unequal variances, but are  still not correlated with each other: Classical regression with hetero-(different) scedastic (variance)  disturbances.     y i   =   β ′ x i  +  ε i ,  E[ ε i ]  =  0,  Var[ ε i ]  =   σ 2   ϖ i ϖ i  > 0.   The classical model arises if  ϖ i  = 1. A normalization:   Σ i   ϖ i  = 1.  Not a restriction, just a scaling that is  absorbed into  σ 2 . A characterization of the heteroscedasticity:  Well defined estimators  and methods for testing hypotheses will be obtainable if the  heteroscedasticity is “well behaved” in the sense that ϖ i   /  Σ i   ϖ i     0  as  n    .   I.e., no single observation becomes dominant. (1/n) Σ i   ϖ i     some stable constant.   (Not a probability limit as such.)
GR Model and Testing Implications for conventional estimation technique  and hypothesis testing: 1 .  b  is still unbiased.  Proof of unbiasedness did  not rely on homoscedasticity 2.  Consistent?  We need the more general proof.   Not difficult. 3.  If plim  b  =  β , then plim s 2   =   σ 2   (with the  normalization).

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Inference Based on OLS     What of s 2 ( X X ) -1  ?  Depends on  X ′ Ω X - X X .  If  they are nearly the same, the OLS        covariance matrix is OK.  When will they be  nearly the same?  Relates to an interesting
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