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Econometrics. Lecture Notes Hayashi, Chapter 5d Panel Data: Classical View. The Classical Model. y i = Z i d + a i + h i y im = z im ’ d + a i + h im y im = f im ’ b + b i ’ g + a i + h im (i=1,…n; m=1,…M) E( h i ) = 0 , E( z im h im ) = 0, E( a i h im ) = 0

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## Econometrics

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**Econometrics**Lecture Notes Hayashi, Chapter 5d Panel Data: Classical View**The Classical Model**• yi = Zid +ai + hiyim = zim’d +ai + himyim = fim’b + bi’g +ai + him(i=1,…n; m=1,…M) • E(hi) = 0, E(zimhim) = 0, E(aihim) = 0 • Note: E(zimai) = 0 is not assumed. • E(hihi’) = sh2IM**Fixed-Effects Estimator**• First Difference EstimatesIf m is a time index, then the model can be transformed as:yim = fim’b + him(i=1,…,n; m=2,…,M)**Fixed-Effects Estimator**• Between Estimates**Fixed-Effects Estimator**• Within EstimatesE(hi*) = 0,E(fim*him*’) = 0,E(hi*hi*’) = sh2[IM-(1/M)1MxM] = sh2Q**Random-Effects Estimator**• Random-Effects Modelyim = zim’d + (ai + him)eim =ai + him • E(ei) = 0,E(zimeim) = 0E(eiei’) = sa21MxM + sh2IM = S**Parameters Estimation**• To obtain the between estimates, OLS is used. • To obtain the within estimates, pooled OLS is used (with correction of the degrees of freedom). • Random-effects model requires the estimate of q from the between and within estimates of variances. Then pooled OLS is used on the transformed model. • Using xtreg in Stata.**Hypotheses Testing**• Testing for fixed-effects • F-test for ai = 0 jointly • Testing for random-effects • Breusch-Pagan LM 2-test for sa2 = 0**R2**• Three concepts of R2 • Between • Within (from pooled OLS) • Totalyim = fim’b + bi’g +ai + him(i=1,…n; m=1,…M)**Parameters Estimation**• Random-effects model can also be estimated with maximum likelihood method. • GLS estimation (with non-classical variance-covariance matrix, see xtgls in Stata) • Mixed fixed-effects and random-effects (xtmixed in Stata) • Autocorrelation in panels (xtregar in Stata) • Dynamic panel data models (xtabond in Stata)**Endogenous Regressors**• yi = Zid +ai + hi • yi = Z1id1 +Z2id2 +ai + hi • Z1iis predetermined, Z2i is endogenous. • Xi = [X1i,X2i] is exogenous.X1iis included, X2i is excluded instruments.Set X1i = Z1i, #X2i #Z2i**Endogenous Regressors**• yi = Zid +ai + hi with instrumental variables Xi.E(Ximhim)=0, E(Zimhim)0, E(aihim)=0E(hi)=0, E(hihi’)=sh2IM • IV method (xtivreg in Stata) can be applied to: • First-Difference Estimates • Between Estimates • Fixed-Effects or Within Estimates • Random-Effects Estimates**Conditional Heteroscedasticity**• Robust (or White) estimates of standard errors can be obtained to improve consistency of the estimates. • GMM methods can be applied to panel data models with conditional heteroscedasticity and random regressors (xtivreg2 in Stata). • Testing for overidentifying restrictions such as Hansen or Sargan test for panel data models is possible (xtoverid in Stata).

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