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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

Econometrics

Lecture Notes Hayashi, Chapter 5d

Panel Data: Classical View


The classical model
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
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 estimator1
Fixed-Effects Estimator

  • Between Estimates


Fixed effects estimator2
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 Estimator

  • Random-Effects Modelyim = zim’d + (ai + him)eim =ai + him

  • E(ei) = 0,E(zimeim) = 0E(eiei’) = sa21MxM + sh2IM = S


Parameters estimation
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
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 estimation1
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
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 regressors1
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
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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