Econometric Methods for Panel Data - univie.ac.at

1 IntroductionFixed effectsRandom effectsTwo-way panelsEconometric Methods for Panel DataBased on the books byBaltagi: Econometric analysis ofPanel Dataand byHsiao: analysis of Panel DataRobert M. of ViennaandInstitute for Advanced Studies ViennaMay 4, 2010 Econometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsOutlineIntroductionFixed effectsThe LSDV estimatorThe algebra of the LSDV estimatorProperties of the LSDV estimatorPooled regression in the FE modelRandom effectsThe GLS estimator for the RE modelfeasible GLS in the RE modelProperties of the RE estimatorTwo-way panelsThe two-way fixed-effects modelThe two-way random-effects modelEconometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe definition of Panel dataThe wordpanelhas a Dutch origin and it essentially means aboard.

2 data for a variable on a board is two-dimensional, thevariableXhas two subscripts. One dimension is anindividualindex (i), and the other dimension istime(t): X11..XNT Econometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsLong and broad boards X11..Xit..XN1..XNT X11..XNT IfT N, the Panel is atime-series Panel , as it is oftenencountered in macroeconomics. IfN T, it is across-sectionpanel, as it is common in Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsNot quite panels longitudinal datasets look like panels but the time index maynot be common across individuals. For examples, growingplants may be measured according to individual time; inpseudo- panels , individuals may change between time points:XitandXi,t+1may relate to different persons; inunbalanced panels ,Tdiffers among individuals and isreplaced byTi: no more matrix or board Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe two dimensions have different quality In the time dimensiont, the Panel behaves like a time series:natural ordering, systematic dependence over time,asymptotics depend on stationarity, ergodicity etc.

3 In the cross-section dimensioni, there is no natural ordering,cross-section dependence may play a role ( secondgeneration ), otherwise asymptotics may also be simpleassuming independence ( first generation ). Sometimes, panels ,imay have structure and natural Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsAdvantages of Panel data analysis panels are more informative than simple time series ofaggregates, as they allow tracking individual histories. A 10%unemployment rate is less informative than a Panel ofindividuals with all of them unemployed 10% of the time orone with 10% always unemployed; panels are more informative than cross-sections, as theyreflect dynamics and Granger causality across Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe heterogeneity problemN= 3,T= 20.

4 For eachi, there is an identical, positive slope in a linearrelationship betweenYandX. For the whole sample, the relationship isslightly falling and nonlinear. If interest focuses on the former model,naive estimation over the whole sample results in aheterogeneity Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsBooks on panelsBaltagi, analysis of Panel data ,Wiley.(textbook)Hsiao, of Panel data ,Cambridge University Press.(textbook)Arellano, data econometrics , Oxford UniversityPress. (very formal state of the art)Diggle, P., Heagerty, P., Liang, , and S. ZegerAnalysis of Longitudinal data ,Oxford University Press.(non-economic monograph) Econometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorThe pooled regression modelConsider the modelyit= + Xit+uit,i= 1.

5 ,N,t= 1, .. , there areKregressors (covariates), such thatdim( ) = models mainly differ in their assumptions acrossiandt,Eu= 0, andvaru= 2define the(usually unrealistic)pooled regression model. It is efficientlyestimated by least squares (OLS).Sometimes, one may consider digressing from the homogeneityassumption i . This entails that most advantages of panelmodelling are Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorThe fixed-effects regressionThe modelyit= + Xit+uit,uit= i+ it,i= 1, .. ,N,t= 1, .. ,T,with the identifying conditionPNi=1 i= 0, and theindividualeffects iassumed as unobserved constants (parameters), is thefixed-effects (FE) regressionmodel. ( it) fulfills the usualconditions on errors: independent,E = 0,var = 2.

6 Econometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorIncidental parametersMost authors call a parameterincidentalwhen its dimensionincreases with the sample size. The nuisance due to such aparameter is worse than for a typical nuisance parameter whosedimension may be , the number of fixed effects iincreases. Thus, thefixed effects are incidental parameters cannot be consistently estimated, andtheymay cause inconsistency in the ML estimation of the Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorOLS in the FE regression modelIn the FE model, the regression equationyit= + Xit+uit,i= 1.

7 ,N,t= 1, .. ,T,does not fulfill the conditions of the Gauss-Markov Theorem, asEuit= i6= 0. OLS may be biased, inconsistent, and even if it isunbiased, it is usually Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorLeast-squares dummy variablesLetZ(j) ,itdenote a dummy variable that is 0 for all observationsitwithi6=jand 1 fori=j. Then, conveningZ ,it= (Z(1) ,it, .. ,Z(N) ,it) and = ( 1, .. , N) , the regressionmodelyit= + Xit+ Z ,it+ it,i= 1, .. ,N,t= 1, .. ,T,fulfills all conditions of the Gauss-Markov Theorem. OLS for thisregression is called LSDV (least-squares dummy variables), thewithin, or the FE estimator. AssumingXas non-stochastic, LSDVis unbiased, consistent, and linear efficient (BLUE).

8 Econometric Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorThe dummy-variable trap in LSDVNote thatPNj=1Z(j) ,it= 1. Inhomogeneous LSDV regression wouldbe multicollinear. Two (equivalent) OLS imposingPNi=1 i= 0; regression with free icoefficients. Then, isrecovered fromPNi=1 i, and i= i .In any case, parameter dimension isK+ Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe LSDV estimatorWithin and betweenBy conditioning on individual ( group ) dummies, the withinorwithin-groups estimator concentrates exclusively on variationwithin the contrast, thebetween estimatorresults from a regressionamongNindividual time averages: + Xi.

9 + ui.,i= 1.. ,N,with 1 PTt=1yitetc. Because ofE i6= 0, it violatesthe Gauss-Markov conditions and is more of theoretical Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe algebra of the LSDV estimatorLSDV regression in matrix formStacking allNTobservations yields the compact formy= NT+X +Z + ,where NT,y, and areNT vectors. Generally, mstands for anm vector of ones. Convention is thatiis the slow index andtthe fast index, such that the firstTobservations belong toi= anNT K matrix, is aK vector, is anN isanNT N Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe algebra of the LSDV estimatorThe matrixZ TheNT N matrix for the dummy regressors looks likeZ = 1 0 00 0 1.

10 This matrix can be written in Kronecker notation asIN theN Nidentity Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe algebra of the LSDV estimatorReview: Kronecker productsThe Kronecker productA Bof two matricesA= [aij] andB= [bkl] of dimensionsn mandn1 m1is defined byA B= a11Ba12B..a1mBa21Ba22B..a2mB..an1 Ban2B..anmB ,which gives a large (nn1) (mm1) matrix. Left factor determines crude form and right factor determines fine form. (Note: someauthors use different definitions, Hendry)I Bis a block-diagonal Methods for Panel DataUniversity of Vienna andInstitute for Advanced Studies ViennaIntroductionFixed effectsRandom effectsTwo-way panelsThe algebra of the LSDV estimatorThree calculation rules for Kronecker products(A B) =A B (A B) 1=A 1 B 1(A B)(C D) = (AC BD)for non-singular matrices (second rule) and fitting dimensions(third rule).