Econometrics Lecture Notes-Panel Data Analysis

1 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsEconometrics Lecture Notes-Panel data AnalysisQingfeng LiuOtaru University of CommerceQingfeng LiuEconometrics Lecture Notes-Panel data Analysis1 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: Extensionsassumptions for OLSyi= + xi+ i = x0x 1x0y1 Var( i)= 2fori=1, ,n. ( i j)=0 fori6=j. No series ( ijxi)=0 (E( ixi)=0). LiuEconometrics Lecture Notes-Panel data Analysis2 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: Extensionsproperties of OLS1E = , ! = , Var ,where is any other linear estimator,e LiuEconometrics Lecture Notes-Panel data Analysis3 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsWhat happens without assumptions assumption 1. (heteroscedasticity), will beine assumption 2. (series correlation), will beine assumption 3.

2 (endogenity), will be LiuEconometrics Lecture Notes-Panel data Analysis4 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsSolutions1 GLS for models with heteroscedasticity or series (2 SLS, GMM) for models with LiuEconometrics Lecture Notes-Panel data Analysis5 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsWhat is Panel DataPanel datarepeated observations on the same cross section,observed for several time periods. (Longitudinaldata). Some observations increase precision estimation of the xed e ects model,solving problem of omitted variables more time series LiuEconometrics Lecture Notes-Panel data Analysis6 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsOutlineSeveral models and corresponding estimation Analysis LiuEconometrics Lecture Notes-Panel data Analysis7 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsPanel data ModelsGeneral Expressionyit= it+xit + it,i=1, ,n,t=1, , Modelsyit= +xit + it,E( jX)=0 Estimation method for Pooled ModelsOLSQ ingfeng LiuEconometrics Lecture Notes-Panel data Analysis8 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models.

3 ExtensionsPanel data ModelsFor now, we assumeExogeneityE[ itj i,xi1, ,xiT]=0,t=1, , E ect Model (RE) i(Individual E ect) is randomvariable and uncorrelated withxit,yit= i+xit + it,Estimatin Method for RE ModelPooled OLS works well for LiuEconometrics Lecture Notes-Panel data Analysis9 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsPanel data ModelsFixed E ect Model (FE) iis random variable and correlated withxit,yit= i+xit + it,Estimation method for FE ModelPooled OLS is inconsistent forFE, and does not work well for FE LiuEconometrics Lecture Notes-Panel data Analysis10 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsBasic Estimation Methods for RE ModelPooled OLSB etween estimator, between estimator is the OLS estimatorof following equation yi= +xi +( i + i)where yi=1/T Tt=1yit,xi=1/T Tt=1xitand i=1/T Tt=1 LiuEconometrics Lecture Notes-Panel data Analysis11 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsBasic Estimation Methods for FE ModelPooled OLSandBetween estimatorsdo not work well forFE model (inconsistent).

4 Within estimator,subtractyit= i+x0it + itfrom yi=ai+x0i + iyieldsyit yi=(xit xi)0 +( it i)(1)LSDVW ithin estimatoris the OLS estimator of eq.(1). Notice thatno iin eq.(1).Within estimator of FE model is LiuEconometrics Lecture Notes-Panel data Analysis12 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsBasic Estimation Method for FE ModelFirst-Di erences estimator,subtractingyi,t 1fromyityieldyit yi,t 1=(xit xi,t 1)0 +( it i,t 1)(2)First-Di erences estimatoris the OLS estimator of theabove erences estimator of FE model is LiuEconometrics Lecture Notes-Panel data Analysis13 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsConsistency and e ciencyQingfeng LiuEconometrics Lecture Notes-Panel data Analysis14 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGLS Estimator for RE Models (RE Estimator) Modelyit= i+X0it + ityit=X0it +( i+ it)yit=X0it +uitwhereuit=( i+ it)uitis series correlated overtOLS is not e cientQingfeng LiuEconometrics Lecture Notes-Panel data Analysis15 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGLS Estimator for RE Models (RE Estimator) EstimatorUse GLS to deal with series correlation 1/2yi= 1/2Wi + 1/2( i+ i)yit yi= 1 + xit xi 0 + it,(3)where it= 1 i+ it i and is consistent estimatorfor =1 / T 2 + 2 1 feasibleGLS (FGLS) estimatoris the OLS estimator of in model (3).

5 FGLSis consistent and full e cient for RE models, but isinconsistent for FE LiuEconometrics Lecture Notes-Panel data Analysis16 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsHausman Test for Panel data Models to distinguish between RE and FEUseHausman the fact thatWithin estimatoris consistentandFGLSis inconsistent for FE LiuEconometrics Lecture Notes-Panel data Analysis17 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsHausman Test for Panel data Models Test statistics1 When iand itboth , use Robust Hausman Test(4)3 Ignoring whether iand itboth , just use RobustHausman Test at all LiuEconometrics Lecture Notes-Panel data Analysis18 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsWithin vs. First Di erences Estimator (FD) for FE yi=(xit xi)0 +( it i)(5)vt=( it i)cov(vt,vt 1)=E(( it i)( i,t 1 i))=0 2 T 2 T+ 2 T= 2 TQingfeng LiuEconometrics Lecture Notes-Panel data Analysis19 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsWithin vs.

6 First Di erences Estimator (FD) for FE yi,t 1=(xit xi,t 1)0 +( it i,t 1)(6)vt=( it i,t 1)cov(vt,vt 1)=E(( it i,t 1)( i,t 1 i,t 2))=0 0 2 0= 2 Qingfeng LiuEconometrics Lecture Notes-Panel data Analysis20 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsTime-invariant variable in FE modelsyit= i+ 1x1t+ 2dit+ itwhenditis time-invariant,dit=di, thenyit= i+ 1x1t+ 2di+ itThe 2can not be estimated by Within or we want to estimate 2, other methods are required (seefollowing slides).Qingfeng LiuEconometrics Lecture Notes-Panel data Analysis21 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsPanel robust estimate of asymptotic varianceGenaral expression of Panel data models~yi=~Wi +uiThree panel robust estimates of asymptotic varianceE(uitjwit)=0,uitare independent overi,V(uit)andcov(uit,uis)both are time-variant (means heteroscedastic andseries dependent) LiuEconometrics Lecture Notes-Panel data Analysis22 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsPanel robust estimate of asymptotic LiuEconometrics Lecture Notes-Panel data Analysis23 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsPanel robust estimate of asymptotic variance estimatorQingfeng LiuEconometrics Lecture Notes-Panel data Analysis24 / 42 Overview of OLS for Linear ModelsLinear Panel data Models.

7 BasicsLinear Panel data Models: ExtensionsConditional MLEA ssuming normal distribution,MLEgets the same results ofWithinestimatorQingfeng LiuEconometrics Lecture Notes-Panel data Analysis25 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsEstimate alpha using Within EstimateTmust be large for consistency of LiuEconometrics Lecture Notes-Panel data Analysis26 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsOther IssuesUnbalanced panel dataMeasurement errorQingfeng LiuEconometrics Lecture Notes-Panel data Analysis27 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsExample of application, labor supply to wagesThe data on 532 males for each of the 10 years from 1979 to1988. lnhrsit= i+ lnwgsit+ itQingfeng LiuEconometrics Lecture Notes-Panel data Analysis28 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsQingfeng LiuEconometrics Lecture Notes-Panel data Analysis29 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsAssumptionEndogenous Variables or lagged dependent variables as regressorsE[ itj i,xi1, ,xiT]6=0,t=1, , , itwill be denoted asuit,E[uitj i,xi1, ,xiT]6=0,t=1, , assumption cause inconsistencies with someaforementioned LiuEconometrics Lecture Notes-Panel data Analysis30 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models.

8 ExtensionsEstimation method under Endogenous settingPanel GMM without iFor simplicity, consider about models without individual e ect iyit=xit +uityi=Xi + LiuEconometrics Lecture Notes-Panel data Analysis31 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsInsrumental VariableAssumeZisatisfyingE[Z0iui]0=0,E[ Z0iXi]06=0 exist,Zican be used as ConditionQingfeng LiuEconometrics Lecture Notes-Panel data Analysis32 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGMM Estimator PGMM=arg minQN( )where Sis a consistent estimator ofQingfeng LiuEconometrics Lecture Notes-Panel data Analysis33 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: Extensions2 SLS (One-Step GMM)2-Step GMM (2 SGMM)where Sis a estimator ofSfrom 2 SLS. 2 SGMM ismore e cientthan One-Step LiuEconometrics Lecture Notes-Panel data Analysis34 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGMM for Models with Individual E ectsFirst-Di erences GMMM odelassume Weak Exogeneity Assumption:E[zis it]=0,s t,yit= i+x0it + rst-di erences2y it=x 0it + it(7)3 Carry outGMMwith model (7).

9 Remark:Fors t 1, we haveE(Zis( it i,t 1))=E(Zis it)=0, but fors=t,E(Zis it)6=0, hence,zi,t 1,zi,t 2, can be used as instrumentforxit, however,zitcan LiuEconometrics Lecture Notes-Panel data Analysis35 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGMM for Models with Individual E ectsWithin GMMU nder the same assumption in last slide, sinceE[zit i]6=0,GMM does not [zis it]=0 for allsandt, GMM works. But theassumptionE[zis it]=0 for allsandt, is too LiuEconometrics Lecture Notes-Panel data Analysis36 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsGMM for Models with Individual E ectsGMM for Random E ect ModelNew meaning of "Random E ect Model"IfZisatisfyingE(Z0i(ai+ i))=0 exists, we call the model asrandom e ect for RE modelsDe ningui=ai+ i, we can useE(Z0iui)=E(Z0i(ai+ i))=0 as momentcondition and carry out GMM LiuEconometrics Lecture Notes-Panel data Analysis37 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsQingfeng LiuEconometrics Lecture Notes-Panel data Analysis38 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsApplication ExampleHours and WagesQingfeng LiuEconometrics Lecture Notes-Panel data Analysis39 / 42 Overview of OLS for Linear ModelsLinear Panel data Models.

10 BasicsLinear Panel data Models: ExtensionsQingfeng LiuEconometrics Lecture Notes-Panel data Analysis40 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsDynamic Panel ModelsDynamic Panel Modelyit= yi,t 1+xit + i+ iti=1, ,N,t=1, , j<1 EndogenitySinceyi,t 1is a function of iyi,t 1= yi,t 2+xi,t 1 + i+ i,t 1E(yi,t 1 i)6=0 Qingfeng LiuEconometrics Lecture Notes-Panel data Analysis41 / 42 Overview of OLS for Linear ModelsLinear Panel data Models: BasicsLinear Panel data Models: ExtensionsEstimator for Dynamic Panel ModelsArellano Bond EstimatorTake di erence(yit yi,t 1)= (yi,t 1 yi,t 2)+(xit xi,t 1) +( it i,t 1)Useyi,t 2as instrumental variablePerform LiuEconometrics Lecture Notes-Panel data Analysis42 / 42