Panel Data Models

1 Panel data ModelsChapter 5 Financial EconometricsMichael HauserWS18/191 / 63 ContentIData structures:Times series, cross sectional, Panel data , pooled dataIStatic linear Panel data Models :fixed effects, random effects, estimation, testingIDynamic Panel data Models :estimation2 / 63 data structures3 / 63 data structuresWe distinguish the following data structuresITime series data :I{xt,t=1,..,T}, univariate series, a price series:Its path over time is modeled. The path may also depend on third , several price series:Their individual as well as their common dynamics is modeled. Third variablesmay be sectional dataare observed at a single point of time for severalindividuals, countries, assets, etc.,xi,i=1,.., interest lies in modeling the distinction of single individuals, theheterogeneity across / 63 data structures: PoolingPooling datarefers to two or moreindependentdata sets of the same time series:We observe return series of several sectors, which are assumed to beindependent of each other, together with explanatory variables.

2 The numberof sectors,N, is usually are viewed as repeated measures at each point of time. Soparameters can be estimated with higher precision due to an increasedsample / 63 data structures: PoolingIPooled cross sections:Mostly these type of data arise in surveys, where people are asked about attitudes to political parties. This survey is repeated,Ttimes, beforeelections every usually we have several cross sections, but the persons asked are chosenrandomly. Hardly any person of one cross section is member of another cross sections are overall questions can be answered, like the attitudes within males orwomen, but no individual (even anonymous) paths can be / 63 data structures: Panel dataApanel dataset (alsolongitudinal data ) has both a cross-sectional and a timeseries dimension, where all cross section units are observed during the whole ,i=1.

3 ,N,t=1,.., usually can distinguish for a balanced Panel :The Mikrozensus in Austria is a household, hh, survey, with the same size each quarter. Each hh has to record its consumption expenditures for 5quarters. So each quarter 4500 members enter/leave the Mikrozensus. This is abalanced / 63 data structures: Panel dataA special case of a balanced Panel is afixed Panel . Here we require that allindividuals are present in all panelis one where individuals are observed a different number oftimes, because of missing are concerned only with balanced/fixed general Panel data Models are more efficient than pooling cross-sections,since the observation of one individual for several periods reduces the variancecompared to repeated random selections of / 63 Pooling time series: estimationWe considerTrelatively large, + xit+uitIn case of heteroscedastic errors, 2i6= 2(= 2u), individuals with large errors willdominate the fit.

4 A correction is necessary. It is similar to a GLS and can beperformed in 2 estimate under assumption of const variance for each indiviand calculatethe individual residual variances, 2 t(yit a b xit)29 / 63 Pooling time series: estimationSecondly, normalize the data withsiand estimate(yit/si) = (1/si) + (xit/si) + uit uit=uit/sihas (possibly) the required constant variance, is :V( uit) =V(uit/si) 1 Dummies may be used for different cross sectional / 63 Panel data modeling11 / 63 ExampleSay, we observe the weekly returns of 1000 stocks in two consecutive pooling model is appropriate, if the stocks are chosen randomly in eachperiod. The Panel model applies, if the same stocks are observed in both could ask the question, what are the characteristics of stocks with high/lowreturns in Panel Models we could further analyze, whether a stock with high/low return inthe first period also has a high/low return in the / 63 Panel data modelThe standard static model withi=1.

5 ,N,t=1,..,Tisyit= 0+x it + itxitis aK-dimensional vector of explanatory variables, without a const term. 0, the intercept, is independent ofiandt. , a(K 1)vector, the slopes, is independent ofiandt. it, the error, varies characteristics (which do not vary over time),zi, may be includedyit= 0+x it 1+z i 2+ it13 / 63 Two problems: endogeneity and autocorr in the errorsIConsistency/exogeneity:Assuming iid errors and applying OLS we get consistent estimates, ifE( it) =0 andE(xit it) =0, if thexitare weakly in the errors:Since individualiis repeatedly observed (contrary to pooled data )Corr( i,s, i,t)6=0withs6=tis very likely. Then,Istandard errors are misleading (similar to autocorr residuals),IOLS is inefficient (cp. GLS).14 / 63 Common solution for individual unobserved heterogeneityUnobserved (const) individual factors, if not allzivariables are available, maybe captured by i.

6 We decompose itin it= i+uitwithuitiid(0, 2u)uithas mean 0, is homoscedastic and not serially this decompositionall individual characteristics- including all observed,z i 2, aswell as all unobserved ones, which do not vary over time - aresummarized in the i distinguishfixed effects(FE), andrandom effects(RE) / 63 Fixed effects model , FEIF ixed effects model , FE: iareindividual intercepts(fixed for givenN).yit= i+x it +uitNo overall intercept is (usually) included in the FE, consistency does not require, that the individual intercepts (whosecoefficients are the i s) anduitare uncorrelated. OnlyE(xituit) =0must areN 1 additional parameters for capturing the / 63 Random effects model , REIR andom effects model , RE: i iid(0, 2 )yit= 0+x it + i+uit,uit iid(0, 2u)The i s arervswith the same variance.

7 The value iis specific for individuali. The s of different indivs are independent, have a mean of zero, and theirdistribution is assumed to be not too far away from normality. The overallmean is captured in 0. iis time invariant and homoscedastic across is only one additional parameter 2 .Only icontributes toCorr( i,s, i,t). idetermines both i,sand i, / 63RE some discussionIConsistency:As long asE[xit it] =E[xit( i+uit)] =0, uncorrelated with iand uit, the explanatory vars are exogenous, the estimates are are relevant cases where this exogeneity assumption is likely to when modeling investment decisions the firm specific heteroscedasticity imight correlate with (the explanatory variable of) the cost of capital of resultinginconsistencycan be avoided by considering a FE :The model can be estimated by (feasible) GLS which is in general more efficient than / 63 The static linear model the fixed effects model3 Estimators: Least square dummy variable estimator, LSDV Within estimator, FE First difference estimator, FD19 / 63[LSDV] Fixed effects model .

8 LSDV estimatorWe can write the FE model usingNdummy vars indicating the j=1 jdjit+x it +uituit iid(0, 2u)with dummiesdj, wheredjit=1 ifi=j, and 0 parameters can be estimated by OLS. The implied estimator for is calledtheLS dummy variable estimator, of exploding computer storage by increasing the number of dummyvariables for largeNthewithin estimatoris / 63[LSDV] Testing the significance of the group effectsApart fromt-tests for single i(which are hardly used) we can test, whether theindivs have the same intercepts wrt some have different intercepts by pooled model (all intercepts arerestrictedto be the same),H0, isyit= 0+x it +uitthe fixed effects model (intercepts may be different, areunrestricted),HA,yit= i+x it +uiti=1,..,NTheFratio for comparing the pooled with the FE model isFN 1,N T N K=(R2 LSDV R2 Pooled)/(N 1)(1 R2 LSDV)/(N T N K)21 / 63[FE] Within transformation, within estimatorThe FE estimator for is obtained, if we use the deviations from the individualmeans as variables.

9 The model in individual means is with yi= tyit/Tand i= i, ui=0 yi= i+ x i + uiSubtraction fromyit= i+x it +uitgivesyit yi= (xit xi) + (uit ui)where the intercepts vanish. Here the deviation ofyitfrom yiis explained (not thedifference between different individuals, yiand yj).The estimator for is called thewithinorFE refers to the variability (over time) among observations of / 63[FE] Within/FE estimator FEThe within/FE estimator is FE=( i t(xit xi)(xit xi) ) 1 i t(xit xi)(yit yi)This expression is identical to the well known formula = (X X) 1X yforN(demeaned wrt individuali) data withT / 63[FE] Finite sample properties of FEFinite samples:IUnbiasedness:if allxitare independent of allujs, strictly :if in additionuitis / 63[FE] Asymptotic samples properties of FEAsymptotically:IConsistency wrtN ,Tfix: ifE[(xit xi)uit] = xiare uncorrelated with the error, (as xi= txi,t/T) implies thatxitisstrictly exogenous:E[xisuit] =0for alls,txithas not to depend on current, past or future values of the error term ofindividuali.

10 This excludesIlagged dependent varsIanyxit, which depends on the history ofyEven largeNdo not mitigate possible normality: under consistency and weak conditions / 63[FE] Properties of theNintercepts, i, andV( FE)IConsistency of iwrtT : ifE[(xit xi)uit] = is no convergence wrt toN, even ifNgets large. Cp. yi= (1/T) estimates for theNintercepts, i, are simply i= yi x i FEReliable estimates forV( FE)are obtained from the LSDV model . A consistentestimate for 2uis 2u=1N T N K i t u2itwith uit=yit i x it FE26 / 63[FD] The first difference, FD, estimator for the FE modelAn alternative way to eliminate the individual effects iis to take first differences(wrt time) of the FE yi,t 1= (xit xi,t 1) + (uit ui,t 1)or yit= x it + uitHere also, all variables, which are only indiv specific - they do not change witht-drop with OLS gives thefirst-difference estimator FD=( iT t=2 xit x it) 1 iT t=2 xit yit27 / 63[FD] Properties of the FD estimatorIConsistency forN.