Econometric Methods for Panel Data - Persönliche …

1 1/21 IntroductionFixed effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsEconometric Methods for Panel DataBased on the books byBaltagi: Econometric Analysis ofPanel Dataand byHsiao:Analysis of Panel DataRobert M. of ViennaandInstitute for Advanced Studies ViennaApril 25, 2018 Econometric Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsOutlineIntroductionFixed effectsRandom effectsTwo-way panelsTests in Panel modelsPoolability testsTesting for the presence of random effectsThe Hausman testCoefficients of determination in panelsEconometric Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsThe hypotheses of poolability testsThese tests help select the

2 Panel model to be estimated, within theframework of fixed-effects models. For example:IAre there individual effects or is it preferable to ignore themand to estimate by pooled OLS?IAre there time effects on top of the individual effects?IAre the coefficients really constant across individuals?These tests follow a simple principle: restricted and unrestrictedmodels are compared via LR or Wald statistics. In most cases, nulldistributions areFor 2, with degrees of freedom matching thenumber of Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsTesting for individual effectsHere, the null model (restricted) isyit= + Xit+ it;withiiderrors, and the alternative (unrestricted) model isyit= + Xit+ i+ it;assuming Ni=1 i=0.

3 The null hypothesis may be written as:H0: i=0;i=1;:::;N: Econometric Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsTesting for individual effects: the statisticThe traditional restriction-test statistic isF1 way=(ESSR ESSU)=(N 1)ESSU=((T 1)N K);as there areN 1 restrictions from the maintained hypothesis tothe null. There are(T 1)N Kdegrees of freedom in theunrestricted model. UnderH0, this statistic will be distributed asFN 1;N(T 1) K, assuming Gaussian Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsOther analogous testsFor the null of the pooled model and the alternative of the two-waymodel, the distribution of theF statistic will be distributed asF(N+T 2;NT N T+1 K)under the null, assuming Gaussian the null of the one-way individual-effects model and thealternative of the two-way model, theF statistic will be distributedasF(T 1.)

4 NT N T+1 K)under the null, assuming Gaussian largeNT, 2versions may be used instead, which areindependent of Gaussian error Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsTesting homogeneity of individual slopesIn one specification, the unrestricted model isyit= + iXit+ i+ it;and the null can be expressed asH0: 1=:::= N= ; 1=:::= N=0;with(K+1)(N 1)restrictions. The unrestricted model has onlyN(T K 1)degrees of freedom. If this test rejects, either theone-way model should be considered or individuals must bemodelled separately.

5 It may make more sense just to test theslopes, withK(N 1) Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsConstellations in Panel models and degrees of freedomcoefficients effectsname i OLSNT K 1N(T K) 1 ione-wayN(T 1) KN(T K 1) ittwo-wayN(T 1) T+1 K N(T K 1) T+1 Econometric Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsPoolability testsThe test byRoyandZellnerSome researchers (among them,Baltagi) criticize that the usualF test checks poolability in an otherwise perfect Gauss-Markovregression withE = 2 I.

6 To cope with this problem, one mayalso test for i= and/or t= in a (one-way or two-way) test is called the Roy-Zellner test. Essentially, it tests for fixed-type poolability of slope coefficients in a Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsTesting for the presence of random effectsLikelihood-ratio tests for variance parametersAssumeLuis the likelihood of an unrestricted model,Lris thelikelihood of a restricted model. A lemma says that thelikelihood-ratio(LR) statisticLR=2(logLu logLr);will be, under the null of the restricted model, asymptoticallydistributed 2with degrees of freedom matching the number lemma requires severalregularity conditions.

7 In testing avariance parameter for zero, these conditions are violated and theproperty is not guaranteed to Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsTesting for the presence of random effectsLR test for random effects in two-way panelsThe unrestricted RE modelyit= + Xit+uit;uit= i+ t+ it;with the restricted nullH0: 2 = 2 =0 violates the regularityconditions. The LR test statistic has the non-standard distribution14 2(0)+12 2(1)+14 2(2);where 2(0)denotes point mass at Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsTesting for the presence of random effectsLR test for random effects in one-way panelsThe unrestricted RE modelyit= + Xit+uit;uit= i+ it;with the restricted nullH0: 2 =0 again violates the regularityconditions.

8 The LR test statistic has the non-standard distribution12 2(0)+12 2(1):An analogous test can be used for random time effects. Thesetests are due toGourieroux, Holly & Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsTesting for the presence of random effectsLM tests for random effectsLagrange-multiplier (LM) tests have standard have usually less power. One may test for two-way randomeffects using pooled-OLS residuals~ubyLM:LM=LM1+LM2;LM1=NT2(T 1){1 ~u (IN JT)~u~u ~u}2;LM2=NT2(N 1){1 ~u (JN IT)~u~u ~u}2;or one may test for one-way effects by usingLM1orLM2.

9 Thistest is due toBreusch & Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsThe Hausman testThe Hausman test principleHausman testscan be used in all situations where two modelspecifications and two estimators are available with the the restricted model (null), the estimator^ is efficient, theestimator~ is consistent though typically not efficient; the unrestricted model (alternative), the estimator^ isinconsistent, the estimator~ is , the differenceq=^ ~ should diverge under the alternativeand it should converge to zero under the null.

10 Moreover, under thenullqand^ should be Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsThe Hausman testThe RE and the FE modelThe null of the RE and the alternative of the FE model correspondto the Hausman the RE model, the GLS-type RE estimator is efficient byconstruction for Gaussian errors, the FE estimator and eventhe OLS estimator are consistent; the FE model, the RE estimator is inconsistent, because ofthe omitted-variable effect, while FE is consistent Methods for Panel DataUniversity of Vienna and Institute for Advanced Studies effectsRandom effectsTwo-way panelsTests in Panel modelsCoefficients of determination in panelsThe Hausman testTwo views on the RE inconsistency in the FE estimator(X 1X) 1X 1yis inconsistent, if the truemodel isy=X +Z + , as the regressorsZ are omitted.