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comxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsDescriptionQuickstartMenuSyntaxOptionsforREmodelOptionsforBEmodelOptionsforFEmodelOptionsforFDmodelRemarksandexamplesStoredresultsMethodsandformulasAcknowledgmentReferencesAlsoseeDescriptionxtivregoffersvedifferentestimatorsforttingpanel-datamodelsinwhichsomeoftheright-hand-sidecovariatesareendogenous.
Theseestimatorsaretwo-stageleast-squaresgeneralizationsofsimplepanel-dataestimatorsforexogenousvariables.
xtivregwiththebeoptionusesthetwo-stageleast-squaresbetweenestimator.
xtivregwiththefeoptionusesthetwo-stageleast-squareswithinestimator.
xtivregwiththereoptionusesatwo-stageleast-squaresrandom-effectsestimator.
Therearetwoimplementations:G2SLSfromBalestraandVaradharajan-Krishnakumar(1987)andEC2SLSfromBaltagi.
TheBalestraandVaradharajan-KrishnakumarG2SLSisthedefaultbecauseitiscomputationallylessexpensive.
Baltagi'sEC2SLScanbeobtainedbyspecifyingtheec2slsoption.
xtivregwiththefdoptionrequeststhetwo-stageleast-squaresrst-differencedestimator.
SeeBaltagi(2013)foranintroductiontopanel-datamodelswithendogenouscovariates.
Forthederivationandapplicationoftherst-differencedestimator,seeAndersonandHsiao(1981).
QuickstartRandom-effectslinearpanel-datamodelwithoutcomey,exogenousx1,andx2instrumentedbyx3usingxtsetdataxtivregyx1(x2=x3)Usexed-effectsestimatorandincludeindicatorsforeachlevelofcategoricalvariableaxtivregyx1i.
a(x2=x3),feUsebetween-effectsestimatorandincludeindicatorsforlevelsofbasinstrumentsxtivregyx1i.
a(x2=x3i.
b),beFirst-differencedmodelofyasafunctionofx1andx2andthelagofyinstrumentedbyitsthirdlagxtivregyx1x2(L.
y=L3.
y),fdMenuStatistics>Longitudinal/paneldata>Endogenouscovariates>Instrumental-variablesregression(FE,RE,BE,FD)12xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsSyntaxGLSrandom-effects(RE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,reREoptionsBetween-effects(BE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,beBEoptionsFixed-effects(FE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,feFEoptionsFirst-differenced(FD)estimatorxtivregdepvarvarlist1(varlist2=varlistiv)ifin,fdFDoptionsREoptionsDescriptionModelreuserandom-effectsestimator;thedefaultec2slsuseBaltagi'sEC2SLSrandom-effectsestimatornosausetheBaltagi–ChangestimatorsofthevariancecomponentsregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsthetareportθdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatisticsxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels3BEoptionsDescriptionModelbeusebetween-effectsestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatisticsFEoptionsDescriptionModelfeusexed-effectsestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatistics4xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsFDoptionsDescriptionModelnoconstantsuppressconstanttermfduserst-differencedestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,anddisplayofomittedvariablescoeflegenddisplaylegendinsteadofstatisticsApanelvariablemustbespecied.
Forxtivreg,fd,atimevariablemustalsobespecied.
Usextset;see[XT]xtset.
varlist1andvarlistivmaycontainfactorvariables,exceptforthefdestimator;see[U]11.
4.
3Factorvariables.
depvar,varlist1,varlist2,andvarlistivmaycontaintime-seriesoperators;see[U]11.
4.
4Time-seriesvarlists.
byandstatsbyareallowed;see[U]11.
1.
10Prexcommands.
coeflegenddoesnotappearinthedialogbox.
See[U]20Estimationandpostestimationcommandsformorecapabilitiesofestimationcommands.
OptionsforREmodelModelrerequeststheG2SLSrandom-effectsestimator.
reisthedefault.
ec2slsrequestsBaltagi'sEC2SLSrandom-effectsestimatorinsteadofthedefaultBalestraandVaradharajan-Krishnakumarestimator.
nosaspeciesthattheBaltagi–ChangestimatorsofthevariancecomponentsbeusedinsteadofthedefaultadaptedSwamy–Aroraestimators.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels5Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,reinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
thetaspeciesthattheoutputincludetheestimatedvalueofθusedincombiningthebetweenandxedestimators.
Forbalanceddata,thisisaconstant,andforunbalanceddata,asummaryofthevaluesispresentedintheheaderoftheoutput.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforBEmodelModelberequeststhebetweenregressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
6xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsThefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforFEmodelModelferequeststhexed-effects(within)regressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforFDmodelModelnoconstant;see[R]Estimationoptions.
fdrequeststherst-differencedregressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels7SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
Remarksandexamplesstata.
comIfyouhavenotread[XT]xt,pleasedoso.
Consideranequationoftheformyit=Yitγ+X1itβ+i+νit=Zitδ+i+νit(1)whereyitisthedependentvariable;Yitisan1*g2vectorofobservationsong2endogenousvariablesincludedascovariates,andthesevariablesareallowedtobecorrelatedwiththeνit;X1itisan1*k1vectorofobservationsontheexogenousvariablesincludedascovariates;Zit=[YitXit];γisag2*1vectorofcoefcients;βisak1*1vectorofcoefcients;andδisaK*1vectorofcoefcients,whereK=g2+k1.
Assumethatthereisa1*k2vectorofobservationsonthek2instrumentsinX2it.
Theorderconditionissatisedifk2≥g2.
LetXit=[X1itX2it].
xtivreghandlesexogenouslyunbalancedpaneldata.
ThusdeneTitobethenumberofobservationsonpaneli,ntobethenumberofpanelsandNtobethetotalnumberofobservations;thatis,N=ni=1Ti.
8xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsxtivregoffersvedifferentestimators,whichmaybeappliedtomodelshavingtheformof(1).
Therst-differencedestimator(FD2SLS)removestheibyttingthemodelinrstdifferences.
Thewithinestimator(FE2SLS)tsthemodelaftersweepingouttheibyremovingthepanel-levelmeansfromeachvariable.
Thebetweenestimator(BE2SLS)modelsthepanelaverages.
Thetworandom-effectsestimators,G2SLSandEC2SLS,treattheiasrandomvariablesthatareindependentandidenticallydistributed(i.
i.
d.
)overthepanels.
Exceptfor(FD2SLS),alloftheseestimatorsaregeneralizationsofestimatorsinxtreg.
See[XT]xtregforadiscussionoftheseestimatorsforexogenouscovariates.
Althoughtheestimatorsallowfordifferentassumptionsaboutthei,alltheestimatorsassumethattheidiosyncraticerrortermνithaszeromeanandisuncorrelatedwiththevariablesinXit.
Justaswhentherearenoendogenouscovariates,asdiscussedin[XT]xtreg,therearevariousperspectivesonwhatassumptionsshouldbeplacedonthei.
Iftheyareassumedtobexed,theimaybecorrelatedwiththevariablesinXit,andthewithinestimatorisefcientwithinaclassoflimitedinformationestimators.
Alternatively,iftheiareassumedtoberandom,theyarealsoassumedtobei.
i.
d.
overthepanels.
IftheiareassumedtobeuncorrelatedwiththevariablesinXit,theGLSrandom-effectsestimatorsaremoreefcientthanthewithinestimator.
However,iftheiarecorrelatedwiththevariablesinXit,therandom-effectsestimatorsareinconsistentbutthewithinestimatorisconsistent.
Thepriceofusingthewithinestimatoristhatitisnotpossibletoestimatecoefcientsontime-invariantvariables,andallinferenceisconditionalontheiinthesample.
SeeMundlak(1978)andHsiao(2014)fordiscussionsofthisinterpretationofthewithinestimator.
Example1:Fixed-effectsmodelForthewithinestimator,consideranotherversionofthewageequationdiscussedin[XT]xtreg.
ThedataforthisexamplecomefromanextractofwomenfromtheNationalLongitudinalSurveyofYouththatwasdescribedindetailin[XT]xt.
Restrictingourselvestoonlytime-varyingcovariates,wemightsupposethatthelogoftherealwagewasafunctionoftheindividual'sage,age2,hertenureintheobservedplaceofemployment,whethershebelongedtounion,whethershelivesinmetropolitanarea,andwhethershelivesinthesouth.
Thevariablesfortheseare,respectively,age,c.
age#c.
age,tenure,union,notsmsa,andsouth.
Ifwetreatallthevariablesasexogenous,wecanusetheone-stagewithinestimatorfromxtreg,yieldingxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels9.
usehttps://www.
stata-press.
com/data/r16/nlswork(NationalLongitudinalSurvey.
YoungWomen14-26yearsofagein1968).
xtregln_wagec.
age#c.
agetenurenot_smsaunionsouth,feFixed-effects(within)regressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
1333min=1between=0.
2375avg=4.
6overall=0.
2031max=12F(6,14867)=381.
19corr(u_i,Xb)=0.
2074Prob>F=0.
0000ln_wageCoef.
Std.
Err.
tP>|t|[95%Conf.
Interval]age.
0311984.
00339029.
200.
000.
0245533.
0378436c.
age#c.
age-.
0003457.
0000543-6.
370.
000-.
0004522-.
0002393tenure.
0176205.
000809921.
760.
000.
0160331.
0192079not_smsa-.
0972535.
0125377-7.
760.
000-.
1218289-.
072678union.
0975672.
006984413.
970.
000.
0838769.
1112576south-.
0620932.
013327-4.
660.
000-.
0882158-.
0359706_cons1.
091612.
052312620.
870.
000.
98907291.
194151sigma_u.
3910683sigma_e.
25545969rho.
70091004(fractionofvarianceduetou_i)Ftestthatallu_i=0:F(4133,14867)=8.
31Prob>F=0.
0000Allthecoefcientsarestatisticallysignicantandhavetheexpectedsigns.
Nowsupposethatwewishtomodeltenureasafunctionofunionandsouthandthatwebelievethattheerrorsinthetwoequationsarecorrelated.
Becausewearestillinterestedinthewithinestimates,wenowneedatwo-stageleast-squaresestimator.
Thefollowingoutputshowsthecommandandtheresultsfromttingthismodel:10xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels.
xtivregln_wagec.
age#c.
agenot_smsa(tenure=unionsouth),feFixed-effects(within)IVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=.
min=1between=0.
1304avg=4.
6overall=0.
0897max=12Waldchi2(4)=147926.
58corr(u_i,Xb)=-0.
6843Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
2403531.
03734196.
440.
000.
1671643.
3135419age.
0118437.
00900321.
320.
188-.
0058023.
0294897c.
age#c.
age-.
0012145.
0001968-6.
170.
000-.
0016003-.
0008286not_smsa-.
0167178.
0339236-0.
490.
622-.
0832069.
0497713_cons1.
678287.
162665710.
320.
0001.
3594681.
997106sigma_u.
70661941sigma_e.
63029359rho.
55690561(fractionofvarianceduetou_i)Ftestthatallu_i=0:F(4133,14869)=1.
44Prob>F=0.
0000Instrumented:tenureInstruments:agec.
age#c.
agenot_smsaunionsouthAlthoughallthecoefcientsstillhavetheexpectedsigns,thecoefcientsonageandnotsmsaarenolongerstatisticallysignicant.
Giventhatthesevariableshavebeenfoundtobeimportantinmanyotherstudies,wemightwanttorethinkourspecication.
Ifwearewillingtoassumethattheiareuncorrelatedwiththeothercovariates,wecantarandom-effectsmodel.
Themodelisfrequentlyknownasthevariance-componentsorerror-componentsmodel.
xtivreghasestimatorsfortwo-stageleast-squaresone-wayerror-componentsmodels.
Intheone-wayframework,therearetwovariancecomponentstoestimate,thevarianceoftheiandthevarianceoftheνit.
Becausethevariancecomponentsareunknown,consistentestimatesarerequiredtoimplementfeasibleGLS.
xtivregofferstwochoices:aSwamy–AroramethodandsimpleconsistentestimatorsfromBaltagiandChang(2000).
BaltagiandChang(1994)derivedtheSwamy–Aroraestimatorsofthevariancecomponentsforunbalancedpanels.
Bydefault,xtivregusesestimatorsthatextendtheseunbalancedSwamy–Aroraestimatorstothecasewithinstrumentalvariables.
ThedefaultSwamy–Aroramethodcontainsadegree-of-freedomcorrectiontoimproveitsperformanceinsmallsamples.
BaltagiandChang(2000)usevariance-componentsestimators,whicharebasedontheideasofAmemiya(1971)andSwamyandArora(1972),buttheydonotattempttomakesmall-sampleadjustments.
Theseconsistentestimatorsofthevariancecomponentswillbeusedifthenosaoptionisspecied.
Usingeitherestimatorofthevariancecomponents,xtivregofferstwoGLSestimatorsoftherandom-effectsmodel.
ThesetwoestimatorsdifferonlyinhowtheyconstructtheGLSinstrumentsfromtheexogenousandinstrumentalvariablescontainedinXit=[X1itX2it].
Thedefaultmethod,G2SLS,whichisfromBalestraandVaradharajan-Krishnakumar,usestheexogenousvariablesaftertheyhavebeenpassedthroughthefeasibleGLStransform.
Inmath,G2SLSusesXitfortheGLSinstruments,whereXitisconstructedbypassingeachvariableinXitthroughtheGLStransforminxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels11(3)giveninMethodsandformulas.
Iftheec2slsoptionisspecied,xtivregperformsBaltagi'sEC2SLS.
InEC2SLS,theinstrumentsareXitandXit,whereXitisconstructedbypassingeachofthevariablesinXitthroughthewithintransform,andXitisconstructedbypassingeachvariablethroughthebetweentransform.
ThewithinandbetweentransformsaregivenintheMethodsandformulassection.
BaltagiandLi(1992)showthat,althoughtheG2SLSinstrumentsareasubsetofthosecontainedinEC2SLS,theextrainstrumentsinEC2SLSareredundantinthesenseofWhite(2001).
Giventheextracomputationalcost,G2SLSisthedefault.
Example2:GLSrandom-effectsmodelHereistheoutputfromapplyingtheG2SLSestimatortothismodel:.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),reG2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0664min=1between=0.
2098avg=4.
6overall=0.
1463max=12Waldchi2(5)=1446.
37corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
1391798.
007875617.
670.
000.
123744.
1546157age.
0279649.
00541825.
160.
000.
0173454.
0385843c.
age#c.
age-.
0008357.
0000871-9.
600.
000-.
0010063-.
000665not_smsa-.
2235103.
0111371-20.
070.
000-.
2453386-.
2016821raceblack-.
2078613.
0125803-16.
520.
000-.
2325183-.
1832044_cons1.
337684.
084498815.
830.
0001.
1720691.
503299sigma_u.
36582493sigma_e.
63031479rho.
25197078(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthWehaveincludedtwotime-invariantcovariates,birthyrand2.
race.
Allthecoefcientsarestatisticallysignicantandareoftheexpectedsign.
12xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsApplyingtheEC2SLSestimatoryieldssimilarresults:.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),re>ec2slsEC2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0898min=1between=0.
2608avg=4.
6overall=0.
1926max=12Waldchi2(5)=2721.
92corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
064822.
002564725.
270.
000.
0597953.
0698486age.
0380048.
00395499.
610.
000.
0302534.
0457562c.
age#c.
age-.
0006676.
0000632-10.
560.
000-.
0007915-.
0005438not_smsa-.
2298961.
0082993-27.
700.
000-.
2461625-.
2136297raceblack-.
1823627.
0092005-19.
820.
000-.
2003954-.
16433_cons1.
110564.
060653818.
310.
000.
99168491.
229443sigma_u.
36582493sigma_e.
63031479rho.
25197078(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels13FittingthesamemodelasabovewiththeG2SLSestimatorandtheconsistentvariancecomponentsestimatorsyields.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),>renosaG2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0664min=1between=0.
2098avg=4.
6overall=0.
1463max=12Waldchi2(5)=1446.
93corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
1391859.
00787317.
680.
000.
1237552.
1546166age.
0279697.
0054195.
160.
000.
0173486.
0385909c.
age#c.
age-.
0008357.
0000871-9.
600.
000-.
0010064-.
000665not_smsa-.
2235738.
0111344-20.
080.
000-.
2453967-.
2017508raceblack-.
2078733.
0125751-16.
530.
000-.
2325201-.
1832265_cons1.
337522.
084508315.
830.
0001.
1718891.
503155sigma_u.
36535633sigma_e.
63020883rho.
2515512(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthExample3:First-differencedestimatorThetwo-stageleast-squaresrst-differencedestimator(FD2SLS)hasbeenusedtotbothxed-effectandrandom-effectmodels.
Iftheiaretrulyxed-effects,theFD2SLSestimatorisnotasefcientasthetwo-stageleast-squareswithinestimatorforniteTi.
Similarly,ifnoneoftheendogenousvariablesarelaggeddependentvariables,theexogenousvariablesareallstrictlyexogenous,andtherandomeffectsarei.
i.
d.
andindependentoftheXit,thetwo-stageGLSestimatorsaremoreefcientthantheFD2SLSestimator.
However,theFD2SLSestimatorhasbeenusedtoobtainconsistentestimateswhenoneoftheseconditionsfails.
AndersonandHsiao(1981)usedaversionoftheFD2SLSestimatortotapanel-datamodelwithalaggeddependentvariable.
ArellanoandBond(1991)developnewone-stepandtwo-stepGMMestimatorsfordynamicpaneldata.
See[XT]xtabondforadiscussionoftheseestimatorsandStata'simplementationofthem.
Intheirarticle,ArellanoandBond(1991)applytheirnewestimatorstoamodelofdynamiclabordemandthathadpreviouslybeenconsideredbyLayardandNickell(1986).
TheyalsocomparetheresultsoftheirestimatorswiththosefromtheAnderson–HsiaoestimatorusingdatafromanunbalancedpanelofrmsfromtheUnitedKingdom.
Asisconventional,allvariablesareindexedoverthermiandtimet.
Inthisdataset,nitisthelogofemploymentinrmiinsidetheUnitedKingdomattimet,witisthenaturallogoftherealproductwage,kitisthenaturallogofthegrosscapitalstock,andysitisthenaturallogofindustryoutput.
Themodelalsoincludestimedummiesyr1980,yr1981,yr1982,yr1983,andyr1984.
InArellanoandBond(1991,table5,columne),theauthorspresenttheresultsfromapplyingoneversionoftheAnderson–Hsiaoestimatortothesedata.
Thisexamplereproduces14xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelstheirresultsforthecoefcients,thoughstandarderrorsareslightlydifferentbecauseArellanoandBondareusingrobuststandarderrorsfromGMMwhileweobtainourrobuststandarderrorsfrom2SLS.
.
usehttps://www.
stata-press.
com/data/r16/abdata.
xtivregnl2.
nl(0/1).
wl(0/2).
(kys)yr1981-yr1984(l.
n=l3.
n),fdvce(robust)First-differencedIVregressionGroupvariable:idNumberofobs=471Timevariable:yearNumberofgroups=140R-sq:Obspergroup:within=0.
0141min=3between=0.
9165avg=3.
4overall=0.
9892max=5Waldchi2(14)=259.
49corr(u_i,Xb)=0.
9239Prob>chi2=0.
0000(Std.
Err.
adjustedfor140clustersinid)RobustD.
nCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]nLD.
1.
4227651.
0199921.
390.
163-.
57638243.
421913L2D.
-.
1645517.
1300598-1.
270.
206-.
4194643.
0903609wD1.
-.
7524675.
2341305-3.
210.
001-1.
211355-.
29358LD.
.
9627611.
78283581.
230.
219-.
57156882.
497091kD1.
.
3221686.
10666453.
020.
003.
1131099.
5312273LD.
-.
3248778.
3933448-0.
830.
409-1.
095819.
4460637L2D.
-.
0953947.
1257672-0.
760.
448-.
3418938.
1511045ysD1.
.
7660906.
31726642.
410.
016.
144261.
387921LD.
-1.
361881.
8980497-1.
520.
129-3.
122026.
3982639L2D.
.
3212993.
42348350.
760.
448-.
5087131.
151312yr1981D1.
-.
0574197.
0323419-1.
780.
076-.
1208088.
0059693yr1982D1.
-.
0882952.
0580339-1.
520.
128-.
2020395.
0254491yr1983D1.
-.
1063153.
0934136-1.
140.
255-.
2894026.
0767719yr1984D1.
-.
1172108.
1150944-1.
020.
308-.
3427917.
1083701_cons.
0161204.
0253760.
640.
525-.
0336155.
0658564sigma_u.
29069213sigma_e.
34152632rho.
42011045(fractionofvarianceduetou_i)Instrumented:L.
nInstruments:L2.
nwL.
wkL.
kL2.
kysL.
ysL2.
ysyr1981yr1982yr1983yr1984L3.
nxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels15Storedresultsxtivreg,restoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(dfrz)residualdegreesoffreedome(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Tcon)1ifpanelsbalanced,0otherwisee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)χ2e(rho)ρe(F)modelF(smallonly)e(mp)p-valuefrommodelteste(thtamin)minimumθe(thta5)θ,5thpercentilee(thta50)θ,50thpercentilee(thta95)θ,95thpercentilee(thtamax)maximumθe(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)g2slsorec2slse(small)small,ifspeciede(clustvar)nameofclustervariablee(chi2type)Wald;typeofmodelχ2teste(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsample16xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsInadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,bestoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(mss)modelsumofsquarese(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedome(dfrz)residualdegreesoffreedomforthebetween-transformedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(rsa)adjustedR2e(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(Nclust)numberofclusterse(chi2)modelWalde(chi2p)p-valueformodelχ2teste(F)Fstatistic(smallonly)e(rmse)rootmeansquarederrore(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)bee(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsamplextivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels17Inadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,festoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedom(smallonly)e(dfrz)residualdegreesoffreedomforthewithin-transformedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(corr)corr(ui,Xb)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)modelWald(notsmall)e(chi2p)p-valueformodelχ2teste(rho)ρe(F)Fstatistic(smallonly)e(Ff)FforH0:ui=0e(Ffp)p-valueforFforH0:ui=0e(dfa)degreesoffreedomforabsorbedeffecte(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)fee(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctions18xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelse(sample)marksestimationsampleInadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,fdstoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedom(smallonly)e(dfrz)residualdegreesoffreedomforrst-differencedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(corr)corr(ui,Xb)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)modelWald(notsmall)e(chi2p)p-valueformodelχ2teste(rho)ρe(F)Fstatistic(smallonly)e(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)fde(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginsMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsamplextivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels19Inadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
MethodsandformulasConsideranequationoftheformyit=Yitγ+X1itβ+i+νit=Zitδ+i+νit(2)whereyitisthedependentvariable;Yitisan1*g2vectorofobservationsong2endogenousvariablesincludedascovariates,andthesevariablesareallowedtobecorrelatedwiththeνit;X1itisan1*k1vectorofobservationsontheexogenousvariablesincludedascovariates;Zit=[YitXit];γisag2*1vectorofcoefcients;βisak1*1vectorofcoefcients;andδisaK*1vectorofcoefcients,whereK=g2+k1.
Assumethatthereisa1*k2vectorofobservationsonthek2instrumentsinX2it.
Theorderconditionissatisedifk2≥g2.
LetXit=[X1itX2it].
xtivreghandlesexogenouslyunbalancedpaneldata.
ThusdeneTitobethenumberofobservationsonpaneli,ntobethenumberofpanels,andNtobethetotalnumberofobservations;thatis,N=ni=1Ti.
Methodsandformulasarepresentedunderthefollowingheadings:xtivreg,fdxtivreg,fextivreg,bextivreg,rextivreg,fdAsthenameimplies,thisestimatorobtainsitsestimatesandconventionalVCEfromaninstrumental-variablesregressionontherst-differenceddata.
Specically,rstdifferencingthedatayieldsyityit1=(ZitZi,t1)δ+νitνi,t1Withtheiremovedbydifferencing,wecanobtaintheestimatedcoefcientsandtheirestimatedvariance–covariancematrixfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2betweenisreportedascorr(Ziδ,yi)2.
R2overallisreportedascorr(Zitδ,yit)2.
20xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsxtivreg,feAttheheartofthismodelisthewithintransformation.
Thewithintransformofavariablewiswit=witwi.
+wwherewi.
=1nTit=1witw=1Nni=1Tit=1witandnisthenumberofgroupsandNisthetotalnumberofobservationsonthevariable.
Thewithintransformof(2)isyit=Zit+νitThewithintransformhasremovedthei.
Withtheigone,thewithin2SLSestimatorcanbeobtainedfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
SupposethatthereareKvariablesinZit,includingthemandatoryconstant.
ThereareK+n1parametersestimatedinthemodel,andtheconventionalVCEforthewithinestimatorisNKNnK+1VIVwhereVIVistheVCEfromtheabovetwo-stageleast-squaresregression.
Therobustandcluster–robustvariance–covariancematricesaretherobustandcluster–robustvariance–covariancematricesfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
Fromtheestimateofδ,estimatesiofiareobtainedasi=yiZiδ.
ReportedfromthecalculatediisitsstandarddeviationanditscorrelationwithZiδ.
Reportedasthestandarddeviationofνitistheregression'sestimatedrootmeansquarederror,s2,whichisadjusted(aspreviouslystated)forthen1estimatedmeans.
R2withinisreportedastheR2fromthemean-deviatedregression.
R2betweenisreportedascorr(Ziδ,yi)2.
R2overallisreportedascorr(Zitδ,yit)2.
Atthebottomoftheoutput,anFstatisticsagainstthenullhypothesisthatalltheiarezeroisreported.
ThisFstatisticisanapplicationoftheresultsinWooldridge(1990).
xtivreg,beAfterpassing(2)throughthebetweentransform,weareleftwithyi=α+Ziδ+i+νi(3)xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels21wherewi=1TiTit=1witforw∈{y,Z,ν}Similarly,deneXiasthematrixofinstrumentsXitaftertheyhavebeenpassedthroughthebetweentransform.
TheBE2SLSestimatorof(3)obtainsitscoefcientestimatesanditsVCE,atwo-stageleast-squaresregressionofyionZiwithinstrumentsXiinwhicheachaverageappearsTitimes.
R2betweenisreportedastheR2fromthettedregression.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2overallisreportedascorr(Zitδ,yit)2.
xtivreg,rePerBaltagiandChang(2000),letu=i+νitbetheN*1vectorofcombinederrors.
Thenundertheassumptionsoftherandom-effectsmodel,E(uu)=σ2νdiagITi1TiιTiιTi+diagwi1TiιTiιTiwhereωi=Tiσ2+σ2νandιTiisavectorofonesofdimensionTi.
Becausethevariancecomponentsareunknown,consistentestimatesarerequiredtoimplementfeasibleGLS.
xtivregofferstwochoices.
ThedefaultisasimpleextensionoftheSwamy–Aroramethodforunbalancedpanels.
Letuwit=yitZitδwbethecombinedresidualsfromthewithinestimator.
Letuitbethewithin-transformeduit.
Thenσν=ni=1Tit=1u2itNnK+1Letubit=yitZitδbbethecombinedresidualfromthebetweenestimator.
Letubi.
bethebetweenresidualsaftertheyhavebeenpassedthroughthebetweentransform.
Thenσ2=ni=1Tit=1u2it(nK)σ2νNr22xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelswherer=traceZiZi1ZiZZZiwhereZ=diagιTiιTiIfthenosaoptionisspecied,theconsistentestimatorsdescribedinBaltagiandChang(2000)areused.
Thesearegivenbyσν=ni=1Tit=1u2itNnandσ2=ni=1Tit=1u2itnσ2νNThedefaultSwamy–Aroramethodcontainsadegree-of-freedomcorrectiontoimproveitsperformanceinsmallsamples.
Givenestimatesofthevariancecomponents,σ2νandσ2,thefeasibleGLStransformofavariablewisw=witθitwi.
(4)wherewi.
=1TiTit=1witθit=1σ2νωi12andωi=Tiσ2+σ2νUsingeitherestimatorofthevariancecomponents,xtivregcontainstwoGLSestimatorsoftherandom-effectsmodel.
ThesetwoestimatorsdifferonlyinhowtheyconstructtheGLSinstrumentsfromtheexogenousandinstrumentalvariablescontainedinXit=[X1itX2it].
Thedefaultmethod,G2SLS,whichisfromBalestraandVaradharajan-Krishnakumar,usestheexogenousvariablesaftertheyhavebeenpassedthroughthefeasibleGLStransform.
Mathematically,G2SLSusesXfortheGLSinstruments,whereXisconstructedbypassingeachvariableinXthoughtheGLStransformin(4).
TheG2SLSestimatorobtainsitscoefcientestimatesandVCEfromaninstrumentalvariableregressionofyitonZitwithinstrumentsXit.
Iftheec2slsoptionisspecied,xtivregperformsBaltagi'sEC2SLS.
InEC2SLS,theinstrumentsareXitandXit,whereXitisconstructedbyeachofthevariablesinXitthroughouttheGLStransformin(4),andXitismadeofthegroupmeansofeachvariableinXit.
TheEC2SLSestimatorobtainsitscoefcientestimatesanditsVCEfromaninstrumentalvariablesregressionofyitonZitwithinstrumentsXitandXit.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels23BaltagiandLi(1992)showthatalthoughtheG2SLSinstrumentsareasubsetofthoseinEC2SLS,theextrainstrumentsinEC2SLSareredundantinthesenseofWhite(2001).
Giventheextracomputationalcost,G2SLSisthedefault.
Thestandarddeviationofi+νitiscalculatedasσ2+σ2ν.
R2betweenisreportedascorr(Ziδ,yi)2.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2overallisreportedascorr(Zitδ,yit)2.
AcknowledgmentWethankMeadOveroftheCenterforGlobalDevelopment,whowroteanearlyimplementationofxtivreg.
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,andS.
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24xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsAlsosee[XT]xtivregpostestimation—Postestimationtoolsforxtivreg[XT]xtabond—Arellano–Bondlineardynamicpanel-dataestimation[XT]xteregress—Extendedrandom-effectslinearregression[XT]xthtaylor—Hausman–Taylorestimatorforerror-componentsmodels[XT]xtreg—Fixed-,between-,andrandom-effectsandpopulation-averagedlinearmodels[XT]xtset—Declaredatatobepaneldata[R]ivregress—Single-equationinstrumental-variablesregression[U]20Estimationandpostestimationcommands
comxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsDescriptionQuickstartMenuSyntaxOptionsforREmodelOptionsforBEmodelOptionsforFEmodelOptionsforFDmodelRemarksandexamplesStoredresultsMethodsandformulasAcknowledgmentReferencesAlsoseeDescriptionxtivregoffersvedifferentestimatorsforttingpanel-datamodelsinwhichsomeoftheright-hand-sidecovariatesareendogenous.
Theseestimatorsaretwo-stageleast-squaresgeneralizationsofsimplepanel-dataestimatorsforexogenousvariables.
xtivregwiththebeoptionusesthetwo-stageleast-squaresbetweenestimator.
xtivregwiththefeoptionusesthetwo-stageleast-squareswithinestimator.
xtivregwiththereoptionusesatwo-stageleast-squaresrandom-effectsestimator.
Therearetwoimplementations:G2SLSfromBalestraandVaradharajan-Krishnakumar(1987)andEC2SLSfromBaltagi.
TheBalestraandVaradharajan-KrishnakumarG2SLSisthedefaultbecauseitiscomputationallylessexpensive.
Baltagi'sEC2SLScanbeobtainedbyspecifyingtheec2slsoption.
xtivregwiththefdoptionrequeststhetwo-stageleast-squaresrst-differencedestimator.
SeeBaltagi(2013)foranintroductiontopanel-datamodelswithendogenouscovariates.
Forthederivationandapplicationoftherst-differencedestimator,seeAndersonandHsiao(1981).
QuickstartRandom-effectslinearpanel-datamodelwithoutcomey,exogenousx1,andx2instrumentedbyx3usingxtsetdataxtivregyx1(x2=x3)Usexed-effectsestimatorandincludeindicatorsforeachlevelofcategoricalvariableaxtivregyx1i.
a(x2=x3),feUsebetween-effectsestimatorandincludeindicatorsforlevelsofbasinstrumentsxtivregyx1i.
a(x2=x3i.
b),beFirst-differencedmodelofyasafunctionofx1andx2andthelagofyinstrumentedbyitsthirdlagxtivregyx1x2(L.
y=L3.
y),fdMenuStatistics>Longitudinal/paneldata>Endogenouscovariates>Instrumental-variablesregression(FE,RE,BE,FD)12xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsSyntaxGLSrandom-effects(RE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,reREoptionsBetween-effects(BE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,beBEoptionsFixed-effects(FE)modelxtivregdepvarvarlist1(varlist2=varlistiv)ifin,feFEoptionsFirst-differenced(FD)estimatorxtivregdepvarvarlist1(varlist2=varlistiv)ifin,fdFDoptionsREoptionsDescriptionModelreuserandom-effectsestimator;thedefaultec2slsuseBaltagi'sEC2SLSrandom-effectsestimatornosausetheBaltagi–ChangestimatorsofthevariancecomponentsregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsthetareportθdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatisticsxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels3BEoptionsDescriptionModelbeusebetween-effectsestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatisticsFEoptionsDescriptionModelfeusexed-effectsestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,displayofomittedvariablesandbaseandemptycells,andfactor-variablelabelingcoeflegenddisplaylegendinsteadofstatistics4xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsFDoptionsDescriptionModelnoconstantsuppressconstanttermfduserst-differencedestimatorregresstreatcovariatesasexogenousandignoreinstrumentalvariablesSE/Robustvce(vcetype)vcetypemaybeconventional,robust,clusterclustvar,bootstrap,orjackknifeReportinglevel(#)setcondencelevel;defaultislevel(95)firstreportrst-stageestimatessmallreporttandFstatisticsinsteadofZandχ2statisticsdisplayoptionscontrolcolumnsandcolumnformats,rowspacing,linewidth,anddisplayofomittedvariablescoeflegenddisplaylegendinsteadofstatisticsApanelvariablemustbespecied.
Forxtivreg,fd,atimevariablemustalsobespecied.
Usextset;see[XT]xtset.
varlist1andvarlistivmaycontainfactorvariables,exceptforthefdestimator;see[U]11.
4.
3Factorvariables.
depvar,varlist1,varlist2,andvarlistivmaycontaintime-seriesoperators;see[U]11.
4.
4Time-seriesvarlists.
byandstatsbyareallowed;see[U]11.
1.
10Prexcommands.
coeflegenddoesnotappearinthedialogbox.
See[U]20Estimationandpostestimationcommandsformorecapabilitiesofestimationcommands.
OptionsforREmodelModelrerequeststheG2SLSrandom-effectsestimator.
reisthedefault.
ec2slsrequestsBaltagi'sEC2SLSrandom-effectsestimatorinsteadofthedefaultBalestraandVaradharajan-Krishnakumarestimator.
nosaspeciesthattheBaltagi–ChangestimatorsofthevariancecomponentsbeusedinsteadofthedefaultadaptedSwamy–Aroraestimators.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels5Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,reinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
thetaspeciesthattheoutputincludetheestimatedvalueofθusedincombiningthebetweenandxedestimators.
Forbalanceddata,thisisaconstant,andforunbalanceddata,asummaryofthevaluesispresentedintheheaderoftheoutput.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforBEmodelModelberequeststhebetweenregressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
6xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsThefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforFEmodelModelferequeststhexed-effects(within)regressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,noemptycells,baselevels,allbaselevels,nofvlabel,fvwrap(#),fvwrapon(style),cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
OptionsforFDmodelModelnoconstant;see[R]Estimationoptions.
fdrequeststherst-differencedregressionestimator.
regressspeciesthatallthecovariatesbetreatedasexogenousandthattheinstrumentlistbeignored.
Specifyingregresscausesxtivregtottherequestedpanel-dataregressionmodelofdepvaronvarlist1andvarlist2,ignoringvarlistiv.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels7SE/Robustvce(vcetype)speciesthetypeofstandarderrorreported,whichincludestypesthatarederivedfromasymptotictheory(conventional),thatarerobusttosomekindsofmisspecication(robust),thatallowforintragroupcorrelation(clusterclustvar),andthatusebootstraporjackknifemethods(bootstrap,jackknife);see[XT]vceoptions.
vce(conventional),thedefault,usestheconventionallyderivedvarianceestimatorforgeneralizedleast-squaresregression.
Specifyingvce(robust)isequivalenttospecifyingvce(clusterpanelvar);seextreg,feinMethodsandformulasof[XT]xtreg.
Reportinglevel(#);see[R]Estimationoptions.
firstspeciesthattherst-stageregressionsbedisplayed.
smallspeciesthattstatisticsbereportedinsteadofZstatisticsandthatFstatisticsbereportedinsteadofχ2statistics.
displayoptions:noci,nopvalues,noomitted,vsquish,cformat(%fmt),pformat(%fmt),sformat(%fmt),andnolstretch;see[R]Estimationoptions.
Thefollowingoptionisavailablewithxtivregbutisnotshowninthedialogbox:coeflegend;see[R]Estimationoptions.
Remarksandexamplesstata.
comIfyouhavenotread[XT]xt,pleasedoso.
Consideranequationoftheformyit=Yitγ+X1itβ+i+νit=Zitδ+i+νit(1)whereyitisthedependentvariable;Yitisan1*g2vectorofobservationsong2endogenousvariablesincludedascovariates,andthesevariablesareallowedtobecorrelatedwiththeνit;X1itisan1*k1vectorofobservationsontheexogenousvariablesincludedascovariates;Zit=[YitXit];γisag2*1vectorofcoefcients;βisak1*1vectorofcoefcients;andδisaK*1vectorofcoefcients,whereK=g2+k1.
Assumethatthereisa1*k2vectorofobservationsonthek2instrumentsinX2it.
Theorderconditionissatisedifk2≥g2.
LetXit=[X1itX2it].
xtivreghandlesexogenouslyunbalancedpaneldata.
ThusdeneTitobethenumberofobservationsonpaneli,ntobethenumberofpanelsandNtobethetotalnumberofobservations;thatis,N=ni=1Ti.
8xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsxtivregoffersvedifferentestimators,whichmaybeappliedtomodelshavingtheformof(1).
Therst-differencedestimator(FD2SLS)removestheibyttingthemodelinrstdifferences.
Thewithinestimator(FE2SLS)tsthemodelaftersweepingouttheibyremovingthepanel-levelmeansfromeachvariable.
Thebetweenestimator(BE2SLS)modelsthepanelaverages.
Thetworandom-effectsestimators,G2SLSandEC2SLS,treattheiasrandomvariablesthatareindependentandidenticallydistributed(i.
i.
d.
)overthepanels.
Exceptfor(FD2SLS),alloftheseestimatorsaregeneralizationsofestimatorsinxtreg.
See[XT]xtregforadiscussionoftheseestimatorsforexogenouscovariates.
Althoughtheestimatorsallowfordifferentassumptionsaboutthei,alltheestimatorsassumethattheidiosyncraticerrortermνithaszeromeanandisuncorrelatedwiththevariablesinXit.
Justaswhentherearenoendogenouscovariates,asdiscussedin[XT]xtreg,therearevariousperspectivesonwhatassumptionsshouldbeplacedonthei.
Iftheyareassumedtobexed,theimaybecorrelatedwiththevariablesinXit,andthewithinestimatorisefcientwithinaclassoflimitedinformationestimators.
Alternatively,iftheiareassumedtoberandom,theyarealsoassumedtobei.
i.
d.
overthepanels.
IftheiareassumedtobeuncorrelatedwiththevariablesinXit,theGLSrandom-effectsestimatorsaremoreefcientthanthewithinestimator.
However,iftheiarecorrelatedwiththevariablesinXit,therandom-effectsestimatorsareinconsistentbutthewithinestimatorisconsistent.
Thepriceofusingthewithinestimatoristhatitisnotpossibletoestimatecoefcientsontime-invariantvariables,andallinferenceisconditionalontheiinthesample.
SeeMundlak(1978)andHsiao(2014)fordiscussionsofthisinterpretationofthewithinestimator.
Example1:Fixed-effectsmodelForthewithinestimator,consideranotherversionofthewageequationdiscussedin[XT]xtreg.
ThedataforthisexamplecomefromanextractofwomenfromtheNationalLongitudinalSurveyofYouththatwasdescribedindetailin[XT]xt.
Restrictingourselvestoonlytime-varyingcovariates,wemightsupposethatthelogoftherealwagewasafunctionoftheindividual'sage,age2,hertenureintheobservedplaceofemployment,whethershebelongedtounion,whethershelivesinmetropolitanarea,andwhethershelivesinthesouth.
Thevariablesfortheseare,respectively,age,c.
age#c.
age,tenure,union,notsmsa,andsouth.
Ifwetreatallthevariablesasexogenous,wecanusetheone-stagewithinestimatorfromxtreg,yieldingxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels9.
usehttps://www.
stata-press.
com/data/r16/nlswork(NationalLongitudinalSurvey.
YoungWomen14-26yearsofagein1968).
xtregln_wagec.
age#c.
agetenurenot_smsaunionsouth,feFixed-effects(within)regressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
1333min=1between=0.
2375avg=4.
6overall=0.
2031max=12F(6,14867)=381.
19corr(u_i,Xb)=0.
2074Prob>F=0.
0000ln_wageCoef.
Std.
Err.
tP>|t|[95%Conf.
Interval]age.
0311984.
00339029.
200.
000.
0245533.
0378436c.
age#c.
age-.
0003457.
0000543-6.
370.
000-.
0004522-.
0002393tenure.
0176205.
000809921.
760.
000.
0160331.
0192079not_smsa-.
0972535.
0125377-7.
760.
000-.
1218289-.
072678union.
0975672.
006984413.
970.
000.
0838769.
1112576south-.
0620932.
013327-4.
660.
000-.
0882158-.
0359706_cons1.
091612.
052312620.
870.
000.
98907291.
194151sigma_u.
3910683sigma_e.
25545969rho.
70091004(fractionofvarianceduetou_i)Ftestthatallu_i=0:F(4133,14867)=8.
31Prob>F=0.
0000Allthecoefcientsarestatisticallysignicantandhavetheexpectedsigns.
Nowsupposethatwewishtomodeltenureasafunctionofunionandsouthandthatwebelievethattheerrorsinthetwoequationsarecorrelated.
Becausewearestillinterestedinthewithinestimates,wenowneedatwo-stageleast-squaresestimator.
Thefollowingoutputshowsthecommandandtheresultsfromttingthismodel:10xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels.
xtivregln_wagec.
age#c.
agenot_smsa(tenure=unionsouth),feFixed-effects(within)IVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=.
min=1between=0.
1304avg=4.
6overall=0.
0897max=12Waldchi2(4)=147926.
58corr(u_i,Xb)=-0.
6843Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
2403531.
03734196.
440.
000.
1671643.
3135419age.
0118437.
00900321.
320.
188-.
0058023.
0294897c.
age#c.
age-.
0012145.
0001968-6.
170.
000-.
0016003-.
0008286not_smsa-.
0167178.
0339236-0.
490.
622-.
0832069.
0497713_cons1.
678287.
162665710.
320.
0001.
3594681.
997106sigma_u.
70661941sigma_e.
63029359rho.
55690561(fractionofvarianceduetou_i)Ftestthatallu_i=0:F(4133,14869)=1.
44Prob>F=0.
0000Instrumented:tenureInstruments:agec.
age#c.
agenot_smsaunionsouthAlthoughallthecoefcientsstillhavetheexpectedsigns,thecoefcientsonageandnotsmsaarenolongerstatisticallysignicant.
Giventhatthesevariableshavebeenfoundtobeimportantinmanyotherstudies,wemightwanttorethinkourspecication.
Ifwearewillingtoassumethattheiareuncorrelatedwiththeothercovariates,wecantarandom-effectsmodel.
Themodelisfrequentlyknownasthevariance-componentsorerror-componentsmodel.
xtivreghasestimatorsfortwo-stageleast-squaresone-wayerror-componentsmodels.
Intheone-wayframework,therearetwovariancecomponentstoestimate,thevarianceoftheiandthevarianceoftheνit.
Becausethevariancecomponentsareunknown,consistentestimatesarerequiredtoimplementfeasibleGLS.
xtivregofferstwochoices:aSwamy–AroramethodandsimpleconsistentestimatorsfromBaltagiandChang(2000).
BaltagiandChang(1994)derivedtheSwamy–Aroraestimatorsofthevariancecomponentsforunbalancedpanels.
Bydefault,xtivregusesestimatorsthatextendtheseunbalancedSwamy–Aroraestimatorstothecasewithinstrumentalvariables.
ThedefaultSwamy–Aroramethodcontainsadegree-of-freedomcorrectiontoimproveitsperformanceinsmallsamples.
BaltagiandChang(2000)usevariance-componentsestimators,whicharebasedontheideasofAmemiya(1971)andSwamyandArora(1972),buttheydonotattempttomakesmall-sampleadjustments.
Theseconsistentestimatorsofthevariancecomponentswillbeusedifthenosaoptionisspecied.
Usingeitherestimatorofthevariancecomponents,xtivregofferstwoGLSestimatorsoftherandom-effectsmodel.
ThesetwoestimatorsdifferonlyinhowtheyconstructtheGLSinstrumentsfromtheexogenousandinstrumentalvariablescontainedinXit=[X1itX2it].
Thedefaultmethod,G2SLS,whichisfromBalestraandVaradharajan-Krishnakumar,usestheexogenousvariablesaftertheyhavebeenpassedthroughthefeasibleGLStransform.
Inmath,G2SLSusesXitfortheGLSinstruments,whereXitisconstructedbypassingeachvariableinXitthroughtheGLStransforminxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels11(3)giveninMethodsandformulas.
Iftheec2slsoptionisspecied,xtivregperformsBaltagi'sEC2SLS.
InEC2SLS,theinstrumentsareXitandXit,whereXitisconstructedbypassingeachofthevariablesinXitthroughthewithintransform,andXitisconstructedbypassingeachvariablethroughthebetweentransform.
ThewithinandbetweentransformsaregivenintheMethodsandformulassection.
BaltagiandLi(1992)showthat,althoughtheG2SLSinstrumentsareasubsetofthosecontainedinEC2SLS,theextrainstrumentsinEC2SLSareredundantinthesenseofWhite(2001).
Giventheextracomputationalcost,G2SLSisthedefault.
Example2:GLSrandom-effectsmodelHereistheoutputfromapplyingtheG2SLSestimatortothismodel:.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),reG2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0664min=1between=0.
2098avg=4.
6overall=0.
1463max=12Waldchi2(5)=1446.
37corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
1391798.
007875617.
670.
000.
123744.
1546157age.
0279649.
00541825.
160.
000.
0173454.
0385843c.
age#c.
age-.
0008357.
0000871-9.
600.
000-.
0010063-.
000665not_smsa-.
2235103.
0111371-20.
070.
000-.
2453386-.
2016821raceblack-.
2078613.
0125803-16.
520.
000-.
2325183-.
1832044_cons1.
337684.
084498815.
830.
0001.
1720691.
503299sigma_u.
36582493sigma_e.
63031479rho.
25197078(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthWehaveincludedtwotime-invariantcovariates,birthyrand2.
race.
Allthecoefcientsarestatisticallysignicantandareoftheexpectedsign.
12xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsApplyingtheEC2SLSestimatoryieldssimilarresults:.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),re>ec2slsEC2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0898min=1between=0.
2608avg=4.
6overall=0.
1926max=12Waldchi2(5)=2721.
92corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
064822.
002564725.
270.
000.
0597953.
0698486age.
0380048.
00395499.
610.
000.
0302534.
0457562c.
age#c.
age-.
0006676.
0000632-10.
560.
000-.
0007915-.
0005438not_smsa-.
2298961.
0082993-27.
700.
000-.
2461625-.
2136297raceblack-.
1823627.
0092005-19.
820.
000-.
2003954-.
16433_cons1.
110564.
060653818.
310.
000.
99168491.
229443sigma_u.
36582493sigma_e.
63031479rho.
25197078(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels13FittingthesamemodelasabovewiththeG2SLSestimatorandtheconsistentvariancecomponentsestimatorsyields.
xtivregln_wagec.
age#c.
agenot_smsa2.
race(tenure=unionbirthsouth),>renosaG2SLSrandom-effectsIVregressionNumberofobs=19,007Groupvariable:idcodeNumberofgroups=4,134R-sq:Obspergroup:within=0.
0664min=1between=0.
2098avg=4.
6overall=0.
1463max=12Waldchi2(5)=1446.
93corr(u_i,X)=0(assumed)Prob>chi2=0.
0000ln_wageCoef.
Std.
Err.
zP>|z|[95%Conf.
Interval]tenure.
1391859.
00787317.
680.
000.
1237552.
1546166age.
0279697.
0054195.
160.
000.
0173486.
0385909c.
age#c.
age-.
0008357.
0000871-9.
600.
000-.
0010064-.
000665not_smsa-.
2235738.
0111344-20.
080.
000-.
2453967-.
2017508raceblack-.
2078733.
0125751-16.
530.
000-.
2325201-.
1832265_cons1.
337522.
084508315.
830.
0001.
1718891.
503155sigma_u.
36535633sigma_e.
63020883rho.
2515512(fractionofvarianceduetou_i)Instrumented:tenureInstruments:agec.
age#c.
agenot_smsa2.
raceunionbirth_yrsouthExample3:First-differencedestimatorThetwo-stageleast-squaresrst-differencedestimator(FD2SLS)hasbeenusedtotbothxed-effectandrandom-effectmodels.
Iftheiaretrulyxed-effects,theFD2SLSestimatorisnotasefcientasthetwo-stageleast-squareswithinestimatorforniteTi.
Similarly,ifnoneoftheendogenousvariablesarelaggeddependentvariables,theexogenousvariablesareallstrictlyexogenous,andtherandomeffectsarei.
i.
d.
andindependentoftheXit,thetwo-stageGLSestimatorsaremoreefcientthantheFD2SLSestimator.
However,theFD2SLSestimatorhasbeenusedtoobtainconsistentestimateswhenoneoftheseconditionsfails.
AndersonandHsiao(1981)usedaversionoftheFD2SLSestimatortotapanel-datamodelwithalaggeddependentvariable.
ArellanoandBond(1991)developnewone-stepandtwo-stepGMMestimatorsfordynamicpaneldata.
See[XT]xtabondforadiscussionoftheseestimatorsandStata'simplementationofthem.
Intheirarticle,ArellanoandBond(1991)applytheirnewestimatorstoamodelofdynamiclabordemandthathadpreviouslybeenconsideredbyLayardandNickell(1986).
TheyalsocomparetheresultsoftheirestimatorswiththosefromtheAnderson–HsiaoestimatorusingdatafromanunbalancedpanelofrmsfromtheUnitedKingdom.
Asisconventional,allvariablesareindexedoverthermiandtimet.
Inthisdataset,nitisthelogofemploymentinrmiinsidetheUnitedKingdomattimet,witisthenaturallogoftherealproductwage,kitisthenaturallogofthegrosscapitalstock,andysitisthenaturallogofindustryoutput.
Themodelalsoincludestimedummiesyr1980,yr1981,yr1982,yr1983,andyr1984.
InArellanoandBond(1991,table5,columne),theauthorspresenttheresultsfromapplyingoneversionoftheAnderson–Hsiaoestimatortothesedata.
Thisexamplereproduces14xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelstheirresultsforthecoefcients,thoughstandarderrorsareslightlydifferentbecauseArellanoandBondareusingrobuststandarderrorsfromGMMwhileweobtainourrobuststandarderrorsfrom2SLS.
.
usehttps://www.
stata-press.
com/data/r16/abdata.
xtivregnl2.
nl(0/1).
wl(0/2).
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0141min=3between=0.
9165avg=3.
4overall=0.
9892max=5Waldchi2(14)=259.
49corr(u_i,Xb)=0.
9239Prob>chi2=0.
0000(Std.
Err.
adjustedfor140clustersinid)RobustD.
nCoef.
Std.
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zP>|z|[95%Conf.
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0199921.
390.
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421913L2D.
-.
1645517.
1300598-1.
270.
206-.
4194643.
0903609wD1.
-.
7524675.
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210.
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.
9627611.
78283581.
230.
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57156882.
497091kD1.
.
3221686.
10666453.
020.
003.
1131099.
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-.
3248778.
3933448-0.
830.
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4460637L2D.
-.
0953947.
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448-.
3418938.
1511045ysD1.
.
7660906.
31726642.
410.
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144261.
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.
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5087131.
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-.
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-.
0882952.
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-.
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140.
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-.
1172108.
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640.
525-.
0336155.
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29069213sigma_e.
34152632rho.
42011045(fractionofvarianceduetou_i)Instrumented:L.
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ysL2.
ysyr1981yr1982yr1983yr1984L3.
nxtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels15Storedresultsxtivreg,restoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(dfrz)residualdegreesoffreedome(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Tcon)1ifpanelsbalanced,0otherwisee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)χ2e(rho)ρe(F)modelF(smallonly)e(mp)p-valuefrommodelteste(thtamin)minimumθe(thta5)θ,5thpercentilee(thta50)θ,50thpercentilee(thta95)θ,95thpercentilee(thtamax)maximumθe(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)g2slsorec2slse(small)small,ifspeciede(clustvar)nameofclustervariablee(chi2type)Wald;typeofmodelχ2teste(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsample16xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsInadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,bestoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(mss)modelsumofsquarese(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedome(dfrz)residualdegreesoffreedomforthebetween-transformedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(rsa)adjustedR2e(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(Nclust)numberofclusterse(chi2)modelWalde(chi2p)p-valueformodelχ2teste(F)Fstatistic(smallonly)e(rmse)rootmeansquarederrore(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)bee(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsamplextivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels17Inadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,festoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedom(smallonly)e(dfrz)residualdegreesoffreedomforthewithin-transformedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(corr)corr(ui,Xb)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)modelWald(notsmall)e(chi2p)p-valueformodelχ2teste(rho)ρe(F)Fstatistic(smallonly)e(Ff)FforH0:ui=0e(Ffp)p-valueforFforH0:ui=0e(dfa)degreesoffreedomforabsorbedeffecte(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)fee(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginse(marginsnotok)predictionsdisallowedbymarginse(asbalanced)factorvariablesfvsetasasbalancede(asobserved)factorvariablesfvsetasasobservedMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctions18xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelse(sample)marksestimationsampleInadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
xtivreg,fdstoresthefollowingine():Scalarse(N)numberofobservationse(Ng)numberofgroupse(dfm)modeldegreesoffreedome(rss)residualsumofsquarese(dfr)residualdegreesoffreedom(smallonly)e(dfrz)residualdegreesoffreedomforrst-differencedregressione(gmin)smallestgroupsizee(gavg)averagegroupsizee(gmax)largestgroupsizee(Nclust)numberofclusterse(sigma)ancillaryparameter(gamma,lnormal)e(corr)corr(ui,Xb)e(sigmau)panel-levelstandarddeviatione(sigmae)standarddeviationofite(r2w)R-squaredforwithinmodele(r2o)R-squaredforoverallmodele(r2b)R-squaredforbetweenmodele(chi2)modelWald(notsmall)e(chi2p)p-valueformodelχ2teste(rho)ρe(F)Fstatistic(smallonly)e(rank)rankofe(V)Macrose(cmd)xtivrege(cmdline)commandastypede(depvar)nameofdependentvariablee(ivar)variabledenotinggroupse(tvar)variabledenotingtimewithingroupse(insts)instrumentse(instd)instrumentedvariablese(model)fde(small)small,ifspeciede(clustvar)nameofclustervariablee(vce)vcetypespeciedinvce()e(vcetype)titleusedtolabelStd.
Err.
e(properties)bVe(predict)programusedtoimplementpredicte(marginsok)predictionsallowedbymarginsMatricese(b)coefcientvectore(V)variance–covariancematrixoftheestimatorse(Vmodelbased)model-basedvarianceFunctionse(sample)marksestimationsamplextivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels19Inadditiontotheabove,thefollowingisstoredinr():Matricesr(table)matrixcontainingthecoefcientswiththeirstandarderrors,teststatistics,p-values,andcondenceintervalsNotethatresultsstoredinr()areupdatedwhenthecommandisreplayedandwillbereplacedwhenanyr-classcommandisrunaftertheestimationcommand.
MethodsandformulasConsideranequationoftheformyit=Yitγ+X1itβ+i+νit=Zitδ+i+νit(2)whereyitisthedependentvariable;Yitisan1*g2vectorofobservationsong2endogenousvariablesincludedascovariates,andthesevariablesareallowedtobecorrelatedwiththeνit;X1itisan1*k1vectorofobservationsontheexogenousvariablesincludedascovariates;Zit=[YitXit];γisag2*1vectorofcoefcients;βisak1*1vectorofcoefcients;andδisaK*1vectorofcoefcients,whereK=g2+k1.
Assumethatthereisa1*k2vectorofobservationsonthek2instrumentsinX2it.
Theorderconditionissatisedifk2≥g2.
LetXit=[X1itX2it].
xtivreghandlesexogenouslyunbalancedpaneldata.
ThusdeneTitobethenumberofobservationsonpaneli,ntobethenumberofpanels,andNtobethetotalnumberofobservations;thatis,N=ni=1Ti.
Methodsandformulasarepresentedunderthefollowingheadings:xtivreg,fdxtivreg,fextivreg,bextivreg,rextivreg,fdAsthenameimplies,thisestimatorobtainsitsestimatesandconventionalVCEfromaninstrumental-variablesregressionontherst-differenceddata.
Specically,rstdifferencingthedatayieldsyityit1=(ZitZi,t1)δ+νitνi,t1Withtheiremovedbydifferencing,wecanobtaintheestimatedcoefcientsandtheirestimatedvariance–covariancematrixfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2betweenisreportedascorr(Ziδ,yi)2.
R2overallisreportedascorr(Zitδ,yit)2.
20xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsxtivreg,feAttheheartofthismodelisthewithintransformation.
Thewithintransformofavariablewiswit=witwi.
+wwherewi.
=1nTit=1witw=1Nni=1Tit=1witandnisthenumberofgroupsandNisthetotalnumberofobservationsonthevariable.
Thewithintransformof(2)isyit=Zit+νitThewithintransformhasremovedthei.
Withtheigone,thewithin2SLSestimatorcanbeobtainedfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
SupposethatthereareKvariablesinZit,includingthemandatoryconstant.
ThereareK+n1parametersestimatedinthemodel,andtheconventionalVCEforthewithinestimatorisNKNnK+1VIVwhereVIVistheVCEfromtheabovetwo-stageleast-squaresregression.
Therobustandcluster–robustvariance–covariancematricesaretherobustandcluster–robustvariance–covariancematricesfromatwo-stageleast-squaresregressionofyitonZitwithinstrumentsXit.
Fromtheestimateofδ,estimatesiofiareobtainedasi=yiZiδ.
ReportedfromthecalculatediisitsstandarddeviationanditscorrelationwithZiδ.
Reportedasthestandarddeviationofνitistheregression'sestimatedrootmeansquarederror,s2,whichisadjusted(aspreviouslystated)forthen1estimatedmeans.
R2withinisreportedastheR2fromthemean-deviatedregression.
R2betweenisreportedascorr(Ziδ,yi)2.
R2overallisreportedascorr(Zitδ,yit)2.
Atthebottomoftheoutput,anFstatisticsagainstthenullhypothesisthatalltheiarezeroisreported.
ThisFstatisticisanapplicationoftheresultsinWooldridge(1990).
xtivreg,beAfterpassing(2)throughthebetweentransform,weareleftwithyi=α+Ziδ+i+νi(3)xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels21wherewi=1TiTit=1witforw∈{y,Z,ν}Similarly,deneXiasthematrixofinstrumentsXitaftertheyhavebeenpassedthroughthebetweentransform.
TheBE2SLSestimatorof(3)obtainsitscoefcientestimatesanditsVCE,atwo-stageleast-squaresregressionofyionZiwithinstrumentsXiinwhicheachaverageappearsTitimes.
R2betweenisreportedastheR2fromthettedregression.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2overallisreportedascorr(Zitδ,yit)2.
xtivreg,rePerBaltagiandChang(2000),letu=i+νitbetheN*1vectorofcombinederrors.
Thenundertheassumptionsoftherandom-effectsmodel,E(uu)=σ2νdiagITi1TiιTiιTi+diagwi1TiιTiιTiwhereωi=Tiσ2+σ2νandιTiisavectorofonesofdimensionTi.
Becausethevariancecomponentsareunknown,consistentestimatesarerequiredtoimplementfeasibleGLS.
xtivregofferstwochoices.
ThedefaultisasimpleextensionoftheSwamy–Aroramethodforunbalancedpanels.
Letuwit=yitZitδwbethecombinedresidualsfromthewithinestimator.
Letuitbethewithin-transformeduit.
Thenσν=ni=1Tit=1u2itNnK+1Letubit=yitZitδbbethecombinedresidualfromthebetweenestimator.
Letubi.
bethebetweenresidualsaftertheyhavebeenpassedthroughthebetweentransform.
Thenσ2=ni=1Tit=1u2it(nK)σ2νNr22xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelswherer=traceZiZi1ZiZZZiwhereZ=diagιTiιTiIfthenosaoptionisspecied,theconsistentestimatorsdescribedinBaltagiandChang(2000)areused.
Thesearegivenbyσν=ni=1Tit=1u2itNnandσ2=ni=1Tit=1u2itnσ2νNThedefaultSwamy–Aroramethodcontainsadegree-of-freedomcorrectiontoimproveitsperformanceinsmallsamples.
Givenestimatesofthevariancecomponents,σ2νandσ2,thefeasibleGLStransformofavariablewisw=witθitwi.
(4)wherewi.
=1TiTit=1witθit=1σ2νωi12andωi=Tiσ2+σ2νUsingeitherestimatorofthevariancecomponents,xtivregcontainstwoGLSestimatorsoftherandom-effectsmodel.
ThesetwoestimatorsdifferonlyinhowtheyconstructtheGLSinstrumentsfromtheexogenousandinstrumentalvariablescontainedinXit=[X1itX2it].
Thedefaultmethod,G2SLS,whichisfromBalestraandVaradharajan-Krishnakumar,usestheexogenousvariablesaftertheyhavebeenpassedthroughthefeasibleGLStransform.
Mathematically,G2SLSusesXfortheGLSinstruments,whereXisconstructedbypassingeachvariableinXthoughtheGLStransformin(4).
TheG2SLSestimatorobtainsitscoefcientestimatesandVCEfromaninstrumentalvariableregressionofyitonZitwithinstrumentsXit.
Iftheec2slsoptionisspecied,xtivregperformsBaltagi'sEC2SLS.
InEC2SLS,theinstrumentsareXitandXit,whereXitisconstructedbyeachofthevariablesinXitthroughouttheGLStransformin(4),andXitismadeofthegroupmeansofeachvariableinXit.
TheEC2SLSestimatorobtainsitscoefcientestimatesanditsVCEfromaninstrumentalvariablesregressionofyitonZitwithinstrumentsXitandXit.
xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodels23BaltagiandLi(1992)showthatalthoughtheG2SLSinstrumentsareasubsetofthoseinEC2SLS,theextrainstrumentsinEC2SLSareredundantinthesenseofWhite(2001).
Giventheextracomputationalcost,G2SLSisthedefault.
Thestandarddeviationofi+νitiscalculatedasσ2+σ2ν.
R2betweenisreportedascorr(Ziδ,yi)2.
R2withinisreportedascorr(ZitZi)δ,yityi2.
R2overallisreportedascorr(Zitδ,yit)2.
AcknowledgmentWethankMeadOveroftheCenterforGlobalDevelopment,whowroteanearlyimplementationofxtivreg.
ReferencesAmemiya,T.
1971.
Theestimationofthevariancesinavariance-componentsmodel.
InternationalEconomicReview12:1–13.
Anderson,T.
W.
,andC.
Hsiao.
1981.
Estimationofdynamicmodelswitherrorcomponents.
JournaloftheAmericanStatisticalAssociation76:598–606.
Arellano,M.
,andS.
Bond.
1991.
Sometestsofspecicationforpaneldata:MonteCarloevidenceandanapplicationtoemploymentequations.
ReviewofEconomicStudies58:277–297.
Balestra,P.
,andJ.
Varadharajan-Krishnakumar.
1987.
Fullinformationestimationsofasystemofsimultaneousequationswitherrorcomponentstructure.
EconometricTheory3:223–246.
Baltagi,B.
H.
2009.
ACompaniontoEconometricAnalysisofPanelData.
Chichester,UK:Wiley.
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EconometricAnalysisofPanelData.
5thed.
Chichester,UK:Wiley.
Baltagi,B.
H.
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Chang.
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Incompletepanels:Acomparativestudyofalternativeestimatorsfortheunbalancedone-wayerrorcomponentregressionmodel.
JournalofEconometrics62:67–89.
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Simultaneousequationswithincompletepanels.
EconometricTheory16:269–279.
Baltagi,B.
H.
,andQ.
Li.
1992.
Anoteontheestimationofsimultaneousequationswitherrorcomponents.
EconometricTheory8:113–119.
Hsiao,C.
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AnalysisofPanelData.
3rded.
NewYork:CambridgeUniversityPress.
Layard,R.
,andS.
J.
Nickell.
1986.
UnemploymentinBritain.
Economica53:S121–S169.
Mundlak,Y.
1978.
Onthepoolingoftimeseriesandcrosssectiondata.
Econometrica46:69–85.
Swamy,P.
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V.
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Arora.
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Theexactnitesamplepropertiesoftheestimatorsofcoefcientsintheerrorcomponentsregressionmodels.
Econometrica40:261–275.
White,H.
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AsymptoticTheoryforEconometricians.
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Wooldridge,J.
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AnoteontheLagrangemultiplierandF-statisticsfortwostageleastsquaresregressions.
EconomicsLetters34:151–155.
24xtivreg—Instrumentalvariablesandtwo-stageleastsquaresforpanel-datamodelsAlsosee[XT]xtivregpostestimation—Postestimationtoolsforxtivreg[XT]xtabond—Arellano–Bondlineardynamicpanel-dataestimation[XT]xteregress—Extendedrandom-effectslinearregression[XT]xthtaylor—Hausman–Taylorestimatorforerror-componentsmodels[XT]xtreg—Fixed-,between-,andrandom-effectsandpopulation-averagedlinearmodels[XT]xtset—Declaredatatobepaneldata[R]ivregress—Single-equationinstrumental-variablesregression[U]20Estimationandpostestimationcommands
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