Regensburger zur Wirtschaftswissenschaft University of Regensburg Wor - PDF document

Regensburger zur Wirtschaftswissenschaft University of Regensburg Working Papers in Business, Economics and Management Information Systems Fractionally Integrated VAR Models with a Fractional Lag Operator and Deterministic Trends: and Two-step Estimation , Roland Weigandng memory, maximum likelihood estimation, fractional Rolf Tschernig holds the chair of Econometrics at the Department of Economics and Econometrics at the University of Regensburg, 93040 Regensburg, Germany. Phone: +49-941-943-2737, E-mail: rolf.tschernig[at]wiwi.uni-regensburg.de Enzo Weber holds the chair of Empirical Economics, especially Macroeconometrics and Labour Markets at the Department of Economics and Econometrics at the University of Regensburg, 93040 Regensburg, Germany, and is head of the research department Forecasts and Structural Analysis of the Institute for Employment Research (IAB), 90478 Nuremberg, Germany. Phone: +49 -911-179-7643, E-mail: enzo.weber[at]iab.de Roland Weigand is a researcher at the research department Forecasts and Structural Analysis of the Institute for Employment Research (IAB), 90478 Nuremberg, Germany. Phone: +49 -911-179-3291, E-mail: roland.weigand[at]iab.de FractionallyIntegratedVARModelswithaFractionalLagOperatorandDeterministicTrends:FiniteSampleIdenticationandTwo-stepEstimationRolfTscherniga,*,EnzoWebera,b,andRolandWeigandbaUniversityofRegensburg,DepartmentofEconomics,D-93040RegensburgbInstituteforEmploymentResearch(IAB),D-90478NurembergJanuary2013AbstractFractionallyintegratedvectorautoregressivemodelsallowtocapturepersistenceintimeseriesdatainavery exibleway.Additional exibilityfortheshortmemoryproper-tiesofthemodelcanbeattainedbyusingthefractionallagoperatorofJohansen(2008)inthevectorautoregressivepolynomial.However,italsomakesmaximumlikelihoodestimationmoredicult.Inthispaperwerstidentifyparametersettingsforunivariateandbivariatemodelsthatsuerfrompooridenticationinnitesamplesandmaythereforeleadtoesti-mationproblems.Second,weproposetoinvestigatetheextentofpooridenticationbyusingexpectedlog-likelihoodsandvariationsthereofwhicharefastertosimulatethanmultivari-atenitesampledistributionsofparameterestimates.Third,weprovidealineofreasoningthatexplainsthendingfromseveralunivariateandbivariatesimulationexamplesthatthetwo-stepestimatorsuggestedbyTschernigetal.(2010)canbemorerobustwithrespecttoestimatingthedeterministiccomponentsthanthemaximumlikelihoodestimator.Keywordsfractionalintegration,longmemory,maximumlikelihoodestimation,fractionallagoperator. Correspondingauthor.Email:Rolf.Tschernig@wiwi.uni-regensburg.de.Phone:(+49)941943-2737Aformerversionwastitled:Conditionalversusapproximateconditionalmaximumlikelihoodestimationinfractionallyintegratedvectorautoregressivemodelswithafractionallagoperatoranddeterministictrends.1 1IntroductionFractionallyintegratedvectorautoregressive(VAR)modelshavebecomeavaluableextensionofVARmodelswithintegerordersofintegration.Recently,Tschernigetal.(2010)introducedthefractionallagoperator(seeJohansen,2008)intostandardfractionallyintegratedVARmodelsinordertoavoidcertainshortcomingsinimpulseresponseanalysisunderlong-runidenticationrestrictions.Theadditionalmodeling exibilityduetothefractionallagoperator,however,makesmaximumlikelihoodestimationmoredicult,inparticularifdeterministiccomponentsareincluded.Inthispaperwerstidentifyparametersettingsforunivariateandbivariatemodelver-sionsthatsuerfrompooridenticationinnitesamplesandmaythereforeleadtoestimationproblems.Second,weproposetoinvestigatetheextentofpooridenticationbyuseofexpectedlog-likelihoodsandvariationsthereofwhicharefastertosimulatethanmultivariatenitesam-pledistributionsofparameterestimates.Third,weprovidealineofreasoningthatexplainsthendingfromseveralunivariateandbivariatesimulationexamplesthatthetwo-stepes-timatorsuggestedbyTschernigetal.(2010)canbemorerobustwithrespecttoestimatingthedeterministiccomponentsthanthemaximumlikelihoodestimator.Withinthemaximumlikelihoodapproachtheestimatorofthedeterministiccomponentsanditspropertiesdependonthesimultaneouslyestimatedfractionalparameters.Ifthelatteraresubjecttopoornitesampleidentication,thedeterministiccomponentsmaybepoorlyestimatedwhichaddstothedicultiesofestimatingthefractionalparameters.Wethereforesuggesttoapplythetwo-stepestimatorinpractice.Section2brie ypresentsfractionallyintegratedVARmodelswithafractionallagoperatorwhileSection3treatsthemaximumlikelihoodestimatorforthesemodels.InSection4wediscussprominentcasesofunivariatedatageneratingprocessesthatmaybepronetopooridentication.Section5explainshowtheexpectedlog-likelihoodandvariationsthereofallowtovisualizethemagnitudeofpooridenticationforagivendatageneratingprocess.InSection6weshowhowtheproblemofpooridenticationworsensoncedeterministiccomponentshavetobeestimatedaswell.Finally,Section7extendstheanalysistobivariatefractionallyintegratedVARprocesseswithafractionallagoperator.2 2FractionallyintegratedVARbmodelsInthissectionweconsiderfractional(vector)autoregressiveprocesseswithafractionallagoperatoranddeterministicterms.Usingthefractionaldierenceoperator(1�L)d=1Xj=0�(j�d) �(�d)�(j+1)Lj;where�(
)denotesthegammafunction,thefractionallagoperatorLbisdened(seeJohansen,2008)asLb=1�(1�L)b=c1L+c2L2+


withb�0:Thedegreeoffractionalintegrationbisrequiredtobepositiveinordertoguaranteethatapplyingthefractionallagoperatordoesnotchangethedegreeofintegration.Forb=1oneobtainsthestandardlagoperatorL.Afractionallyintegratedk-variatevectorautoregressiveprocesswithfractionallagoperator(FIVARb)processforxtisgivenbyA(Lb)
(L;d)xt=ut;utWN(0;);t=1;2;:::(1a)
(L;d):=diag(1�L)d1;(1�L)d2;:::;(1�L)dk:(1b)Heretheerrorsutaremultivariatewhitenoise(WN)withmeanzeroandhomoscedasticcovariancematrix.Forthepth-ordervectorautoregressivelagpolynomialA(z)=I�A1z�


�ApzpwerequirethestabilityconditionofJohansen(2008,Corollary6)tohold.ItprovidesaconditionsuchthateachelementinthevectorprocesstgivenbyA(Lb)t=ut;t=:::;�2;�1;0;1;2;:::;(2)isI(0).TherootsofjA(z)j=0havetobeoutsideCb,whichistheimageoftheunitcircleunderthemappingf:z7!1�(1�z)b.ThisconditiondependsbothonA(.)andonbandcaneasilybecheckedonceparametervaluesaregiven.Sincez=1liesonCbregardlessofthevalueofb,stabilityoftheA(Lb)polynomialexcludestheunitrootcaseandalsoimpliesnonsingularityofA(1).Underthestabilityconditiontheparameterbaddssome exibilitytotheshort-runprop-ertiesoftheprocessratherthanhavingin uenceontheintegrationorders.Forb=1oneobtainsastandardfractionallyintegratedVARprocess(e.g.Nielsen,2004a).Inthesequeldenotetheparametervectorwithallparametersofthemodelby2SwhereSistheparameterspacethatcontainsallparametersthatfullthestabilitycondition.3 2.1TreatmentofpresamplevaluesInordertoobtainasolutiontotheprocess(1)fornonstationaryxtwemakeuseofthetrun-catedoperatornotation(Johansen,2008,AppendicesA.4,A.5).Let(L)denoteaninnitematrixpolynomialandI(
)theindicatorfunction.Then+(L)xt=I(t1)Pt�1i=0ixt�iand�(L)xt=(L)xt�+(L)xt.1Notealsothat+(L)canalwaysbeinvertedbyexpanding(z)�1aroundzeroandtakingthersttterms.Thepresamplevaluesxt,t=0;�1;:::,arenotmodeledalthoughtheyareallowedtobestochastic.Undermildconditionsonthepresamplevalues,seee.g.JohansenandNielsen(2012a),anequivalentrepresentationoftheFIVARbprocess(1)isgivenbyA+(Lb)
+(L;d)xt=ut+mt;t=1;2;:::;(3)mt=[A+(Lb)
�(L;d)+A�(Lb)
+(L;d)+A�(Lb)
�(L;d)]xtwithitssolutiongivenbyxt=A+(Lb)�1
+(L;d)�1ut+t;t=A+(Lb)�1
+(L;d)�1mt;(4)wheretcapturestheimpactofthepresamplevalues.Notethattheforecasterrorimpulseresponses hforhorizonhcanbecomputedfromthetruncatedlagpolynomial +(L)=A+(Lb)�1
+(L;d)�1,replacingin(4)tbyt+h.2.2DeterministiclineartrendsThemodelingoflineardeterministictimetrendsisrestrictedtothesampleofmodeleddatabyassumingthatyt=8�&#x]TJ ;� -1; .63; Td;&#x [00;:0+1t+xt;ift1;xtift0:(5)Therefore,theFIVARbmodelwithdeterministictrendsisgivenbyA(Lb)
(L;d)(yt�0�1t)=ut;t=1;2;::::(6) 1Notethatforaproduct(L)=(L)(L)ofVARpolynomials(L),(L)onehas+(L)=+(L)+(L);�(L)=+(L)�(L)+�(L)+(L)+�(L)�(L):4 ByexplicitlystatingthedependenceonpresamplevaluesoneobtainstherepresentationsA+(Lb)
+(L;d)yt=ut+A+(Lb)
+(L;d)(0+1t)+mt;t=1;2;:::;(7)yt=A+(Lb)�1
+(L;d)�1ut+(0+1t)+t;t=1;2;::::(8)Ifitisassumedthatallpresamplevaluesarezero,xt=0,t0,thenmt=t=0,t=1;2;:::.3MaximumlikelihoodestimationInthefollowingwestatetheconditionalmaximumlikelihoodestimatorforgivenpresamplevaluesxt,t=0;�1;:::;Tp.Forbrevityitiscalledmaximumlikelihoodestimatorthroughoutthepaper.Let=vec(A1;:::;Ap)denotethevectorofallVARcoecientsandY=�y�Tp;:::;y�1;y0;y1;:::;yT
thevectorofobservablepresampleandsamplevalues.Foranobservedtimeseries,themaximumlikelihoodestimatorsforthegeneralmodel(6)allowingfordeterministictrendsisgivenby^d;^b;^0;^1;^;^=argmaxd;b;0;1;;L(d;b;0;1;;;Y);(9)wherethemaximizationiscarriedoutoveranappropriateparameterspace.Further,weassumenormallydistributederrorsinordertoderivethelog-likelihoodfunctionexplicitly:L(d;b;0;1;;;Y)=�Tk 2log2�T 2logjj�1 2TXt=1ut(d;b;0;1;)0�1ut(d;b;0;1;);(10)whereut(d;b;0;1;)isobtainedbyrearranging(6)asut(d;b;0;1;)=
(L;d)(yt�(0+1t))| {z }zt(d;0;1)�A1Lb
(L;d)(yt�(0+1t))| {z }zt�1(d;b;0;1)�


�ApLpb
(L;d)(yt�(0+1t))| {z }zt�p(d;b;0;1);t=1;2;::::(11)Sincezt(d;0;1)doesnotdependonitfollowsfrom(11)thatforgivend,b,0,1theVARcoecientmatricesA1,...,Ap,ifunrestricted,areobtainedbyleastsquares,regressingzt(d;0;1)onzt�1(d;b;0;1);:::;zt�p(d;b;0;1).Togetherwithconcentratingoutthissimpliesthemaximizationconsiderablyandleadstotheconcentratedlog-likelihoodLc(d;b;0;1;Y)=�Tk 2(log2+1)�1 2logTXt=1ut(d;b;0;1)ut(d;b;0;1)0;(12)5 thatcanbemaximizedinsteadof(10).Forexample,inthebivariatecasethemaximizationofthisconcentratedlog-likelihoodfunctiononlyrequiresanonlinearoptimizationonsevenparametersindependentlyoftheVARorderp.AvailablerelatedasymptoticresultsarederivedbyJohansenandNielsen(2012a)inaframeworkoffractionalcointegrationundertheassumptionofnodeterministictrends.Whileintheirsettingidenticaldisareassumed,thisdoesnotimplythattheindividualfractionalordersofintegrationareidenticalduetothepossibilityoftrivialcointegrationwith=(1;0)0.However,weexcludethepossibilityoffractionalcointegrationbythestabilityrestrictionontheVARpolynomialA(Lb).ForthestandardFIVARmodelunderthestabilityconditionwithb=1asymptoticresultsarederivedbyNielsen(2004a)andimpliedbyHualdeandRobinson(2011).ItremainstobecheckedifHualdeandRobinson(2011)coverstheFIVARbmodelforb6=1andzeropresamplevalues.Withrespecttounivariateprocessestheasymptoticbehaviorofthemaximumlikelihoodestimator(9)isinvestigatedbyNielsen(2004b),whorequiresb=1,astableARpolynomial,andzeropresamplevaluesbutallowsforadeterministictrend,andbyJohansenandNielsen(2010),whoallowfornonzeropresamplevalues,b6=1aswellasforunitrootsintheARpolynomial.Theyexcludedeterministictrends,however.JohansenandNielsen(2012b)derivetheasymptoticsecond-orderbiasduetopresamplevaluesforapureunivariatefractionalprocess
dxt=ut.WiththeexceptionofJohansenandNielsen(2010)andJohansenandNielsen(2012a)allmentionedresultsrequiretheA(Lb)polynomialtobestable.4Poornitesampleidenticationincaseofunivariatepro-cessesItiswellknownthatparameterestimationmaybemoredicultiftheparametervaluesofthedatageneratingprocessareclosetotheboundaryoftheparameterspacewhereallorsomeparametersarenotidentied.Togiveanexample,letthedatageneratingprocessbeasmoothtransitionautoregressiveprocessthatisveryclosetoalinearautoregressiveprocess.Sincetheparametersofasmoothtransitionautoregressivemodelarenotidentiedifthedatageneratingprocessisinfactlinear,aweaknonlinearstructuremaynotbedetectableinmanysamplessuchthatestimationtakesplaceasifparametersarenotidentied.AnotherwellknownexamplearecommonrootsinARMA(p;q)models.Ifthetrueorders6 arep0andq0,thenestimatinganARMA(p0+1;q0+1)modelsuersfromcommonrootswhichcausestheparametersofthemodeltobenotidentied.Incontrasttothepreviousexamplewhereaweaknonlinearstructureinthedatageneratingprocesscausesestimationproblems,itistoolargeamodelorderinthelatterexample.WhenestimatingFIVARbprocessessimilarscenariosmayoccurwheretheorderoftheautoregressivepolynomialistoolarge,p�p0:a)p0=0andp=1:AssumethatthedatageneratingprocessisaunivariatewhitenoisebutaunivariateFIVARbmodeloforderoneistted,(1�a1Lb)
dxt=ut;utWN(0;2);t=1;2;::::(13)ThismodelwillhenceforthbecalledaFARbmodeloforderone.i)Insertingthetrueparametervaluesa1;0=0andd0intothelagpolynomialsdelivers(1�0Lb)
d0=
d0sothatbcantakeanypositivevalue.Thus,bisnotidentiedbutdis.ii)However,onemayalsoinserta1;0=1.Then(1�Lb)
d=
b+dwhichhastobeequalto
d0.Thenthereisacontinuumofcombinationsforbanddforwhichb+d=d0holds.b)p0=1andp=2:Since(1�a1Lb�a2L2b)
d=(1�1Lb)(1�2Lb)
d,insertingthetrueparameters0a1;01,b0&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0,andd0&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;0intoaFARbmodelwithorder2deliverstworepresentationsusing1=a1;0:i)2=0andthusa2=0,beingequivalenttoaFARbmodeloforderone.ii)2=1.Then(1�a1;0Lb0)
d0=(1�a1;0Lb0)
b0
d0�b0=(1�a1;0Lb0)(1�1Lb0)
d0�b0=(1�a1Lb0�a2L2b0)
d0�b0wherea1=a1;0+1,a2=�a1;0.Notethatinii)thestabilityconditionisviolatedsince2=1.IncontrasttoCasea),theparametersforeachscenarioarelocallyidentiedbutnotglobally.Thus,onemayexpectthelog-likelihoodtobebimodal.7 Eveniftheparametersareidentied,theymaybeclosetotheboundaryof(partial)non-identication.Assumingp0=1and0a1;01suchthatthestabilityconditionholds,itcanstillhappenthatthesampleinformationisnotsucienttokeeptheestimateofa1reasonablyfarawayfromzerosothatanidenticationproblemmayresultinnitesamples,resemblingcasea)i)above.Similarly,ifa1;0issmallerthanunitybutnotdistinguishablefromoneinanitesample,thend=d0+b0�b(14)givestheapproximatelocationsofestimateddandb.Onlyd+bisappropriatelyidentied.Wecallthesescenariospoornitesampleidentication.ForhigherorderFARbmodels,p1,poornitesampleidenticationmayalsoresultfromthepossibilitythatanestimateofbisclosetozero.ThenonehasajLjb0=aj1�(1�L)b00(15)independentlyofthevalueofaj.Therefore,ifaDGPthatiswhitenoiseismodeledbyaFARbprocesswithorderp,theARparametersaj,j=1;2;:::;parepoorlyidentiedforvaluesofbclosetozero.Hencealsoforthisreasonitisimportanttoaimatusingcorrectlagorders.Dependingonthepurposeofthemodel,poornitesampleidenticationmayormaynotbeharmful.Ifoneisinterestedinthelong-rundynamicsimpliedbythedegreeofintegration,thenitcanbeveryproblematic.Asanexample,comparethedegreeofintegrationimpliedbyaprocesswitha1closetoonetothecasea1=1foridenticald0andb0:whiletheformerisanI(d0)process,thelatterisanI(d0+b0)process.Whetherpoornitesampleidenticationisanissueforagivensamplemaybecheckedbyinvestigatingthelog-likelihoodfunctionontherelevantrangeoftheparametersofinterest.FortheFARbmodel(13)oforderone,onemayplottheconcentratedlog-likelihoodonagridfordandbandvisuallycheckwhethertherearetwopeaksormountainridgesthatindicatepoornitesampleidentication.Incaseonewantstocheckthepotentialofpoornitesampleidenticationpriortosampling,onemaysimply\average"thelog-likelihoodfunction(10)overpossiblesamplesbytakingexpectationsofthelog-likelihoodfunctionatthetrueparametervector.Thisdeliverstheexpectedlog-likelihoodfunction(16)furtherdescribedinthenextsection.Finally,ifthetrueparametersarefarenoughawayfromtheboundaryof(partial)non-identication,poornitesampleidenticationshouldnotbeamajorissue.Thus,incaseofthe8 FARbmodel(13),onemayexpectreasonablenitesampleidenticationincaseofa1;0=0:6.5Visualizingtheexpectedlog-likelihoodIntheprevioussectionitwasarguedthatpoornitesampleidenticationmaybecheckedwithoutreferringtoanysamplebyinvestigatingtheexpectedlog-likelihoodfunction.Thissuggestiondiersfromthecommonlyusedmethodforinvestigatingnitesampleestimationpropertiesbysimulatingthenitesampledistributionof^.Theexpectedlikelihoodcomputa-tionstakeintoaccountadditionalinformationabouttheshapeofpossiblelikelihoodfunctionsawayfromtheirmaxima.Additionally,whenjointlyconsideringmorethantwoparameters,theexpectedlikelihoodcanbesimulatedfasterthanthejointdensityoftheparameterestimator.5.1Theexpectedlog-likelihoodLetE0[
]indicatethattheexpectationistakenwithrespecttothedatageneratingprocess.Thentheexpectedlog-likelihoodisgivenbyE0[L(;Y)]=Zlogf(Y;)f(Y;0)dY:(16)Notethat0maynotbeuniquewithouttherestriction02S.Plottingcontourlinesorsurfacesoftheexpectedlog-likelihood(16)isonlypossibleifisoflengthl=2.Onewaytodealwiththecasel�2istosplitthe(l1)vectorintoa(21)vectorIthatcontainsthetwoparametersofinterestanda((l�2)1)vectorIIofallotherparametersandthenmaximize(16)withrespecttoII:E0[L(I;mII(I);Y)];wheremII(I)=argmaxIIE0[L(I;II;Y)]:(17)FortheunivariateFARbprocess(13)onemaydeneI=(d1;b)0andII=�d2;a1;2
0.Ifaconcentratedlog-likelihoodisavailable,itmaybepreferabletoconsidertheexpectedconcentratedlog-likelihood.IfIIcanbeconcentratedoutcompletely,theexpectedconcen-tratedlog-likelihoodisdenedbyE0hLI;^II(I;Y);Yi(18)throughconcentrating^II(I;Y)=argmaxIIL(I;II;Y):(19)9 Consideringtheexpectedconcentratedlog-likelihoodallowstostudytheindirecteectsthatestimatingIIhasontheestimationofI,whilesuchindirecteectsareignoredinthemaximizationapproach(17).Further,onemayuse(18)tocomparevariousestimatorsfor^II(Y)withrespecttotheirin uenceontheestimationproblemofI.Ifanalternativeestimatorto(19)isused,thenoneobtainsdierentobjectivefunctionsfortheestimationofIwhichcanbecompared.FinallyonemaycombinebothapproachesbyconcentratingoutsomeparametersofIIwhiletakingthemaximumwithrespecttotheremainingones.Asanexamplewithrespectto(10)onemaydeneI=(d1;b)0,II=d2,andIII=(vec(A)0;00;01;vech()0)0.Thentheexpectedconcentratedlog-likelihooddependingonIisgivenbyE0hLI;mII(I);^III(I;mII(I;Y);Yi(20)throughconcentrating^III(I;II;Y)=argmaxIIIL(I;II;III;Y)and'optimizingout'mII(I)=argmaxIIE0hLI;II;^III(I;II;Y);Yi:5.2Expectedconcentratedlog-likelihoodsforFARbprocessesoforderoneInthissectionweusesimulationstocomputetheexpectedconcentratedlog-likelihoodsforvariousFARbprocesses(13)oforderone.Wechoosed0=b0=0:8andconsidera1;0=0:1;0:6;0:9.ForgivenI=(d;b)0andII=(a1;2)0theconcentratedlog-likelihoodiseasilycomputedasdescribedinSection3.Inordertoapproximatetheexpectedconcentratedlog-likelihood(18)wedraw100realizationsforgivendandb.Forobtainingcontourplotswevarytheparametersd2[�1;1:5]andb=[0:02;1:5]usinggridswithstepsizeof0.02.Themagnitudeofpoornitesampleidenticationofdandbisvisualizedbytheshapeandsizeoftheareawiththelargestvaluesoftheexpectedconcentratedlog-likelihood,whichcanwellbeseenfromaplotwithcontourlines.ForsamplesizeT=250anda1;0=0:9thecontourlinesoftheexpectedconcentratedlog-likelihoodareshowninthetoppanelofFigure1.Ifa1;0wereexactly1,onewouldexpectfrom(14)thatthelocationofthelargestvaluesof10 theexpectedconcentratedlog-likelihoodisdescribedby^dd0+b0�^b=1:6�^b.FromthetoppanelofFigure1itisseenthatbylettinga1;0deviateslightlyfromone,thislocationisshiftedsomewhatto^d1:8�^b.AsarguedinSection4,pooridenticationinsmallsamplesislessofaproblemifa1;0isneitherclosetozeronortoone,saya1;0=0:6,ascanbeseenfromthemiddlepanelofFigure1.ThelowerpanelofFigure1showsthatthepoornitesampleidenticationissueisagainprominentifa1;0=0:1andthusclosetozero.FromCasea)ii)inSection4itfollowsthatifa1wereexactlyzero,^bcan oatarbitrarilywhile^d0:8.Thisexplainstheupperridgeifa1isestimatedclosetozero.Thelowerridgeisexplainedbyestimatesa11leadingtoanegativetrade-obetweendandb.6DealingwithdeterministictrendsInthissectionweinvestigatewhytheproblemofpoornitesampleidenticationworsensifdeterministictrendsareallowedinthemodel.InordertoestimatetheparametersoftheFIVARbmodel(6)withlineartrendsonemayusethemaximumlikelihoodestimatorbasedon(10)inSection3.Inthefollowingweprovideargumentsandafewsimulationresultsthatatwo-stepestimationismorerobust.6.1PitfallsinthemaximumlikelihoodestimationOnemayrewrite(6)sothat0and1canbeestimatedbyleastsquaresifalltheotherparametersd,b,andaregiven:A(Lb)
(L;d)yt=A+(Lb)
+(L;d)10+A+(Lb)
+(L;d)t1+ut;t=1;2;:::;T:(21)Thereforetheestimatesofthedeterministiccomponentsarein uencedbyboththelongmem-oryparametersdandtheparametersandbdeterminingtheI(0)dynamics.Asaconse-quence,ifoneofthecasesinvestigatedinSections4and5.2occurswhereforagivensamplesizeandDGPonlyd+biswellidentiedwhiledandbarenot,bmaybeestimatedtoolargeanddtoosmall.Toseethepossibleimplications,consideraunivariateDGPwithd0=1,0;06=0,1;0=0,anda1;0closetoone.Suppose^dtakesthetruevalued0=1,thentheregressionforestimating0and1correspondingto(21)is(1�a1Lb)
yt=(1�a1Lb)I(t=1)0+(1�a1Lb)I(t1)1+ut;t=1;2;:::;T:(22)11 Then0isestimatedfromonlyoneobservation,t=1,andVar(^0)1forT!1.Thisisnotproblematicastheimpactoftheestimated0vanisheswithgrowingsamplesize.Duetopoornitesampleinformationa1maybeestimatedclosetoone.Sincefora1;0=1,onlyd+bisidentiedby(14),^dmaybeclosetozeroand^bcloseto1+b0.Then,settingd=0,no(fractional)dierencesaretakenandtheerrorsin(1�a1Lb)yt=(1�a1Lb)I(t1)0+(1�a1Lb)tI(t1)1+ut;t=1;2;:::;T(23)exhibitaunitroot.InthiscaseitcanbeshownthatVar(^0)increaseswithsamplesizeT.Then,theestimateof0canbeexpectedtoremainin uential.Thus,theimplicitestimationpropertiesfor0anditsimpactontheotherestimatescruciallydependonthedestimate.Itcanbeshownthatthisalsoholdsforthe1estimate.Suchdicultiesduetoagrosslywrongdestimatescanbeavoidedifthepoornitesampleidenticationproblemiscircumventedwhenestimating0and1.Toachievethis,weinvertA(Lb)in(21),whichisalwayspossibleifthestabilityconditionholds,andconsidertheregression
(L;d)yt=
+(L;d)10+
+(L;d)t1+"t;t=1;2;:::;T;(24)wheretheerrors"t=A(Lb)�1utareautocorrelatedbutI(0).Inthisregressionbdoesneitherentertheregressandnortheregressor.Usingtheregression(24)workssincetheregressorsaredeterministicandthereforetheautocorrelatederrorsdonotmattermuchforestimating0and1.Therefore,estimatingtheparametersofthedeterministictermsonbasisof(24)onlyrequiresknowledgeofdwhichcanbeestimatedbysomesemiparametricestimatorthatdoesnotsuerfromthenitesampleidenticationproblems.Thisleadsdirectlytothetwo-stepestimatordescribednext.6.2Two-stepestimationWithinthetwo-stepestimationprocedure,thedeterministiccomponentsareestimatedintherststep.Inthesecondstepthelog-likelihoodfunctionismaximizedafterreplacingthedeterministiccomponentsbytheirestimatesfromtherststep.Therststepinvolvesrunningtheregression(24).Thisamountstocomputingtheleast12 squaresestimatorfrom(1�L)dsys;t=(1�L)ds+10;s+(1�L)ds+t1;s+"s;t;s=1;2;:::;k;t=1;2;:::;T:(25)Weobtainthefollowingtwo-stepestimatorsuggestedbyTschernigetal.(2010):Firststep:Foreachseriess=1;2;:::;kestimatethememoryparameterdswiththesemipara-metricexactlocalWhittleestimatorofShimotsu(2010)thatallowsfordeterministictrends.Inordertoobtain~0;sand~1;s,runtheregression(25)aftertakingfractionaldif-ferencesbasedonthe~dsestimate.Dothisfors=1;2;:::;k.Secondstep:Maximizethelog-likelihoodfunction(10)with0=~0and1=~1.Notethatinthebivariatecasethenonlinearoptimizationinthesecondsteponlyincludesthethreeparametersd1;d2;b.6.3Expectedconcentratedlog-likelihoodsforFARbprocessesoforderonewithdeterministictrendsWeconsiderthesamethreeDGPsasinSection5.2butestimateaFARbmodelallowingforadeterministictrend.Figures2,4,and6displaythecontourlinesoftheexpectedconcentratedlog-likelihoodwherethemaximumlikelihoodestimatorisusedforconcentratingout0and1.Figures3,5,and7displaythecontourlinesoftheexpectedconcentratedlog-likelihoodwherethetwo-stepestimatorofSection6.2isusedforestimatingthedeterministicterms.Comparingthecontourlinesofbothestimatorsfora1;0=0:9inFigures2and3showsthattheregionofhighestexpectedconcentratedlog-likelihoodisclosertothetruevaluesforthetwo-stepestimatorthanforthemaximumlikelihoodestimatorasconjecturedintheprevioussubsection.FortheotherFARbwitha1;0=0:1whoseestimationmayalsosuerfrompoornitesampleidentication(closetoCasea)inSection4),themaximumlikelihoodestimatorinFigure6isevenmoreoincomparisontothetwo-stepestimatorinFigure7.Evenforthecaseofa1;0=0:6whichshowstheleastnitesampleidenticationproblemsincaseofnodeterministiccomponentthetwo-stepprocedureseemssuperiorascanbeseenfromcomparingFigures4and5.Whencomparingthecontourlinesfromthetwo-stepprocedure13 withtheresultsincaseofknowndeterministictermsoneobservesthatestimatingdeterministiccomponentsiscostlysincetheregionsofhighestexpectedconcentratedlog-likelihoodnolongerincludethetrueparameters.Inordertocheckwhethertheidenticationproblemreallydependsonthesamplesizewealsocomputedtheexpectedconcentratedlog-likelihoodsfortheFARbwitha1;0=0:9forsamplesizeT=1000.NowbothestimationproceduresworkmuchbetterascanbeseenfromFigures8and9althoughthetwo-stepconcentratedlikelihoodstillshowsahighercurvature.Insum,thetwo-stepestimatorworksbetteraswasexpectedfromthereasoninginSections6.1and6.2.ItremainstobeinvestigatedhowthesendingscarryovertothebivariateFIVARbmodel(6)whichisdonenext.7PoornitesampleidenticationinbivariateFIVARbpro-cessesInthissectionweconsidertheFIVARbprocesses(1)and(6).7.1ProcesseswithpoornitesampleidenticationIfthebivariateDGPhasadiagonalVARcoecientmatrix0@I�0@10021ALb1A0@(1�L)d100(1�L)d21Axt=ut;t=1;2;:::;(26)thenmaximizingthelikelihoodcanbeexpectedtobeclosealthoughnotidenticaltomaximizingthelikelihoodofeachunivariateseries.Thelattergenerallydeviatesfromthejointestimationsincebisestimatedforeachseriesseparately.Ifdandbareknown,bothestimatorscoincide.However,sincebisidenticalacrosstheindividualseries,oneeigenvalueneitherbeingclosetoonenorzeroshouldhelptoestimatebsucientlywellsothattheothereigenvaluedoesnolongercauseproblemsevenifitisclosetooneorzero.Forthisreason,weexpectestimationproblemsfrompoornitesampleidenticationifneitherindividualprocesshelpstodeterminebandweexpectthatsuchaFIVARbprocessinheritsthepoornitesampleidenticationproblemsfromtheindividualFARbprocesses.Inthefollowingwewillinvestigateprocesses(26)with=1=2.Sinceistheeigenvalue(withmultiplicity2)oftheautoregressiveparametermatrixA,itisinterestingtoinvestigatewhetherthepoornitesampleidenticationproblemsdiminishonce14 dependencebetweenthetwoprocessesthroughtheautoregressivepolynomialisintroducedwhilekeepingtheeigenvalueconstant.InordertoobtainamatrixAthatexhibitsthesameeigenvaluebutnonzeroo-diagonalelementsweapplytheJordandecompositionforrealmatrices(Lutkepohl,1996,Section6.2.1(2)).UsinganonsingularJ=(acbd),a(22)matrixAwithoneeigenvaluecanbewrittenasA=J0@101AJ�1=0@�ac ad�bca2 ad�bc�c2 ad�bc+ac ad�bc1A:(27)Weconjecturethatthepoornitesampleidenticationproblemsdiminishforacommoneigenvalueoncethedependencebetweenthetwoprocessesisincreasedthrougho-diagonalelementsinA.Wealsoconjecturethattheimpactoftheseproblemsmattersmoreforthemaximumlikelihoodestimatorthanforthetwo-stepestimatorlikeintheunivariatecase.Inordertochecktheseclaimsweinvestigatetheexpectedconcentratedlog-likelihoodfunctionsfortwoDGPsnext.7.2Expectedconcentratedlog-likelihoodsforFIVARbprocessesoforderoneAllDGPsconsideredhavenormallydistributederrorswith0=(10:50:51);nodeterministictrend,0;0=1;0=0,andzeropresamplevalues.Tochecktheclaimthatidenticaleigenvaluesclosetoonewithoutautoregressivedependenceposeproblemsweconsider:DGPdiagthediagonalprocess(26)withd1;0=b0=0:8andd2;0=1:8.ThefractionalparametervaluesresemblevaluesthatwereestimatedbyTschernigetal.(2010)forUSrealGDPandpricedata.Inordertoinvestigatetheeectofautoregressivedependence,wechooseJin(27)suchthatthecorrelationbetweeny1tandy2tisabout-0.5when=I.ThisisthecaseforJ0=�1�111
.ThisdeliversthefollowingDGP:DGPdep0@I�0@+1 2+1 2�1 2�1 21AL0:81A0@(1�L)0:800(1�L)1:81Axt=ut;t=1;2;:::(28)Supposethattherearenowthreeparametersofinterest:d1,d2andb.Forcomputingtheexpectedlog-likelihoodswerstconcentrateoutallparametersexceptthosethreeandthen15 plotcontourlineswherewetakethemaximumoftheexpectedconcentratedlog-likelihoodwithrespecttothethirdparameter.Asanexamplechoosein(20)I=(d1;b)0,II=d2,andIII=(0;00;01;vech()0)0.WerstdiscusstheresultsofthediagonalDGPdiag.Ifnodeterministiccomponentsareestimated,Figure10showsthatthetrueparametersareclosetothepointofhighestexpectedconcentratedlog-likelihood.Oncedeterministiccomponentsareallowedforintheestimationmodel,thisisnolongerthecaseforthemaximumlikelihoodestimatorascanbeseenfromtheleftcolumninFigure11.Incontrast,thetwo-stepestimatorstilldeliversreasonableresults.ThisndingsupportsthereasoningofSection7.1.Ifdynamicspill-oversbetweentheseriesarepresentasintheDGPdep,thenitturnsoutfrominspectingFigures12and13(forestimationwithoutandwithdeterministiccomponents,respectively)thattheweaknitesampleidenticationproblemislesspronouncedthaninthediagonalcasealthoughbseemstobeestimablelesspreciselythand1andd2.NotablydierenteectsoftheestimatorsforthedeterministictermsarenotpresentinFigure13.8ConclusionWediscussednitesampleestimationpropertiesoffractionallyintegratedVARmodelswherehigh exibilityisintroducedthroughthefractionallagoperatoranddeterministictrends.Weidentifysituationswhereidenticationmaybepoorinnitesamplesandverifytheseclaimsbyplottingexpected(concentrated)likelihoods.Deterministictrendsaggravatetheproblems.Atwo-stepestimatorhelpstocircumventatleastpartofthe awswhicharefacedifthemaximumlikelihoodestimatorisused.Subsequentworkmaybeconcernedwiththeasymptoticpropertiesoftheestimators.Asageneralrecommendation,futureempiricalresultsusingfractionallyintegratedtimeseriestechniquesshouldbecheckedwithrespecttonitesampleidenticationissuesbothtoassessrobustnessoftheresultsandtosupportanappropriateestimatorchoice.16 ReferencesHualde,J.andRobinson,P.M.(2011),\GaussianPseudo-MaximumLikelihoodEstimationofFractionalTimeSeriesModels,"TheAnnalsofStatistics,39,3152{3181.Johansen,S.(2008),\ARepresentationTheoryforaClassofVectorAutoregressiveModelsforFractionalProcesses,"EconometricTheory,24,651{676.Johansen,S.andNielsen,M..(2010),\LikelihoodInferenceforaNonstationaryFractionalAutoregressiveModel,"JounalofEconometrics,158,51{66.|(2012a),\LikelihoodInferenceforaFractionallyCointegratedVectorAutoregressiveModel,"Econometrica,80,2667{2732.|(2012b),\Theroleofinitialvaluesinnonstationaryfractionaltimeseriesmodels,"WorkingPaper1300,Queen'sEconomicsDepartment.Lutkepohl,H.(1996),HandbookofMatrices,Wiley&Sons.Nielsen,M..(2004a),\EcientInferenceinMultivariateFractionallyIntegratedTimeSeriesModels,"EconometricsJournal,7,63{97.|(2004b),\EcientLikelihoodInferenceinNonstationaryUnivariateModels,"EconometricTheory,20,116{146.Shimotsu,K.(2010),\ExactLocalWhittleEstimationofFractionalIntegrationwithUnknownMeanandTimeTrend,"EconometricTheory,26,501{540.Tschernig,R.,Weber,E.,andWeigand,R.(2010),\Long-runIdenticationinaFraction-allyIntegratedSystem,"RegensburgerDiskussionsbeitragezurWirtschaftswissenschaft447,UniversityofRegensburg.17 Figure1:Theexpectedconcentratedlog-likelihoodfunctionforaFAR0:8process(13)with250observations,d=0:8anda1=0:9(above),a1=0:6(middle)anda1=0:1(below)18 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -3.1 -3.1 -3.1 -3.1 -3.1 -3.1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -2.9 -2.9 -2.9 -2.9 -2.9 -2.9 -2.9 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.7 -2.7 -2.7 -2.7 -2.6 -2.6 -2.5 -2.5 -2.4 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -3.6 -3.6 -3.6 -3.6 -3.6 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.5 -3.4 -3.4 -3.4 -3.4 -3.4 -3.4 -3.4 -3.4 -3.4 -3.3 -3.3 -3.3 -3.3 -3.2 -3.1 -3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -2.25 -2.25 -2.2 -2.2 -2.2 -2.2 -2.2 -2.2 -2.2 -2.2 -2.2 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.1 -2.05 -2.05 -2.05 -2.05 -2.05 -2.05 -2.05 -2.05 -2.05 -2 -2 -2 -2 -2 -2 -2 -1.95 -1.95 -1.95 -1.95 -1.95 -1.9 -1.9 -1.9 -1.9 -1.9 -1.9 -1.85 -1.85 -1.85 -1.85 -1.85 -1.8 -1.8 -1.8 -1.8 -1.75 -1.75 -1.75 -1.7 -1.7 -1.7 -1.65 -1.65 -1.65 -1.6 -1.6 -1.6 -1.55 -1.55 -1.55 -1.5 -1.5 -1.5 -1.45 -1.45 -1.45 -1.4 -1.4 -1.4 -1.35 -1.35 -1.35 -1.3 -1.3 -1.25 -1.25 -1.2 -1.2 -1.15 -1.1 Figure2:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:9with250observations. Figure3:Theexpectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:9with250observations.19 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -2 -2 -2 -2 -2 -2 -2 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.6 -1.6 -1.6 -1.6 -1.6 -1.4 -1.4 -1.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -2.9 -2.9 -2.9 -2.9 -2.9 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.7 -2.7 -2.7 -2.7 -2.7 -2.7 -2.7 -2.6 -2.6 -2.6 -2.6 -2.6 -2.6 -2.5 -2.5 -2.5 -2.5 -2.5 -2.4 -2.4 -2.4 -2.4 -2.3 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 Figure4:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:6with250observations. Figure5:Theexpectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:6with250observations.20 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -2.2 -2.2 -2.2 -2.2 -2.2 -2.2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -1.8 -1.8 -1.8 -1.8 -1.8 -1.6 -1.6 -1.6 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -3.2 -3.2 -3.2 -3.2 -3.2 -3 -3 -3 -3 -3 -3 -3 -3 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.8 -2.6 -2.6 -2.4 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 Figure6:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:1with250observations. Figure7:Theexpectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:1with250observations.21 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.4 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 0 0 0 0 0 0 0 0 0 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -1.3 -1.2 -1.2 -1.2 -1.2 -1.2 -1.2 -1.2 -1.2 -1.1 -1.1 -1.1 -1.1 -1.1 -1.1 -1.1 -1.1 -1.1 -1.1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.8 -0.8 -0.8 -0.8 -0.8 -0.7 -0.7 -0.7 -0.6 -0.6 -0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure8:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:9with1000observations. Figure9:Theexpectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrendforaFAR0:8process(13)withd=0:8anda1=0:9with1000observations.22 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -9 -9 -9 -9 -9 -9 -9 -9 -9 -9 -9 -8.5 -8.5 -8.5 -8.5 -8.5 -8.5 -8.5 -8 -8 -8 -8 -8 -8 -8 -8 -7.5 -7.5 -7.5 -7.5 -7.5 -7 -7 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d -10 -10 -10 -10 -10 -9.5 -9.5 -9.5 -9.5 -9.5 -9.5 -9 -9 -9 -9 -9 -9 -9 -9 -9 -8.5 -8.5 -8.5 -8.5 -8.5 -8.5 -8 -8 -7.5 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 Figure10:Theexpectedconcentratedlog-likelihoodfunctionfortheFIVAR0:8processDG-Pdiag(26)withd1=0:8,d2=1:8anda1=0:8with250observations.23 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 24.2 24.8 25.3 25.3 25.3 25.8 25.8 25.8 26.4 26.4 26.4 26.4 26.4 26.9 26.9 26.9 26.9 27.4 27.4 27.4 27.4 27.9 27.9 27.9 27.9 28.5 28.5 29 29 29 29.5 29.5 30.1 30.1 30.6 30.6 30.6 31.1 31.1 31.6 31.6 32.2 32.2 32.7 32.7 33.2 33.2 33.8 34.3 34.8 35.3 35.9 36.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 24.3 24.3 27.9 27.9 27.9 27.9 31.5 31.5 31.5 35.1 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 31.4 32 32 32 32 32 32.7 32.7 32.7 32.7 32.7 32.7 32.7 33.3 33.3 33.3 33.3 33.3 33.3 33.9 34.6 35.2 35.8 36.5 Figure11:Leftcolumn:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrends.Rightcolumn:expectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrends.EachfortheFIVAR0:8processDG-Pdiag(26)withd1=0:8,d2=1:8anda1=0:8with250observations.24 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 27.9 27.9 28.5 28.5 29 29.5 29.5 30.1 30.1 30.6 30.6 31.1 31.1 31.1 31.6 31.6 32.2 32.2 32.2 32.7 32.7 32.7 33.2 33.2 33.8 33.8 33.8 34.3 34.8 34.8 35.3 35.9 36.4 36.4 36.9 36.9 37.5 37.5 38 38.5 39.1 39.6 40.1 40.6 41.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 25.3 25.8 25.8 26.4 26.4 26.9 26.9 27.4 27.4 27.9 27.9 27.9 28.5 28.5 28.5 29 29 29.5 29.5 29.5 30.1 30.1 30.6 30.6 30.6 31.1 31.1 31.6 31.6 31.6 32.2 32.2 32.7 33.2 33.2 33.8 33.8 34.3 34.3 34.8 34.8 35.3 35.9 36.4 36.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 31.5 31.5 31.5 31.5 35.1 35.1 38.7 38.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 27.9 27.9 27.9 27.9 27.9 27.9 31.5 31.5 31.5 35.1 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 36.5 37.1 37.1 37.1 37.1 37.8 37.8 37.8 37.8 38.4 38.4 38.4 39 39.7 40.3 41 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 32.7 32.7 32.7 33.3 33.3 33.3 33.3 33.9 33.9 33.9 33.9 33.9 34.6 34.6 34.6 35.2 35.8 36.5 37.1 Figure12:Theexpectedconcentratedlog-likelihoodfunctionfortheFIVAR0:8processDG-Pdep(28)withd1=0:8,d2=1:8and=0:8butautoregressivedependencewith250observations.25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 -46.9 -43.7 -43.7 -40.4 -40.4 -40.4 -40.4 -37.1 -37.1 -33.8 -33.8 -33.8 -33.8 -30.5 -30.5 -30.5 -27.2 -27.2 -27.2 -23.9 -20.6 -17.3 -14 -10.7 -7.4 -4.1 -0.8 -0.8 2.5 2.5 5.8 5.8 9.1 9.1 12.4 12.4 15.7 15.7 19 19 22.3 22.3 25.6 28.9 32.1 35.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 5.3 5.3 5.3 8.4 8.4 8.4 8.4 8.4 8.4 11.4 11.4 11.4 11.4 11.4 14.4 14.4 17.5 17.5 20.5 20.5 23.6 26.6 29.6 32.7 35.7 35.7 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 1.6 1.6 1.6 1.6 1.6 1.6 1.6 3.4 3.4 3.4 3.4 5.3 5.3 5.3 5.3 5.3 7.1 7.1 7.1 7.1 7.1 9 9 9 9 10.9 10.9 10.9 12.7 12.7 14.6 16.4 18.3 20.2 22 23.9 25.7 27.6 29.4 31.3 33.2 35 Figure13:Leftcolumn:Theexpectedconcentratedlog-likelihoodfunctionwithestimateddeterministictrends.Rightcolumn:expectedconcentratedtwo-stepapproximatelog-likelihoodfunctionwithestimateddeterministictrends.EachfortheFIVAR0:8processDG-Pdep(28)withd1=0:8,d2=1:8and=0:8butautoregressivedependencewith250observations.26 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 -17.3 -14 -14 -10.7 -10.7 -10.7 -7.4 -7.4 -4.1 -4.1 -0.8 -0.8 2.5 2.5 5.8 5.8 9.1 9.1 12.4 12.4 15.7 15.7 19 19 22.3 22.3 25.6 25.6 28.9 32.1 35.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -1.0 -0.5 0.0 0.5 1.0 1.5 b d1 -56.8 -46.9 -46.9 -43.7 -43.7 -37.1 -37.1 -33.8 -30.5 -27.2 -23.9 -20.6 -17.3 -14 -10.7 -7.4 -7.4 -4.1 -4.1 -0.8 -0.8 2.5 2.5 5.8 5.8 9.1 9.1 12.4 12.4 15.7 15.7 19 19 22.3 22.3 25.6 28.9 32.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 11.4 11.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 17.5 17.5 17.5 20.5 20.5 23.6 23.6 26.6 26.6 29.6 32.7 35.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 b d2 2.3 5.3 5.3 5.3 5.3 8.4 8.4 8.4 8.4 8.4 11.4 11.4 11.4 14.4 14.4 14.4 17.5 17.5 20.5 20.5 23.6 23.6 26.6 29.6 32.7 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 7.1 7.1 7.1 7.1 7.1 9 9 9 9 9 9 10.9 10.9 10.9 10.9 12.7 12.7 14.6 14.6 14.6 14.6 16.4 16.4 18.3 20.2 22 23.9 25.7 27.6 29.4 31.3 33.2 35 36.9 -1.0 -0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 d1 d2 -0.3 1.6 1.6 1.6 3.4 3.4 3.4 3.4 3.4 5.3 5.3 5.3 5.3 5.3 7.1 7.1 7.1 7.1 9 9 10.9 10.9 12.7 14.6 16.4 18.3 20.2 22 23.9 25.7 27.6 29.4 31.3 33.2