Abstract. Estimation of the autoregressive coefficient ? in linear models with firstorder autoregressive disturbances has been broadly studied in the literature. Based on C.R. Rao’s MINQE-theory, Aza?s et al. (1993) gave a new general approach for computi

enianae

Vol.LXV,1(1996),pp.129–139129

ONVARIANCE–COVARIANCECOMPONENTSESTIMATION

INLINEARMODELSWITHAR(1)DISTURBANCES

´V.WITKOVSKY

Abstract.Estimationoftheautoregressivecoe cient inlinearmodelswith rst-

orderautoregressivedisturbanceshasbeenbroadlystudiedintheliterature.Based

onC.R.Rao’sMINQE-theory,Aza¨ setal.(1993)gaveanewgeneralapproachfor

computinglocallyoptimumestimatorsofvariance-covariancecomponentsinmodels

withnon-linearstructureofthevariance-covariancematrix.Asaspecialcase,in

thelinearmodelwithAR(1)errors,wediscussanewalgorithmforcomputing

locallyoptimumquadraticplusconstantinvariantestimatorsoftheparameters

andσ2,respectively.Moreover,simpleiterationofthisestimationproceduregivesa

maximumlikelihoodestimatesofboth,the rstorderparameters,andthevariance-

covariancecomponents.

1.Introduction

Linearmodelswithdisturbancesthatfollow rst-orderautoregressivescheme,abbreviatedasAR(1),arefrequentlyconsideredineconometricalapplications.Greatimportanceisgiventothesecond-orderparameters—ly,totheautoregressivecoe cient andtothevarianceσ2,whichareusuallyunknownandaretobeestimated.

Estimationofvariance-covariancecomponentsinmodelswithAR(1)errorshasalonghistory.ThesimplestestimatorsarebasedonverynaturalideatochangetheunobservabledisturbancesbythecorrespondingleastsquaresresidualsinthegeneratingschemeforAR(1)process.Ontheotherside,undernormalityassump-tions,fullMLE’s,MaximumLikelihoodEstimators,arestudied.Generally,mostofthoseestimatorsreachsomeofoptimalityproperties,atleasttheyareconsistentestimatorsof andσ2.Formoredetailssee,e.g.,PraisandWinsten(1954),Durbin(1960),Magnus(1978),andKmenta(1986).

Inarecentpaper,Aza¨ setal.(1993)gaveageneralizationofC.R.Rao’sMin-imumNormQuadraticEstimationTheoryofestimationvarianceand/orReceivedMarch20,1996;revisedMay13,1996.

1980MathematicsSubjectClassi cation(1991Revision).Primary62J10,62F30.

Keywordsandphrases.Autoregressivedisturbances,variance-covariancecomponents,min-imumnormquadraticestimation,maximumlikelihoodestimation,Fisherscoringalgorithm,MINQE(I),LMINQE(I),AR(1),MLE,FSA.

Abstract. Estimation of the autoregressive coefficient ? in linear models with firstorder autoregressive disturbances has been broadly studied in the literature. Based on C.R. Rao’s MINQE-theory, Aza?s et al. (1993) gave a new general approach for computi

130´V.WITKOVSKY

covariancecomponentstothemodelswithnon-linearvariance-covariancestruc-ture.TheirnewLMINQE,LinearizedMINQE,isde nedasaMINQE,(moreparticularlyasaninvariantMINQE(I)oranunbiasedandinvariantMINQE(U,I)),afterlinearizationofthevariance-covariancematrixatapriorvalueoftheparam-eters.Rao’sMINQE-method,incontrasttoMLE-method,involvesonlyslightdistributionalassumptions,especially,theexistenceofthe rstfourmomentsoftheprobabilitydistribution.Moreover,MINQE’sarelocallyoptimumforapri-orichosenEuclideannormandtheiriterationarenumericallyequivalenttothewellknownFSA,FisherScoringAlgorithm—anumericalmethodfor nd-ingsingularpointsofthelikelihoodfunction,whichleadstoGaussianmaximumlikelihoodestimates.FormoredetailsonMINQE-theoryandvariance-covariancecomponetsestimationseee.g.C.R.Rao(1971a),C.R.Rao(1971b),C.R.RaoandJ.Kle e(1980),C.R.RaoandJ.Kle e(1988),S.R.Searleetal.(1992),J.Volaufov´aandV.Witkovsk´y(1992),andJ.Volaufov´a(1993a).

Inthispaper,basedontheabovementionedresults,anewalgorithmforesti-mationthevariance-covariancecomponentsisgiven,i.e. andσ2,inlinearmodelwithAR(1)disturbances.Theveryspecialcharacterofthismodelallowusto ndclosed-formformulasforLMINQE’s,locallyoptimuminvariantestimatorsof andσ2.Moreover,becauseofthelinkbetweeniteratedLMINQE’sandFisherScoringAlgorithm,wegetdirectlyaniterativemethodfor ndingGaussianMLE’s.Assuch,thesuggestedalgorithmserves,inthespecialcaseofthemodelwithconstantmean,asaspecialalternativetotheclassicalalgorithmsforcomputingMLE’softheparametersoftheAR(1)process,seee.g.J.P.BrockwellandR.A.Davis(1987).Finally,wenotethattheapproachintroducedbyAza¨ setal.(1993)isquitegeneralandcanbeused,e.g.,foranylinearmodelwithARMA(p,q)disturbances.However,theclosed-formformulasarestillasubjectoffurtherinvestigation.

2.ModelwithLinearizedVariance-CovarianceStructure

Weconsiderlinearregressionmodel

(1)yt=xtβ+εt,t=1,...,n,

whereytrepresentsobservationintimet,xt=(x1t,...,xkt)isavectorofknownconstants,andβ=(β1,...,βk) isavectorofunknown rst-orderparameters.WeassumethatthedisturbancesfollowastationaryAR(1)processwhichstartedattimet= ∞,i.e.theerrorsaregeneratedbythefollowingscheme:

(2)ε1=u1/,

εt= εt 1+ut,t=2,...,n,

where| |<1,isanautoregressivecoe cient,generallyunknownparameter,andut,t=1,...,n,representuncorrelatedrandomerrorswithzeromeanandthe

Abstract. Estimation of the autoregressive coefficient ? in linear models with firstorder autoregressive disturbances has been broadly studied in the literature. Based on C.R. Rao’s MINQE-theory, Aza?s et al. (1993) gave a new general approach for computi

MODELSWITHAUTOREGRESSIVEDISTURBANCES131

varianceσ2>0,whichisalsosupposedtobeanunknownparameter.Generally,wedonotassumenormalityoftheprobabilitydistributionoftheerrors.However,weassumetheexistenceofthethirdandfourthmoments.

Themodelcanberewrittentothematrixform

(3)y=Xβ+ε,

withexpectationE(y)=Xβandthevariance-covariancematrixVar(y)=V( ,σ2),wherey=(y1,...,yn) ,Xis(n×k)-dimensionalmatrixwithxtbeeingthet-throw,andε=(ε1,...,εn) .Weusuallydenotethemodelas

(4)(y,Xβ,V( ,σ2)).

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