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)).

Itcanbeeasilyshownthatundergivenassumptionstheexplicitformofthevariance-covariancematrixV( ,σ2)isgivenby

(5) σ 2V( ,σ)= 21 2...

n 1 1 ... n 2 2 1... n 3·········...··· n 1 n 2 n 3 . ... 1

Notethenon-linearstructureofthematrixV( ,σ2)initsparameters,especiallyintheautoregressivecoe cient .Undertheconstraints| |<1andσ2>0,matrixV( ,σ2)alwaysremainspositivede nite.Asafunctionoftheparameters andσ2,V( ,σ2)belongstotheclassC2,i.e.totheclassoftwicedi erentiablefunctions.

Togetamodelwithlinearvariance-covariancestructureweconsider rst-order

22TaylorexpansionofV( ,σ2)aroundapriorvalue( 0,σ0).Let 0andσ0,| 0|<1

22andσ0>0,denotepriorvaluesfortheparameters andσ.Thenthelinearized

2variance-covariancematrixV( ,σ2)around( 0,σ0)isapproximatelygivenby

(6)2V( ,σ2)≈V0+( 0)V1+(σ2 σ0)V2,

2whereV0=V( 0,σ0),and

(7) V( ,σ2) V1= 0,σ20and V( ,σ2) V2=. 0,σ20

2RelationV0=σ0V2implies

(8)V( ,σ2)≈( 0)V1+σ2V2.

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

132´V.WITKOVSKY

IfwedenoteW( ,σ2)=( 0)V1+σ2V2,thelinearapproximationofthevariance-covariancematrix,thenthelinearmodel

(y,Xβ,W( ,σ2))

isthelinearmodelwithvarianceandcovariancecomponents,asusuallyconsideredinMINQE-theory,i.e.W( ,σ2)isalinearcombinationofknownsymmetricalmatrices,V1andV2,andunknownvariance-covariancecomponents,( 0)andσ2.Itshouldbeemphasized,however,thatwecannotgenerallyensurethepositivede nitenessofthematrixW( ,σ2).Itispositivede niteonlyinsu cientlyclose

2neighborhoodofapriorichosenpointoftheparameterspace( 0,σ0).

3.LocallyOptimumEstimatorsof andσ2

FollowingAza¨ setal.(1993),MINQE’s,( 0)andσ 2,ofthevariance-

covariancecomponents( 0)andσ2,respectively,computedforthepriorvalues0andσ0inthelinearizedmodel(y,Xβ,W( ,σ2)),leadsdirectlytoLMINQE’s, andσ 2,oftheparameters andσ2intheoriginalmodel(y,Xβ,V( ,σ2)):

(10) = 0+( 0)andσ 2=σ 2.

Moreparticularly,MINQE(U,I)inthelinearizedmodelleadstoLMINQE(U,I)inoriginalmodel,andsimilarly,MINQE(I)leadstoLMINQE(I).Inthepresentpaperweconcentrateourattentiontoinvariant(withrespecttothegroupoftranslationsy→y+Xα,forarbitraryα)estimationonly,i.e.toLMINQE(I)’s.

ThefollowingtheoremgivestheexplicitformoftheLMINQE(I)’sinthemodel(y,Xβ,V( ,σ2)).

Theorem1.Considerthelinearmodel(4)withautoregressivedisturbances.

2>0denotethepriorvaluesfortheautoregressivecoe cient Let| 0|<1andσ022andthevarianceσ.Further,lete=(e1,...,en) denotesthevectorof( 0,σ0)-

locallybestleastsquaresresiduals:e=y X(X V0 1X) X V0 1y,whereV0=

2).V( 0,σ0ThentheLMINQE(I)’s, andσ 2,oftheparameters andσ2,respectively,computedforthepriorvalue( 0,σ0),aregivenby

(11)

(12)nn n δ22 2 0 = 0+et+κetet 1 0(κ 0),et 10t=3t=1t=22nn n1 22 2σ =(1 δ)et 2 0etet 1+(1+δ) 0,et 1t=3t=1t=2

2whereκ=n (n 2) 20,andδ=(1 0)/κ.

Proof.Accordingto(10),wecomputetheMINQE(I)’sfortheparameters(

2 0)andσ2,respectively,atpriorvalue(0,σ0)inthemodel(y,Xβ,W( ,σ2)).

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

MODELSWITHAUTOREGRESSIVEDISTURBANCES133

BytheMINQE-theory,thesolutionisalinearcombinationoftwoquadratics,givenby

(13) 0q1 1=K(I),2σq2

whereK(I)denotesthe(2×2)criterionmatrixforinvariantquadraticestimation,withitselementsgivenby

(14){K(I)}ij=tr(V0 1ViV0 1Vj),i,j=1,2,

2whereV0=V( 0,σ0).(“tr”standsforthetraceoperatorwhichsumsthediago-

nalelementsofamatrix).Thequadraticsq1andq2arede nedas

(15)q1=e V0 1V1V0 1e,

q2=e V0 1V2V0 1e,

2wheree=y X(X V0 1X) X V0 1yisthevectorof( 0,σ0)-locallybestwighted

leastsquaresresiduals.

ConsideringthespecialformoftheinversionofthematrixV0,seee.g.Kmenta(1986),theformofthematricesV1andV2,de nedby(7),andaftersomealgebra,weget

(16) n(1 2)2 021 0(1 2 10)σ0K(= 4,I)2220 0(1 0)σ0n 1 (n 3) 0σ0and

(17) n n 2q1=etet 1 0e2t 1,0t=2t=3 n nn 1q2=e2etet 1+ 2e2t 2 00t 1.0t=1t=2t=3

TheexplicitformsoftheLMINQE(I)’s,givenby(11)and(12),areadirectcon-sequenceofthepreviouscalculationsandtheequation(13). Remark1.Generally,LMINQE’sarequadraticplusconstantestimators,i.e.theyareofthetypec+y Ay,wherecisaconstantandAisagivensymmetricalmatrix.Theoptimalitypropertiesof andσ aredirectconsequenceoftheresultsgiveninAza¨ setal.(1993):

2“LMINQE(I)’sof andσ2are( 0,σ0)-locallyoptimuminQCE(I)—the

classofquadraticplusconstantandinvariantestimators”.

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

134´V.WITKOVSKY

SimilarresultholdstrueforLMINQE(U,I),seeProposition5inAza¨ setal.(1993).

Remark2.Followingthelinesoftheproof,wecaneasily ndthattheex-

2 pectationoftheLMINQE(I)-vector( ,σ 2) ,locallyatthepoint( 0,σ0),isequal

to

(18) 12 E( ,σ 2) =( 0,0) +K(I)K(UI)(0,σ0) n rank(X) 1+=( 0,0) +K(,I)tr(V1(MV0M)),0

1whereK(I)isde nedby(16),K(UI)denotesthecriterionmatrixforunbiasedandinvariantquadraticestimation,withitselementsgivenby

(19){K(UI)}ij=tr((MV0M)+Vi(MV0M)+Vj),i,j=1,2,

andwhere(MV0M)+=V0 1 V0 1X(X V0 1X) X V0 1.

Undernormalityassumptions,i.e.iftheAR(1)processisGaussian,thevarian-

2 ce-covariancematrixofthevector( ,σ 2) ,locallyatthepoint( 0,σ0),isequal

to

(20) 1 1Var( ,σ 2) =2K(I)K(UI)K(I).

Formoredetailssee,e.g.J.Volaufov´aandV.Witkovsk´y(1992).

Remark3.Thestatisticalpropertiesoftheestimatorofthelinearfunctionpβofthe rst-orderparametersβinlinearmodelwithvarianceandcovariance

β=p (X V 1X) X V 1y,theso-calledplug-inortwo-stagecomponents,p =V( estimator,basedontheestimateofthevariance-covarincematrixV ,σ 2)

aregenerallyunknown.However,oneapproachtodeterminigtheupperboundforthedi erenceinvariancesoftheplug-inestimatorandtheBLUE,undertheassumptionofsymetryofthedistributionofεandtheexistenceof nitemomentsuptothetenthorder,wasproposedbyJ.Volaufov´a,(1993b).

TheLMINQE(I)’softheautoregressivecoe cient andthevarianceσ2are

2sensitivetothechoiceofthepriorvalues 0andσ0,respectively.Aninappropriate

choiceofthepriorparameters,i.e.inconsistentwiththeobserveddata,mayleadstotheestimate outoftheparameterspace| |<1.

Forpracticalapplications,ifthereisnostrongevidenceaboutthepriorval-ues,wesuggesttheso-calledtwo-stageLMINQE(I)’s,whicharebasedontwoiterationsofthefollowingtwo-stepmethod:

1.Firststep:Computeσ 2,accordingto(12),forthepriorchoice 0.(Note,that

2theestimatorσ 2doesnotdependonapriorvalueσ0,itdependsonlyon 0).

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

MODELSWITHAUTOREGRESSIVEDISTURBANCES135

2.Secondstep:Compute ,accordingto(11),atthepriorvalue( 0,σ 2).

Note,thatthistwo-stepmethodcomputedforthepriorvalue 0=0,leadstotheestimators

nn 1nt=2etet 1e2and =(21)σ 2=,t=1tt=1t

withet,t=1,...,n,beingtheordinaryleastsquaresresidualsgivenbye=y X(X X) X y.

Theseconditerationofthetwo-stepmethod,computedforthepriorvalue 0= , givenby(21),givesthetwo-stageLMINQE(I)’sofσ2and ,respectively.

4.MaximumLikelihoodEstimates

Thetwo-stepmethodcouldbenaturallyexpandedtotheiterativealgorithmbyarepeatedapplicationofthosetwostepsuntilconvergenceisreached.

2 NdenotetheestimatesaftertheNthstageoftheProvidedthat Nandσ

iterationprocedure,the(N+1)ststageofthealgorithmisgivenby:

1.Firststep:Computethenewvectorofresiduals

(22) 1 1e=y X(X VN+1X)XVN+1y,

22 N,σ N).Further,computeσ NwhereVN+1=V( +1followingtheformula(12),

andreplacing 0by N.

N2.Secondstep:Compute N+1,followingtheformula(11),asafunctionof

2andσ N+1.

Note,thattheeventuallimitpointsoftheaboveiterativealgorithmareequiva-lenttotheeventuallimitpointsofiteratedLMINQE(I)’s,denotedasILMINQE(I)’s,oftheparameters andσ2,respectively.Thelimitpointswedenoteby and σ2,respectively.

Aza¨ setal.(1993)haveprovedtheequivalencebetweentheFisherScoringAlgorithmfor ndingsingularpointsoftheGaussianlikelihood,and ILMINQE(I)’s.Followingthoseresults,thepoint(β, , σ2),withILMINQE(I)’s

2 suchthatand σ,andβ

1 2 1 (23)Xβ=XXV( , σ)XX V( , σ2)y,

isthesingularpointfortheGaussianlog-likelihoodfunction

(24)L(β, ,σ2)=const 111log|V( ,σ2)| (y Xβ) V( ,σ2)(y Xβ), i.e.L(β, , σ2)=maxL(β, ,σ2).Hence,thefollowingtheoremholdstrue:

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

136´V.WITKOVSKY

Theorem2.Eventuallimitpointsoftheaboveiterationprocedure, and σ,arethemaximumlikelihoodestimates,MLE’s,oftheautoregressivecoe -cient andthevarianceσ2inthelinearmodelwithautoregressivedisturbances(y,Xβ,V( ,σ2)).2

5.Discussion

Toillustratethepropertiesoftheproposedestimatorsweconsiderthesimpleformmodelofthequantitytheoryofmoney,originallydisscusedbyFriedmanandMeiselman(1963),seeTable1:

(25)Ct=α+βMt+εt,

withC—consumerexpenditureandM—stockofmoney,bothmeasuredinbillionsofcurrentdollars.Itisassumedthatthedisturbancesfollowa rst-orderautoregressivescheme.YearQuarter

1952I

II

III

IV

I

II

III

IV

I

IIConsumerMoneyexpenditurestock214.6217.7219.6227.2230.9233.3234.1232.3233.7236.5159.3161.2162.8164.6165.9167.9168.3169.7170.5171.6YearQuarter19541955IIIIVIIIIIIIVIIIIIIIVConsumerMoneyexpenditurestock238.7243.2249.4254.3260.9263.3265.6268.2270.4275.6173.9176.1178.0179.1180.2181.2181.6182.5183.3184.3195319561954

Table1.Consumerexpenditureandstockofmoney,1952(I)—1956(IV),bothmeasuredinbillionsofcurrentdollars.Source:M.Frie-dmanandD.Meiselman:“TherelativestabilityofmonetaryvelocityandtheinvestmentmultiplierintheUnitedStates,1897–1958”,In:CommisiononMoneyandCredit,StabilizationPolicies(EnglewoodCli s,NJ:Prentice-Hall,1963),p.266.

WewillconsiderthreetypesofLMINQE(I)’softheautoregressiveparameter inthemodel(25),whichdi erinthechoiceofthepriorvaluesoftheparameters

2 0andσ0,respectively:

2),whereσ 21.EstimatorI:LMINQE(I)of computedforallpriorvalues( 0,σ

isgivenby(12),and| 0|<1.

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

MODELSWITHAUTOREGRESSIVEDISTURBANCES137

Thisestimatorwegetasasingleiterationofthetwo-stepmethodforcomputingLMINQE(I)oftheparameter .Theestimatorseemstobehighlysensitivetothechoiceofthepriorvalueof .

2),where2.EstimatorII:LMINQE(I)of computedforallpriorvalues( 0,σLSE2σLSEistheestimatebasedontheordinaryleastsquaresresiduals,givenby(21),

andwith| 0|<1.

Thisestimatorseemstobequiterobusttoallpossiblechoicesoftheprior

2valueof 0.ThereasonisthatσLSEistheupperboundfortheallpossible

estimatesofσ2.

3.EstimatorIII:LMINQE(I)of computedforallpriorvalues( 0,(1 20)

22),whereσLSEistheestimatebasedontheordinaryleastsquaresresiduals,σLSEgivenby(21),and| 0|<1.

Theestimatorisamodi cationtothepreviousone.Hereweputthevariance

22ofthedisturbance,undergivenpriorvalues 0andσ0,Var(εt)=σ0/(1 20),

2tobeconstantandequaltoσLSE.

Figure1.Estimatesoftheautoregressivecoe cient fordi erent

priorvaluesof 0.EstimatorI—solidline;EsimatorII—dashedline;

EstimatorIII—dasheddottedline;MLE—dottedline.

ThevaluesoftheconsideredestimatorscanbeeasilyseenfromtheFigure1.Note,thattheEstimatorI,i.e.LMINQE(I)computedbytwo-stepmethod,ishighlysensitivetothechoiceofthepriorvalue 0oftheparameter .Eachofthoseestimators,ifiterated,leadstothemaximumlikelihoodestimate.

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

138

Stage

1

2

3

4

5

6 σ2´V.WITKOVSKYαβLog-likelihood-48.7363-44.0237-44.0010-44.0003-44.0003-44.00030.791514.2593-154.79292.30080.85884.5273-157.79192.32550.84834.4785-156.23562.31920.84704.4771-156.60782.32090.84704.4771-156.64742.32100.84704.4771-156.64962.3211

Table2.IterationstepsofthealgorithmforcomputingtheMLE’softheparameters ,σ2,αandβinthemodel(25).

Finally,intheTable2weillustratethespeedofconvergenceoftheproposedalgorithmforcomputingMLE’softheparametersofthemodel(25).Foreachstagethetablegivestheestimatesof ,σ2,α,andβ,togetherwiththevalueoflog-likelihoodfunction.

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