Multivariate Portmanteau test for Autoregressive models with uncorrelated but nonindependent errors

HAL Id: hal-00517078

https://hal.archives-ouvertes.fr/hal-00517078

Submitted on 13 Sep 2010

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

models with uncorrelated but nonindependent errors

Christian Francq, Hamdi Raïssi

To cite this version:

Christian Francq, Hamdi Raïssi. Multivariate Portmanteau test for Autoregressive models with un-

correlated but nonindependent errors. Journal of Time Series Analysis, Wiley-Blackwell, 2007, 28,

pp.454-470. �hal-00517078�

DENT ERRORS

CHRISTIANFRANCQ,

∗

GREMARSUniversitéLille3

HAMDIRAÏSSI,

∗∗

GREMARSUniversitéLille3

Abstrat

Inthispaperweonsiderestimationandtestoftformultipleautoregressive

timeseriesmodelswithnonindependentinnovations.Wederivetheasymptoti

distributionoftheresidualautoorrelations. It isshownthatthis asymptoti

distributionanbequitedierentformodelswithiidinnovationsandmodelsin

whihtheinnovationsexhibitonditionalheterosedastiity orother formsof

dependene. Consequently,theusual hi-squaredistributiondoesnotprovide

adequate approximation to the distributionof the Box-Piere goodness-of-t

portmanteau test in the presene of nonindependent innovations. We then

proposeamethodtoadjusttheritialvaluesoftheportmanteautests. Monte

Carlo experiments illustrate the nite sample performane of the modied

portmanteautest.

Keywords: VetorweakARmodel,Goodness-of-ttest,Residualautoorrela-

tion,Diagnosti Cheking,Box-PiereandLjung-Boxportmanteautests.

1. Introdution

In multivariate time series analysis,the Vetorial AutoRegressive (VAR) models are

muhemployed(seeLütkepohl(1993)). TheVARmodelspostulatethatthed^-dimen-

sionalvetorXt^oftheobservationsattimet^anberepresentedasalinearombination of p^{past values}Xt−1, . . . , Xt−p ^{plus anerror} ǫt^{. A theoretial argument in favor of}

the VAR models is that any stationary proess an be approximated by a VAR(p)

model with suiently largep^and unorrelatederrors. Thereason ofthe suessof these modelsishoweverofpratialnature,andisertainly dueto thefatthat itis

relativelyeasytodealwithalinearfuntionofanitenumberofpastvalues.

ItishoweverlearthattheVARmodelsarenotuniversalandthatthehoieofthe

order p^{is ruial. Thus it isimportantto hekthe validity of aVAR}(p)^{model, for}

agivenorderp^{. In}multivariate,thehoieofp^ispartiularlyimportantbeausethe numberofparameters,pd^{2,quiklyinreaseswith}p^{,whihentailsstatistialdiulties.}

Thispaperisdevotedtotheso-alledportmanteautestsonsideredbyChitturi(1974)

andHosking(1980)forhekingtheoverallsignianeoftheresidualautoorrelations

of a VAR(p) ^{model (see also Ahn (1988), Hosking (1981a, 1981b), Li and MLeod}

(1981)).

∗

Postaladdress: GREMARS,UFRMSES,UniversitéLille3,Domaine duPontdebois,BP149,

59653Villeneuved'AsqCedex,Frane

∗∗

Postaladdress: GREMARS,UFRMSES,UniversitéLille3,DomaineduPontdebois,BP149,

59653Villeneuved'AsqCedex,Frane

ForthestatistialanalysisofVARmodels,theerrorsǫt^{aregenerallysupposedtobe}

independent(as in Lütkepohl (1993),Denition 3.1). Thisindependene assumption

is restritive beause it preludes onditional heterosedastiity and/or other forms

ofnonlinearity(see Franq, Royand Zakoïan (2005)andFranqand Zakoïan (2005)

forthestatistialinfereneofunivariateARMAmodelswith unorrelatedbut nonin-

dependent errors,and see Dufour andPelletier (2005)forweak VARMA modelling).

Themaingoalofthepresentpaperisto studythebehaviouroftheabove-mentioned

goodness-of-tportmanteautestswhentheǫt^'sarenotindependent. Wewillseethat theresultsobtainedbythestandardportmanteautestsanbequitemisleadinginthis

framework.A modiedversionofthese testsisproposed.

Brüggemann, Lütkepohl and Saikkonen (2004) is an important reent referene

dealing with portmanteau tests and other tests for residual autoorrelation in VAR

models with iid innovations when somevariables are ointegrated. Inthis paperwe

donotonsiderointegratedvariables,buttheindependeneassumptionoftheǫt^'sis

relaxed.

Therest of thepaperis organizedas follows. Setion 2 provides examplesof AR

models withunorrelatedbut nonindependenterrors,presentsseveralexpressions for

theleastsquares (LS)estimatorofthe ARoeients,and givesonditionsensuring

theonsisteny andasymptoti normality oftheLS estimator. Setion 3studies the

asymptotibehaviourof the ARresiduals. Weobtainthe asymptotidistribution of

vetorsof residual autoorrelations, under the assumption of the tted AR(p) ^{is an}

adequatelinearmodel with awellhosenorder p^{. The resultsareapplied in Setion}

4to obtainthe asymptoti distribution of the portmanteau tests and to modify the

ritialvaluesofthesetestswhentheyareappliedtoVARmodelswithnonindependent

errors. Setion5isdevotedtothepratialimplementationofthemodiedversionof

thetests. Setion6proposesnumerialillustrations. Thetehnialproofsarerelegated

totheappendix.

Thefollowing notationswill be used throughout. For amatrix A ^{of generi term} A(i, j)^{weuse thenorm}kAk=P

|A(i, j)|^{. Thespetralradiusof asquarematrix}A

is denoted by ρ(A)^{, its traeis denoted by Tr. We denote by} A⊗B ^{the Kroneker}

produt oftwomatriesA ^andB^{, ve}A ^{denotesthevetorobtainedbystakingthe}

olumnsofA^,andA^{⊗2standsfor}A⊗A^{(seee.g. Harville(1997)formoredetailsabout}

thesematrixoperators). Thesymbol⇒^{denotestheonvergenein}distribution.

2. LSestimator of weak VARmodels

ConsiderthevetorialAR(p^)model Xt=

Xp

i=1

A0iXt−i+ǫt ^forallt∈Z={0,±1,±2, . . .} ^(2.1)

where theǫt^'sare d-dimensionalerror terms,theXt^'sared-dimensional vetors,and theA0i^'sared×d^{matries. Itisustomarytosaythat} (Xt)^isastrongAR(p^)model

if(ǫt)^{isastrongwhitenoise,thatis, ifitsatises}

TheauthorsaregratefultotheProfessorLütkepohlwhosequestionsheaskedduringtheMadrid

A1: (ǫt) ^{isa sequeneof} independent and identially distributed (iid)random vetors,Eǫt= 0^andVar(ǫt) = Σǫ^.

Wesaythat(2.1)isaweak AR(p^{)model if}(ǫt)^{isaweakwhitenoise,thatis, ifit}

satises

A1': Eǫt= 0^,Var(ǫt) = Σǫ^{,and Cov}(ǫt, ǫt−h) = 0^forallt∈Z^andallh6= 0^.

Assumption A1 is learly stronger than A1'. The lass of strong AR models is

oftenonsidered toorestritivebypratitioners. IndeedAssumption A1amountsto

assume that the best preditor of Xt ^{is a linearombinationof its} p ^{pastvalues. If} p^{is hosenlargeenough,it is reasonableto onsider that thebest preditor of}Xt ^is

wellapproximatedbyafuntion ofXt−1, . . . , Xt−p^,butitisquestionabletoassumea linearformforthisfuntion. Itiswellknownthatnumerousnonlinearproessesadmit

weaklinearrepresentations(seeExample2.1below). Weaklinearrepresentationsare

alsoobtainedfromtransformationsofstronglinearproesses(seeExample2.2below).

In Examples 2.1-2.3 below, Assumption A1' holds but A1 is not satised. Other

examplesofunivariateweak linearmodels an befound in Franq, Roy andZakoïan

(2005),andreferenestherein.

For the statistial analysis of multivariate AR time series models, it is therefore

of interest to replae the standard strong white noise assumption A1 by the more

exible weak white noiseassumption A1'. Inthis paperwe fous on theestimation

andvalidationstagesofthestatistialanalysisoftheseweakmultivariateARmodels.

2.1. Examples of weak VARmodels

Theexamplesgivenin thissetionaremainly hosenfortheirsimpliity. Therst

oneisaweakwhitenoiseinspiredbyexamplesgivenbyRomanoandThombs(1996)

intheunivariatease. Theseondissimplytheausalrepresentationofanonausal

AR(1) proess. This example shows that A1 must by replaed by A1' when one

wantsto make,withoutlossofgenerality, theusualassumption that therootsof the

ARpolynomialareoutsidetheunit irle. Thethird belongstothelass ofGARCH

models.

Example2.1. Intheunivariatease,Romanoand Thombs(1996)builtweak white

noises(ǫt) ^bysetting ǫt =ηtηt−1· · ·ηt−k ^where (ηt) ^{is astrongwhite noiseand} k ≥ 1^{. This approah an be extended to the} d-multivariate framework by setting ǫt = B(ηt)· · ·B(ηt−k+1)ηt−k^,where{ηt= (η1t, . . . , ηd t)^′}t^isad-dimensionalstrongwhite noise, and B(ηt) = {Bij(ηt)} ^{is a} d×d ^{random matrix whose elements are linear}

ombinationsof theomponentsof ηt^{. It isobvioustohekthat} ǫt^{is awhitenoise,}

butingeneralthisnoiseisnotastrongone. Indeed,assumingforsimpliitythatk= 1

andB11(ηt) =η1t ^andB1j(ηt) = 0^forallj >1^,wehave

Cov ǫ²_1t, ǫ²_1t−1

=

Eη²_1t ^2Var

η²_1t 6= 0,

whihshowsthattheǫt^'sarenotindependent.

Example2.2. (NonausalAR(1))LettheAR(1) model

Xt=AXt−1+ǫt, ǫt ^iidand E(ǫt) = 0, E(ǫtǫ^′_t) = Σǫ,

where A ^{is an invertiblematrix whose all the} eigenvalues λi,1 ≤i ≤d, ^{are greater}

thanone in modulus. Thisequation hasastationary andantiipativesolutionof the

formXt=−P∞

i=1A⁻ⁱǫt+i. ^Theautoovarianefuntion of(Xt)^{isthengivenby} ΓX(h) =A^−h

X∞

i=1

A⁻ⁱΣǫA⁻ⁱ, h≥0.

Let ǫ^∗_t = Xt−A⁻¹Xt−1^{. We have} E(ǫ^∗_t) = 0^{, Var}(ǫ^∗_t) = ΓX(0) + ΓX(1)(A⁻¹)^′ + A⁻¹ΓX(1) +A⁻¹ΓX(1)(A⁻¹)^{′, andCov}(ǫ^∗_t, ǫ^∗_t−h) = 0 ^forh6= 0^{. Thus} Xt ^{admits the}

ausal AR(1) representation Xt = A⁻¹Xt−1+ǫ^∗_t^{. However, in general,} ǫ^∗_t ^{is not a}

martingale dierene. To see this, assume for simpliity that the matrix A ^{is suh}

that |a11| >1 ^and a1j =aj1 = 0, ∀j ∈ {2,· · · , d}. ^{Wethen have} EX₁³_t−1 = (1− a³₁₁)⁻¹Eǫ³_1t^, EX1tX₁²_t−1=a⁻²₁₁(1−a³₁₁)⁻¹Eǫ³_1t^andE(ǫ^∗_1tX₁²_t−1) =EX1tX₁²_t−1− a⁻¹₁₁EX₁³_t−16= 0^whenEǫ³_1t6= 0^{. Inthisase}ǫ^∗_t ^{isnotamartingaledierenebeause} E(ǫ^∗_1tX₁²_t−1) =E

X₁²_t−1E ǫ^∗_1t|ǫ^∗_t−1, . . . 6= 0^{. Thusthewhitenoise}ǫ^∗_t ^isnotstrong.

Example2.3. Intheunivariatease,theGARCHmodelsonstituteimportantexam-

plesofweakwhitenoises. Thesemodelshavenumerousextensionstothemultivariate

framework. The simplest of these extensions is ertainly the multivariate GARCH

model with onstant orrelation proposed by Jeantheau (1998). In this model the

proess(ǫt)^{veriesthefollowingrelation}ǫt= ∆tηt^whereηt= (η1t, . . . , ηd t)^{′ isaniid}

entered proesswith Var(ηi t) = 1^{, and}∆t ^{isadiagonal matrixwhose elements}σii t

verify



 σ²_11t

.

.

.

σ_{dd t}²



=



 c1

.

.

.

cd



+ Xq

i=1

Ai



 ǫ²_1t−i

.

.

.

ǫ²_{d t−i}



+ Xp

j=1

Bj



 σ²_11t−j

.

.

.

σ_{dd t−j}²



.

TheelementsofthematriesAi^andBj^,aswellastheci^{'s,aresupposedtobepositive.}

In addition suppose that the stationarity onditions hold (see Jeantheau (1998) for

more details). The stationary solution of this GARCH equation satises A1^′, but doesnotsatisfyA1ingeneral. ConsiderforinstanetheARCH(1)asewithA1 ^suh

that a11 6= 0^and a12 =· · · =a1n = 0^{. Thenit iseasy to seethat Cov}(ǫ²_1t, ǫ²_1t−1) =

Cov

(c1+a11ǫ²_1t−1)η₁²_t, ǫ²_1t−1 =a11^Varǫ²_1t6= 0^{,whihshowsthattheiidassumption} A1is notsatised.

2.2. Derivation of theLS estimator

Itiswellknownthat(Xt)^{an bewrittenas aMA}(∞)^oftheform Xt=

X∞

i=0

ψ0iǫt−i, ψ00=Id, X∞

i=0

kψ0ik<∞, ^(2.2)

undertheassumption

A2: detA0(z)6= 0^forall|z| ≤1^,where A0(z) =Id−Pp

i=1A0iz^i.

Denote by θ0 = ^ve (A01· · ·A0p) ^{the vetor of the unknown AR} parameters. For any θ ∈ R^{d2p, let} A1 = A1(θ), . . . , Ap = Ap(θ) ^be d×d ^{matries suh that} θ =

ve(A1, . . . , Ap)^{. With thisnotation(2.1)an berewrittenas} Xt=

X_t−1^′ , . . . , X_t−p^′

⊗Id θ0+ǫt ^(2.3)

usingtheelementaryrelationve(ABC) = (C^′⊗A)^veB ^formatriesofappropriate dimensions.

One of the most popular estimation proedure is that of the least squares (LS)

estimator. For linear proesses of the form (2.1), the LS estimator of θ ^oinides

withthegaussianquasi-maximumlikelihood(QML)estimator. Giventheobservations

X1, . . . , Xn^{,theLS/QMLestimatorsof}θ^andΣǫ ^aredenedby

(ˆθn,Σˆǫ) = arg min

θ,Σǫ

(

nlog (det Σǫ) + Xn

t=1

ǫ^′_t(θ)Σ⁻¹_ǫ ǫt(θ) )

where

ǫt(θ) =Xt− Xp

i=1

AiXt−i.

To give an expliit expression for these estimators, the d-dimensional AR(p^{) model}

(2.1)anberewrittenasthedp-dimensionalAR(1)model

X˜t= ˜A0X˜t−1+ ˜ǫt, ^(2.4)

where

A˜0=







A01 · · · A0p−1 A0p

Id 0

.

.

.

0 Id 0





, X˜t=





 Xt

Xt−1

.

.

.

Xt−p+1





, ˜ǫt=





 ǫt

0

.

.

.

0





.

NotethatA2isequivalenttoρ A˜0

<1^{. Let}ΣˆX˜t,X˜t−h= _n¹Pn

t=1X˜tX˜_t−h^{′ andwrite} ΣˆX˜t−1

instead of ΣˆX˜t−1,X˜t−1

. Note that ΣˆX˜t−1

is a onsistent estimator of ΣX˜t = EX˜tX˜_t^{′,whihisgivenby}

ve ΣX˜t

=

I(dp)²−A˜0⊗A˜0

−1

ve (Σ˜ǫt), Σǫ˜t=

Σǫ 0^′_d(p−1)

0d(p−1) 0d(p−1)×d(p−1)

.

It iseasy tosee that theLSestimatorsof theAR parametersofmodels(2.1)and

(2.4)aregivenby

b˜ A=







Ab1 · · · Abp−1 Abp

Id 0

.

.

.

0 Id 0





= ˆΣX˜t,X˜t−1

Σˆ⁻¹_˜

Xt−1, ^(2.5)

provided ΣˆX˜t−1

is non singular, and that the LS estimators of the varianes of the

noises(˜ǫt)^and(ǫt)^aregivenby Σbǫ˜= Σbǫ 0^′_d(p−1)

0d(p−1) 0d(p−1)×d(p−1)

!

= ˆΣX˜t−ΣˆX˜t,X˜t−1Σˆ⁻¹_˜

Xt−1

ΣˆX˜t−1,X˜t, ^(2.6)

withtheonventionthat Xt= 0^whent≤0 ^ort > n^.

2.3. Asymptoti behaviorofthe LS estimator

Toestablishthestrongonsistenyoftheestimatorsdenedin (2.5)and(2.6),we

needthefollowingassumptions.

A3: MatrixΣǫ ^{ispositivedenite.}

A4: Thesequene(ǫt)^{isstritlystationary andergodi.}

Note that A4 isentailedby A1, but notby A1^′. A straightforwardonsequeneof theergoditheoremisstatedinthefollowingproposition.

Proposition 2.1. UnderassumptionsA1-A2-A3orA1'-A2-A3-A4,thematrixΣˆX˜t

isalmost surelynonsingular, andalmost surely

b˜

A→A˜0, Σb˜ǫ→Σ˜ǫ ^and θˆn→θ0

asn→ ∞^.

ToobtaintheasymptotinormalityoftheLSestimatorofθ^,additionalassumptionsare neededwhen(ǫt)^{isnotiid. Intheunivariatease,FranqandZakoïan(1998)showed}

the asymptoti normality under mixing assumptions. We will extend this result to

VARmodels. ThemixingoeientsofastationaryproessX = (Xt)^aredenotedby αX(h) = sup

A∈σ(Xu,u≤t),B∈σ(Xu,u≥t+h)|P(A∩B)−P(A)P(B)|.

Thereader isreferredto Davidson(1994)for details aboutmixing assumptions. Let

kXk^r= (EkXk^r)^1/r,where kXk^{denotestheEulideannorm.}

A5: TheproessX= (Xt)^issuhthatP∞

h=0{αX(h)}^ν/(2+ν)<∞^andkXtk4+2ν

<∞^forsomeν >0^.

Theasymptotidistribution oftheLSestimatorisgiveninthefollowingproposition.

Proposition 2.2. Under assumptionsA1A3 orunderA1' andA2A5,

√n^ve b˜ A−A˜0

⇒ N(0,Ω), ^(2.7)

where

Ω = X∞

h=−∞

En Σ⁻¹_X˜

t

X˜t−1X˜_t−h−1^′ Σ⁻¹_X˜

t ⊗˜ǫt˜ǫ^′_t−ho

. ^(2.8)

Moreover

√n θˆn−θ0

⇒ N 0,Σθˆn

, ^(2.9)

where

Σθˆn = X∞

h=−∞

En Σ⁻¹_X˜

t

X˜t−1X˜_t−h−1^′ Σ⁻¹_X˜

t ⊗ǫtǫ^′_t−ho

. ^(2.10)

Notethatunder A1,wehave

Ω = Σ⁻¹_˜

Xt ⊗Σ˜ǫt.

and

Σθˆn = Σ⁻¹_X˜

t ⊗Σǫt. ^(2.11)

ThisstandardresultanbefoundinJohansen(1995,Theorem2.3p. 19).

Example2.4. The following example illustrates the dierene between the asymp-

totivariane(2.10)intheweakaseandtheasymptotivariane(2.11)ofthestrong

AR ase. Consider a bivariate AR(1) model Xt = AXt−1 +ǫt^{, with true AR(1)}

parameterA0= 0^andǫt=B(ηt). . . B(ηt−k+1)ηt−k ^{oftheformgiveninExample2.1.}

WehaveXt= ˜Xt=ǫt^{. For simpliity assumethat} B(ηt) = ^Diag(η1t, η2t)^{and that} η ^{is gaussianwith Var}(η1t) =^Var(η2t) = 1 ^{and Cov}(η1t, η2t) =ρ ∈(−1,1)^{. Thus,}

using Eη₁³_tη2t = 3ρ ^and Eη₁²_tη²_2t = 1 + 2ρ^{2, we ndthat Var}(ǫ1t) = ^Var(ǫ2t) = 1^,

Cov(ǫ1t, ǫ2t) =ρ^{k+1,and,using(2.10),} Σθˆn = E

Σ⁻¹_ǫt ǫt−1ǫ^′_t−1Σ⁻¹_ǫt ⊗ǫtǫ^′_t

= Σ⁻¹_ǫt ⊗I2

E

(ǫt−1⊗ǫt) (ǫt−1⊗ǫt)^′ Σ⁻¹_ǫt ⊗I2

,

withΣǫ=

1 ρ^k+1 ρ^k+1 1

andE

(ǫt−1⊗ǫt) (ǫt−1⊗ǫt)^{′ equalto}







3^k ρ(3ρ)^k ρ(3ρ)^k ρ²(1 + 2ρ²)^k ρ(3ρ)^k (1 + 2ρ²)^k ρ²(1 + 2ρ²)^k ρ(3ρ)^k ρ(3ρ)^k ρ²(1 + 2ρ²)^k (1 + 2ρ²)^k ρ(3ρ)^k ρ²(1 + 2ρ²)^k ρ(3ρ)^k ρ(3ρ)^k 3^k





.

Whenk= 1^thematrixΣθˆn

isequalto

ΣW := 1

(ρ²+ 1)(1−ρ⁴)×





−2ρ⁴+ 3ρ²+ 3 ρ²(ρ²+ 3) −3ρ⁴−ρ² ρ²(−2ρ⁴−3ρ²+ 1) ρ²(ρ²+ 3) 3ρ²+ 1 ρ²(−2ρ⁴−3ρ²+ 1) −ρ²(3ρ²+ 1)

−ρ²(3ρ²+ 1) ρ²(−2ρ⁴−3ρ²+ 1) 3ρ²+ 1 ρ²(ρ²+ 3) ρ²(−2ρ⁴−3ρ²+ 1) −3ρ⁴−ρ² ρ²(ρ²+ 3) −2ρ⁴+ 3ρ²+ 3



.

Byomparison, forthestrongAR(1)withaniidnoisewithvariane

Σǫ=

1 ρ² ρ² 1

,

theasymptotivarianeΣθˆn

isgivenby

ΣS := 1 1−ρ⁴







1 ρ² −ρ² −ρ⁴ ρ² 1 −ρ⁴ −ρ²

−ρ² −ρ⁴ 1 ρ²

−ρ⁴ −ρ² ρ² 1





.

Figure 2.1displays theratioΣW(1,1)/ΣS(1,1) ^{asfuntion of}ρ^{. Itis learfrom this}

examplethattheasymptotivarianeoftheLSestimatormaybequitedierentwhen