TESTING THE COINTEGRATING RANK WHEN THE ERRORS ARE UNCORRELATED BUT NONINDEPENDENT

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THE ERRORS ARE UNCORRELATED BUT NONINDEPENDENT

Hamdi Raïssi

To cite this version:

Hamdi Raïssi. TESTING THE COINTEGRATING RANK WHEN THE ERRORS ARE UNCORRE-

LATED BUT NONINDEPENDENT. Stochastic Analysis and Applications, Taylor & Francis: STM,

Behavioural Science and Public Health Titles, 2009, 27, pp.24-50. �hal-00517094�

HAMDIRAÏSSI,

∗

Université Lille3

Abstrat

We study the asymptoti behaviour of the redued rank estimator of the

ointegratingspaeandadjustmentspaeforvetorerrororretiontimeseries

modelswithnonindependentinnovations. Itis shownthatthedistributionof

the adjustment spae an be quite dierent for models with iid innovations

and models with nonindependent innovations. It is also shown that the

likelihoodratiotestremainsvalidwhentheassumptionofiidGaussianerrors

is relaxed. MonteCarlo experimentsillustratethe nitesampleperformane

ofthelikelihoodratiotestusingvariouskindsofweakerrorproesses.

Keywords:Cointegration,reduedrankregression,likelihoodratiotest,strong

mixingondition,vetorerrororretionmodel.

1. Introdution

Multivariateproessesareoftenusedineonometriappliationsbeausetheyallow

tounderstandtheinterationsbetweendierentvariables. Inordertodesribelongrun

eonomirelationships,theointegrationtheoryhasbeendevelopedbyGranger(1981),

Engle and Granger(1987), Ahnand Reinsel (1990). This theory postulatesthat, in

someases,astationary proess oflowerdimensionis obtainedbyonsidering linear

ombinationsoftheomponentsofamultivariatenonstationaryproess. Thenumber

ofindependentlinearombinationsistheointegratingrankandisanimportantpiee

ofinformationfortheanalysisofeonomidata.

The dominant test for the ointegrating rank is the likelihood ratio (LR) test

developed by Johansen (1988, 1991), Perron and Campbell (1993), Lütkepohl and

Saikkonen (1999) in the framework of vetor error orretion models (VECM). For

theointegrationanalysis,theerrorstermsaregenerallysupposed tobeindependent

and identially distributed (iid). When applied to eonomi data (see for instane

Johansen andJuselius (1990),Clementsand Hendry (1996)orTrenkler(2003)),this

iidassumptionseemstoorestritivebeausemaroeonomitimeseries oftenexhibit

onditionalheterosedastiityand/orother formsofnonlinearity.

Rahbek,HansenandDennis(2002)studied theeetof ARCH innovations onthe

LRtest. AnimportantoutputoftheirworkisthattheLRtestremainsvalidwhenthe

errorproessisamartingaledierene. Howevertheassumptionthattheerrorproess

isamartingaledierenepreludesotherformsofdependene. Indeedthereexistmany

exampleswheretheassumptionofiidormartingalediereneontheinnovationsisnot

satised(see for instane Franq, Roy and Zakoïan (2005) in the univariateARMA

aseor Franq and Raïssi(2005)in theVARase). The rstaim ofthis paper isto

studythevalidityoftheLRtest inageneralontextofunorrelatederrors.

∗

Postaladdress: UniversitéLille 3,EQUIPPE UniversitésdeLille BP60149 VilleneuveD'Asq

Cedex,Frane. E-mail:hamdi.raissietu.univ-lille3.fr.

The seond aim is to study the asymptoti behaviour of the usual estimators of

the ointegration and adjustment spaes, in the general framework of VECM with

unorrelated, but possibly dependent errors. We will ompare our ndings to the

usualiidaseandresultsofSeo (2007)whihshowsin partiularthattheasymptoti

distribution of the redued rank estimator of the ointegrating spae is robust to

onditional heteroskedastiity. We will use the standardredued rank proedure to

estimatetheointegrationspae,relaxingtheassumptionofiidgaussianinnovations.

Thestrutureofthepaperisasfollows. InSetion2wepresentthemodelandwe

derivetheestimatorsoftheparameters.InSetion3wegivetheasymptotibehaviour

oftheLRtest. Insetion4westatetheonsistenyoftheointegrationspaeandthe

adjustment spae. In Setion 5Monte Carloexperimentsare performed. Theproofs

arerelegatedtotheappendix.

Inthesequelthefollowingnotationsareused. Weak onvergene isdenoted by⇒

andwedenote by

→P ^{theonvergenein}probability. For afullolumn rankmatrixA

of dimensiond×r ^withd > r^{, wedene theorthogonal omplement} A⊥^{, whih is a}

fullolumnrankmatrixofdimensiond×(d−r)^andsuhthatA^′A⊥= 0^{. Thesymbol}

⊗ ^{denotes the usual Kroneker produt and ve(}A^{) denotes the vetor obtained by}

stakingtheolumnofthematrixA^{. Wedenoteby}tr(B)^{thetraeofasquarematrix} B^{. Wedenoteby}[m]^{theintegerpartofagivenreal}m^.

2. Charaterization ofthe model

WeonsiderthefollowingVECMwithlineartrend

∆Xt= Π0Xt−1+

p−1

X

i=1

Γ0i∆Xt−i+µo0+µo1t+ǫt ^(2.1)

where µo0 ^and µo1 ^are d-dimensional parametervetors. The proess (ǫt) ^{is usually}

assumediidwithmeanzeroand positivedenite ovarianematrixΣǫ^{. Inthesequel}

wewillonsideraweakerassumptionfortheerrorproess. TheΓ0i^,i∈ {1, ..., p−1}^,

ared×d^{shortrunparametersmatries. Byonventionthesumvanishesin(2.1)when} p= 1^{. Thefollowingassumption givesusthegeneralframeworkofourstudy.}

AssumptionA1 (Cointegrationandrestritiononthetrendparameters)

(a) The matrixΠ0 ^{is ofrank}r0 (0≤r0< d)^{. If}r0 >0 ^thenΠ0 ^{an bewritten as}

Π0=α0β₀^{′ where}α0 ^andβ0 ^{arefullolumnrankmatriesofdimension}d×r0^.

(b) The autoregressivepolynomialA(z) = (1−z)Id−Π0z−Pp−1

i=1 Γ0i(1−z)zⁱ, ^is

suhthat|A(z)|= 0 ^impliesthat |z|>1^orz= 1^.

() Thematrixα^′_0⊥Γ0β0⊥ ^isoffullrankd−r0^,whereΓ0=Id−Pp−1 i=1Γ0i^.

(d) The vetor µo1 ^{is suh that} µo1 = −α0τ0^{, where} τ0 6= 0 ^{is an} r0-dimensional vetor.

Note that if r0 = 0 ^{the relation (2.1) is a vetor} autoregressive model for the proess(∆Xt)^{. Condition(d)isthelessrestritiveonditionontheparametersofthe}

deterministipartof(2.1)whihallowsfortrendingbehaviourfor(Xt)^{. Indeedunder}

representation

Xt=C

t

X

i=1

ǫi+ρo1t+ρo0+Yt+A, ^(2.2)

where C =β0⊥(α^′_0⊥Γ0β0⊥)⁻¹α^′_0⊥^{. Theterm}A ^{dependson initialvaluesand is suh}

thatβ₀^′A= 0^{. Thestationaryproess}(Yt)^isoftheform Yt=

∞

X

i=0

ϕ0iǫt−i,

whereC(z) =P∞

i=0ϕ0iz^{iisonvergentfor}|z|≤1 +δ^,forsomeδ >0^{. Notethat(2.2)}

impliesthat(Xt)^isanI(1)^{proess. From}(a)^and(d)^{weanwrite (2.1)as}

∆Xt=ν0+α0β₀^′∗Z1t+

p−1

X

i=1

Γ0i∆Xt−i+ǫt ^(2.3)

whereZ1t= (X_t−1^′ ,−t+ 1)^{′ and}β₀^∗= (β₀^′, τ0)^{′. The}d-dimensionalvetorofonstants

ν0 ^{and the} r0-dimensional vetor τ0 ^{are funtions of the parameters in (2.1). Note}

that in (2.2) the vetorρo1 ^{is suh that} β^′₀ρo1 =τ0^{. Then it an beseen from (2.2)}

that (β₀^′Xt−E(β₀^′Xt))^{is trendstationary and the} r0-dimensional proess (β₀^′∗Z1t− E(β₀^′∗Z1t))^isstationary. Wesayin thisasethattheointegratingrankisr0^{. Inthis}

studywetest,forsomer(0≤r < d)^{,thenullhypothesis} H0:r0=r ^vs. H1:r0> r.

Notethatin(2.3)theparametersα0^,β0^andτ0^{arenotidentied. Indeedforagiven}

α01^, β01^{, and sine we assumed that these matries havefull rank, wean takeany}

nonsingularmatrixζ ^ofdimensionr0×r0^suhthat β02=β01ζ ^andα02=α01(ζ^′)⁻¹

willgivethesamematrixΠ0^{. Togetridofthisproblemoneanonsiderthefollowing}

normalization

β_0c^∗ = (β_0c^′ , τ0c)^′ = ((β0(c^′β0)⁻¹)^′,(β₀^′c)⁻¹τ0)^{′ and} α0c =α0β₀^′c,

wherethedimensionald×r0^matrixc^issuhthatc^′β0^{hasfullrank. This}normalization ensuresidentiabilityin thesense thatwehaveβ01c=β02c^{. Toseethis, notethat}

c^′β01c =c^′β02c =Ir0 ⇒ c^′β01(c^′β01)⁻¹=c^′β01ζ(c^′β01ζ)⁻¹

⇒c^′β01

(c^′β01)⁻¹−ζ(c^′β01ζ)⁻¹

= 0. ^(2.4)

Thensinec^′β01^{isafullrankmatrix,thisimpliesthat}

(c^′β01)⁻¹−ζ(c^′β01ζ)⁻¹= 0. ^(2.5)

Multiplying(2.5)byβ01 ^{ontheleft,weobtain}β01c=β02c^{. Onetheparameter}β0c ^is

identied,itiseasytoseethatα0c ^andτ0c ^{arealsoidentied. It shouldbealsonoted}

that theointegrationspaeandtheadjustmentspae,that isthespaesspanned by

respetivelyβ0c ^andα0c^{, donotdependonthehoieofthematrix}c^.

In general the assumption that (ǫt) ^{is iid gaussian may appear to be toostrong.}

IndeeditisquestionabletoassumethatalinearombinationofXt−1, . . . , Xt−p ^isthe

bestpreditorofXt^{. Inadditionnotethat, fromapratialpointofview,theorder}p

isoften identiedusing teststhat are onlybasedontheautoorrelationsof(ǫt)^{. For}

instaneletusonsiderthedailyexhangeratesofU.S.DollarstooneBritishPound

andofU.S.DollarstooneEurofromJanuary2,2001toApril12,2007. Thelengthof

theseriesisT = 1578^{. TheanalyzeddataareplottedinFigure7.10. Weadjustedthe}

model (2.1) to theseries with r0 = 1 ^and p= 2 ^{using thesoftware JMulTi. Figures}

7.11-7.12display the autoorrelations and rossorrelationsof the residuals. Figures

7.13-7.14displaytheautoorrelationsand rossorrelationsofthesquaredomponent

oftheresiduals. InviewofFigures7.11-7.12thehypothesisofunorrelatederrorsseems

plausible. Indeed most of the autoorrelations and rossorrelations are inside the

5%^{signianelimits. Howeversinemany}autoorrelationsandrossorrelationsare outsidethe5%^{signiane limitsin Figures 7.13-7.14,thehypothesisof} independent errorsis learlyrejeted.

Rahbeket al (2002)onsidered VECMwith martingaledierene innovations. In

ourframeworkwewillonsideramoregeneralassumptionallowingforalargelassof

errorproesses.

Assumption A2 The error proess (ǫt) ^{is stritly stationary and suh that} Cov(ǫt, ǫt−h) = 0^forallt∈Z^andallh6= 0^.

Suh errorproessesareommonly namedweakwhitenoise. Note that Granger's

representationtheorem stillholds whenthe assumptionofiid gaussianinnovationsis

replaed by A2. Thefollowingare examplesof error proesseswhih verifyA2 but

arenotiid.

Example2.1. Considertheproess(ǫt)^{dened bytherelation}

ǫt=at+ Φ{ǫt−1⊙at}, ^(2.6)

where ⊙^{denotesthe Hadamardprodut,} (at)^{is a}d-dimensional iidentered proess suhthat |E(aitajt)|≤1^{, andthematrix}Φ^{is diagonalofdimension}d×d^{and suh}

that|Φii |<1^{. Taking}Φ⁰=Id^{,theequation(2.6)hasastationarysolutionoftheform}

ǫt=P∞

i=0Φⁱat−i⊙ · · · ⊙at.^{Itis easyto seethatthe}ǫt^'sareunorrelated. However

Cov(ǫ²_it, ǫ²_it−1) =E(a²_it)Cov((1 + Φiiǫit−1)², ǫ²_it−1)6= 0,

ingeneral,showingthattheproess(ǫt)^isnotiid.

Example2.2. The univariate all-pass models (see for instane Breidt, Davis and

Trindade (2001)) onstitute an important lass whih an be extended to the mul-

tivariate ase. Assume that the proess (ǫt) ^{is the unique solution to the following}

equation

ǫt−φ01ǫt−1− · · · −φ0qǫt−q =wt+φ0q−1φ⁻¹_0q wt−1+· · ·+φ01φ⁻¹_0qwt−q+1−φ⁻¹_0qwt−q,

where φ(z) = Id−φ01z· · · −φ0qz^{q is suh that} φ(z)6= 0 ^for| z |≤ 1^{. The entered}

proess (wt)^{is iidwith variane}Σw^{. Assume alsothat the matries}φ01, . . . , φ0q ^are

diagonal. Writingthespetraldensityforeahomponent(ǫit)^{,it anbeshownthat}

theproess(ǫt)^isunorrelated(seeAndrews,DavisandBreidt(2006)). Howeverify0

isnotgaussiantheproess (ǫt)^isnotindependent. Toseethis onsiderthefollowing bivariatesimpleexample

ǫt−φǫt−1=wt−φ⁻¹wt−1

where φ =

φ1 0 0 φ2

and | φ1 |< 1^, | φ2 |< 1^{. Let us introdue} ϑt = ǫ1t− φ1ǫ1t−1^{. Sine} (ǫt)^is unorrelated, the proess (ϑt) ^{followsan ausal}M A(1)^{. Then}

we have ǫ1t = P

i≥0φⁱϑt−i^. Straightforwardomputations show that E(ǫ1tϑ²_t−1) = E[ǫ1t(ǫ1t−1−ǫ1t−2)²] =Ew³_t(1−φ⁻²₁ )(1+φ1)^andE(ǫ1tϑ³_t−1) =E[ǫ1t(ǫ1t−1−ǫ1t−2)³] = (Ew_t⁴−3)(1−φ⁻²₁ )²φ1^{. Usingthefat that} ϑt−1 ^{belongsto the}σ^{-eld generated by} {ǫ1u, u < t}^,wehaveE{ϑ²_t−1E(ǫ1t|ǫ1t−1,· · ·)} 6= 0^forEw_t³6= 0^andE{ϑ³_t−1E(ǫ1t| ǫ1t−1,· · ·)} 6= 0^for Ew⁴_t 6= 0^{. Thusthe} (ǫt)^{proessis notamartingaledierene in}

general.

2.1. Derivation of thequasi maximumlikelihood(QML) estimators

Now we turn to the derivation of the QML estimators of α0c ^and β_0c^{∗. We use}

here the QML method beause we assume that the errors terms are unorrelated

but not neessary gaussian independent. Note that the estimation proedure we

will desribe is performed under H0^{. In the framework of the VECM we shall see}

that the methodology in Johansen (1988,1991) in the iid ase remains valid under

unorrelatederrorsassumption. Wewill usethe followingnotation. Let Z0t= ∆Xt^,

Z2t = (∆X_t−1^′ , . . . ,∆X_t−p+1^′ ,1)^′, Ψ0 = (Γ01, . . . ,Γ0p−1, ν0) ^where Xt = 0 ^for t ≤ 0^.

Theexpression(2.3)beomeswiththesenotations

Z0t=α0cβ_0c^′∗Z1t+ Ψ0Z2t+ǫt. ^(2.7)

HereweanremarkthatsineXt^isI(1)^{thentheproesses}Z0t^andZ2t^arestationary.

Using(2.7)andgiventheobservationsX1, . . . , XT ^{wewritethequasi}log-likelihoodas follows

logL(Ψ, αc, βc,Σǫ) =−1

2Tlog|Σǫ|

−1 2tr

( _T X

t=1

Σ⁻¹_ǫ (Z0t−αcβ_c^′∗Z1t−ΨZ2t)(Z0t−αcβ^′∗_c Z1t−ΨZ2t)^′ )

,

where

β_c^∗= (β^′_c, τc)^′= ((β(c^′β)⁻¹)^′,(β^′c)⁻¹τ)^{′ and} αc=αβ^′c.

Themaximum likelihood estimationmethod fortheVECM with unorrelatederrors

impliatesseveralsteps. WerstestimatetheparametersinthematrixΨ0^andobtain

Ψ(αˆ c, β^∗c) =M02M₂₂⁻¹−αcβ^′∗c M12M₂₂⁻¹

where

Mij =T⁻¹

T

X

t=1

ZitZ_jt^′ .

Now dening by R0t ^and R1t ^{the residualsof} respetivelytheregressionsof Z0t ^and

Z1t^onZ2t^{,wegettheonentrated}log-likelihood

logL(αc, β^∗_c,Σǫ) = −1

2Tlog|Σǫ|

−1 2tr

( _T X

t=1

Σ⁻¹_ǫ (R0t−αcβ_c^′∗R1t)(R0t−αcβ_c^′∗R1t)^′ )

(2.8)

where

R0t=Z0t−M02M₂₂⁻¹Z2t ^and R1t=Z1t−M12M₂₂⁻¹Z2t.

Sine the R1t^{'s are the residuals of the regressionof the} Z1t^{'s on the} Z2t^{'s, and}

noting that the proess (Z1t) ^is I(1) ^{and the proess} (Z2t) ^is I(0)^{, then theproess} (R1t) ^is I(1)^{. The expression of the onentrated} log-likelihood orresponds to the regressionequation

R0t=α0cβ_0c^′∗R1t+ ˜ǫt, ^(2.9)

sothatweobtainthefollowingunfeasibleestimatorsofα0c^andΣǫ^{in(2.9)byordinary}

leastsquares

ˆ

αc(β^∗_0c) =S01β^∗_0c(β_0c^′∗S11β^∗_0c)⁻¹, ^(2.10) Σˆǫ(β^∗_0c) =S00−αˆc(β_0c^∗)(β_0c^′∗S11β_0c^∗)ˆα^′_c(β^∗_0c)

where

Sij =T⁻¹

T

X

t=1

RitRjt^′ .

Notethatreplaingαc ^andΣǫ ^{bytheirestimatesin(2.8)wewrite}

logL(ˆα(β_c^∗), β^∗_c,Σˆǫ(β^∗_c)) =−1

2Tlog|Σˆǫ(β_c^∗)| −1 2dT.

Finallytheparametersin β_0c^{∗ an beestimatedusingtheresultsofthewellknown}

reduedrankmethodofAnderson(1951). Inthisendweshallminimizethefollowing

expression

|Σˆǫ(β_c^∗)|=|S00−S01β_c^∗(β_c^′∗S11β^∗_c)⁻¹β^′∗_c S10|.

Usingtherelation

A11 A12

A21 A22 =|A11||A22−A21A⁻¹₁₁A12|=|A22||A11−A12A⁻¹₂₂A21|,

wend

|S00−S01β_c^∗(β_c^′∗S11β^∗_c)⁻¹β^′∗_c S10|=|S00| |β^′∗_c (S11−S10S₀₀⁻¹S01)β^∗_c |

|β_c^′∗S11β_c^∗| .

UnderthenullhypothesisandusingLemma7.1theexpression|β_c^′∗(S11−S10S₀₀⁻¹S01)β_c^∗| /|β_c^′∗S11β^∗_c |^{isminimizedforthefollowingnormalizedexpression}

βˆ_c^∗= ( ˆβ_c^′,τˆc)^′= (( ˆβ(c^′β)ˆ ⁻¹)^′,(( ˆβ^′c)⁻¹τ))ˆ ^′,

where

βˆ^∗= ( ˆβ^′,τ)ˆ ^′ =S⁻

1 2

11 (v1, . . . , vr)

andv1, . . . , vr ^areeigenvetorsorrespondingto ther ^{largestsolutions}λˆ1≥ · · · ≥λˆr

oftheeigenvalueproblem

|λI−S⁻

1 2

11 S10S₀₀⁻¹S01S⁻

1 2

11 |= 0. ^(2.11)

Inadditionthematrixc^′βˆ^{isoffullrank. Weobtain}αˆc=S01βˆ_0c^∗( ˆβ_0c^′∗S11βˆ^∗_0c)^{−1. Noting}

thatwehave|Σˆǫ( ˆβ_c^∗)|=Qr

i=1(1−λˆi)^{,thelikelihood ratiotestfor}r^{is givenby} Q⁻

2

rT = Qr

i=1(1−λˆi) Qd

i=1(1−λˆi)=

d

Y

i=r+1

(1−λˆi)⁻¹.

Thentotestthenullhypothesis,weonsider theLR teststatisti

−2 logQr=−T

d

X

i=r+1

log(1−ˆλi),

where ˆλ1 ≥ · · · ≥λˆd ^arethed ^{greatersolutionsof theeigenvalueproblem (2.11). In}

thenextsetionwewillstudy theasymptotibehaviouroftheLR teststatisti.

3. Asymptotipropertiesof the LR statisti

To state the main results of the paper, the assumption that the proess (ǫt) ^is

unorrelated is not enough. Indeed we haveto ontrol the serial dependene of the

proess (ǫt)^{. To this end we introdue the mixing oeients} αξ(h) ^{for a given}

stationaryproess(ξt)

αξ(h) = sup

A∈σ(ξu,u≤t),B∈σ(ξu,u≥t+h)

|P(A∩B)−P(A)P(B)|,

whihmeasuresthetemporaldependeneoftheproess(ξt)^{. Dene}kξtkq = (Ekξtk^q)^1/q,

wherek.k^{denotestheEulideannorm. Thenweneedtomakethefollowingassumption}

ontheproess(ǫt)^.

Assumption A3 The proess (ǫt) ^satises kǫtk_2+ν+η < ∞ ^{and the mixing oef-}

ientsof theproess (ǫt)^{are suh that} P∞

h=0{αǫ(h)}^ν/(2+ν)<∞ ^{for some} ν >

0 ^and η >0.

Note that the kind of dependene induedby A3 is mild for the error proess (ǫt)^.

ThefollowingpropositiongivesustheasymptotidistributionoftheLRteststatisti.

Proposition 3.1. UnderA1,A2andA3,theLRteststatistihasthesameasymp-

totidistributionasin the iidgaussianase, that is

−2 logQr0 ⇒tr (Z 1

0

F(dB)^{′ ′}Z 1

0

F F^′du

⁻¹Z 1 0

F(dB)^′ )

, ^(3.1)