Idescat. SORT. A construction of continuous-time ARMA models by iterations of Ornstein

36 Pages·2016·1.87 MB·English

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Statistics&OperationsResearchTransactions
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SORT40(2)July-December2016,267-302 © Institutd’Estad´ısticadeCatalunya
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ISSN:1696-2281 [email protected]
eISSN:2013-8830
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A construction of continuous-time ARMA models
by iterations of Ornstein-Uhlenbeck processes
ArgimiroArratia1, AlejandraCaban˜a2 and EnriqueM.Caban˜a3
Abstract
Wepresentaconstructionofafamilyofcontinuous-timeARMAprocessesbasedon piterations
ofthelinearoperatorthatmapsa Le´vyprocessontoanOrnstein-Uhlenbeckprocess. Thecon-
structionresemblestheproceduretobuildanAR(p)fromanAR(1).Weshowthatthisfamilyisin
facta subfamilyofthewell-knownCARMA(p,q) processes,withseveralinterestingadvantages,
including a smaller number of parameters. The resulting processes are linear combinations of
Ornstein-UhlenbeckprocessesalldrivenbythesameLe´vyprocess.Thisprovidesastraightfor-
wardcomputationofcovariances,astate-spacemodelrepresentationandmethodsforestimating
parameters.Furthermore,the discrete and equallyspaced samplingof the process turnsto be
an ARMA(p,p−1) process. We propose methods for estimating the parameters of the iterated
Ornstein-UhlenbeckprocesswhenthenoiseiseitherdrivenbyaWieneroramoregeneralLe´vy
process,andshowsimulationsandapplicationstorealdata.
MSC: 60G10,62M10,62M9960M99.
Keywords: Ornstein-Uhlenbeckprocess,Le´vyprocess,ContinuousARMA,stationaryprocess.
1. Introduction
Thelink betweendiscrete time autoregressivemovingaverage(ARMA) processesand
stationaryprocesseswithcontinuous-timehasbeenofinterestformanyyears,seeforin-
stance,Doob(1944),Durbin(1961),Bergstrom(1984,1996)andmorerecentlyBrock-
well (2009), Thornton and Chambers (2013). Continuous time ARMA processes are
better suited than their discrete counterpartsfor modelling irregularly spaceddata, and
when the white noise is driven by a non-Gaussian process it becomes a more realistic
modelinfinanceandotherfieldsofapplication.
1UniversitatPolite`cnicadeCatalunya,Barcelona,[email protected]
SupportedbySpain’sMINECOprojectAPCOM(TIN2014-57226-P) andGeneralitatdeCatalunya2014SGR
890(MACDA).
2UniversitatAuto`nomadeBarcelona,[email protected]
SupportedbySpain’sMINECOprojectMTM2015-69493-R.
3UniversidaddelaRepu´blica,Montevideo,[email protected]
Received: November2015
Accepted: May2016
268 Aconstructionofcontinuous-timeARMAmodelsbyiterationsofOrnstein-Uhlenbeckprocesses
A popular continuous-time representation of ARMA(p,q) process (known as
CARMA(p,q))canbeobtainedviaastate-spacerepresentationoftheformalequation
a(D)Y(t)=σb(D)DΛ(t),
where σ > 0 is a scale parameter, D denotes differentiation with respect to t, Λ is a
second-orderLe´vyprocess,a(z)=zp+a zp−1+...+a isapolynomialoforder pand
1 p
b(z)=b +b z+...+b zqapolynomialoforderq≤ p−1withcoefficientb 6=0(see,
0 1 q q
e.g., Brockwell, 2004, 2009, Thornton and Chambers, 2013). The parameters of this
modelare estimated by adjusting first an ARMA(p,q), q< p to regularly spaceddata.
Then obtain the parameters of the continuous version whose values at the observation
timeshavethesamedistributionofthefittedARMA.Hence, p+q+1parametershave
tobeestimated.
We proposein this work a parsimoniousmodelfor continuousautoregression,with
fewer parameters (as we shall see exactly p plus the variance). Our construction de-
partsfromtheobservationthataOrnstein-Uhlenbeck(OU)processcanbethoughtofas
continuous-timeinterpolationofanautoregressiveprocessoforderone(i.e.anAR(1)).
Thisis shownin Section2,wherewealsoreviewsomewellknownfactsonLe´vypro-
cesses,ARMAmodelsandtheirrepresentations.Themodelisobtainedbyaprocedure
that resembles the one that allows to build an AR(p) from an AR(1). Departing from
thisanalogy,wedefineandanalysetheresultofiteratingtheapplicationoftheoperator
that maps a Wiener process onto an OU process. This operator is defined in Section 3
anddenotedOU,withsubscriptsdenotingtheparametersinvolved.
The p iterations of OU, for each positive integer p, give rise to a new family of
processes, the Ornstein-Uhlenbeck processes of order p, denoted OU(p). They can be
used as models for either stationary continuous-time processes or the series obtained
by observing these continuous processes at equally spaced instants. We show that an
OU(p) process can be expressedas a linear combination of ordinary OU processes,or
generalizedOUprocesses,alsodefinedinSection3.Thisresultresemblestheaggrega-
tionsofGaussian(andnon-Gaussian)processesstudiedwiththeideaofdeconstructing
a complicated economic model into simpler constituents. In the extensiveliterature on
aggregations (or superpositions) of stochastic processes the aggregated processes are
driven by independent Le´vy processes (see, e.g., Granger and Morris, 1976, Granger,
1980,Barndorff-Nielsen,2001,Eliazar andKlafter, 2009,amongmanyothers). A dis-
tinctivepointofourconstructionisthatthestochasticprocessesobtainedbyconvolution
oftheOU operatorresultin a linearcombinationcomprisedofprocessesdrivenbythe
sameLe´vyprocess.
Another consequence of writing the OU(p) process as the aggregation of simpler
onesisthederivationofaclosedformulaforitscovariance.Thishasimportantpractical
implications since it allows to easily estimate the parameters of a OU(p) process by
matchingcorrelations(aprocedureresemblingthemethodofmoments,tobedescribed
inSection6.2),andbymaximumlikelihood.
ArgimiroArratia,AlejandraCaban˜aandEnriqueM.Caban˜a 269
In Section 4 we show how to write the discrete version of a OU(p) as a state-space
model, and from this representation we show in Section 5 that for p>1, a OU(p) be-
haves like an aggregation of AR processes (in the manner considered in Granger and
Morris (1976)), that turns outto be anARMA(p,q),with q≤ p−1.Consequentlythe
OU(p)processesareasubfamilyoftheCARMA(p,q)processes.Notwithstandingthis
structural similarity, the family of discretized OU(p) processes is more parsimonious
than the family of ARMA(p,p−1) processes, and we shall see empirically that it is
abletofitwelltheautocovariancesforlarge lags.Hence,OUprocessesofhigherorder
appear as a new continuous model, competitive in a discrete time setting with higher
order autoregressive processes (AR or ARMA). The estimation of the parameters of
OU(p)processesisattemptedinSection6.Simulationsandapplicationstorealdataare
providedinSection6.5.OurconcludingremarksareinSection7.
2. Preliminaries
LetusrecallthataLe´vyprocessΛ(t)isaca`dla`gfunction,withindependentandstation-
ary increments, that vanishesint =0. As a consequence,Λ(t) is, for eacht, a random
variable with an infinitely divisible law (Sato, 1999). A Wiener process W is a cen-
tred Gaussian process, with independent increments and variance E(W(t)−W(s))2 =
σ2|t−s|. Wiener processes are the only Le´vy processes with almost surely continu-
ous paths. For parameter λ>0 the classical Ornstein-Uhlenbeckprocess is defined as
t e−λ(t−s)dW(s)(UhlenbeckandOrnstein,1930).
−∞
Wiener processcanbereplacedbya secondorderLe´vyprocessΛ to definea Le´vy
R
drivenOrnstein-Uhlenbeckprocessas
t
x(t)(=x (t)):= e−λ(t−s)dΛ(s) (1)
λ,Λ
Z−∞
Thepreviousequationcanbeformallywrittenindifferentialform
dx(t)=−λx(t)dt+dΛ(t) (2)
We may think of x as the result of accumulating a random noise dΛ, with reversion to
themean(thatweassumetobe0)ofexponentialdecaywithrateλ.
WhentheOrnstein-Uhlenbeckprocessxissampledatequallyspacedtimes{hτ :h=
0,1,2,...,n},τ >0,theseriesX =x(hτ)obeysanautoregressivemodeloforder1(i.e.
h
(h+1)τ
an AR(1)), because X =e−λτX +Z , where Z = e−λ((h+1)τ−s)dΛ(s),
h+1 h h+1 h+1
hτ
isthestochasticinnovation. Z
Hence,wecanconsidertheOUprocessascontinuous-timeinterpolationofanAR(1)
process. Notice that both models are stationary. This link between AR(1) and OU(1)
suggeststhedefinitionofiteratedOUprocessesintroducedinSection3.
270 Aconstructionofcontinuous-timeARMAmodelsbyiterationsofOrnstein-Uhlenbeckprocesses
An ARMA(p,q) or autoregressive moving average process of order (p,q) has the
followingform
x =φ x +···+φ x +θ ǫ +θ ǫ +···+θ ǫ
t 1 t−1 p t−p 0 t 1 t−1 q t−q
where φ , ...,φ are the autoregressiveparameters,θ , ...,θ are the movingaverage
1 p 0 q
parameters,andthewhite-noiseprocessǫ hasvarianceone.DenotebyB thebackshift
t
operatorthatcarries x into x .By consideringthepolynomialsin thebackshiftoper-
t t−1
ator,
φ(B)=1−φ B−···−φ Bp and θ(B)=θ +θ B+···+θ Bq
1 p 0 1 q
theARMA(p,q)modelcanbewrittenas
φ(B)x =θ(B)ǫ (3)
t t
This compact expression comes in handy for analysing structural properties of time
series. It also links to the representation of ARMA processes as a state-space model,
useful for simplifying maximum likelihood estimation and forecasting. A state-space
modelhasthegeneralform
Y =AY +ηηη (4)
t t−1 t
x =KTY +N (5)
t t t
where (4) is the state equation and (5) is the observation equation, with Y the m-
t
dimensionalstatevector,AandK arem×mandm×kcoefficientmatrices,KT denotes
thetransposeofK,ηηη andN are mandk dimensionalwhite noises.N wouldbepresent
onlyiftheprocessx isobservedsubjecttoadditionalnoise(seeBox,Jenkins,andRein-
t
sel,1994forfurtherdetails).WepresentinSection4astate-spacemodelrepresentation
ofourgeneralizedOUprocess.
3. Ornstein-Uhlenbeck processes of order ppp
The AR(1) process X = φX +ǫ , where ǫ , t ∈Z, is a white noise, can be written
t t−1 t t
as (1−φB)X = ǫ using the back-shift operator B. Equivalently, X can be written
t t t
as X = MA ǫ, where MA is the moving average that maps ǫ onto MA ǫ ,=
t 1/ρ t 1/ρ t 1/ρ t
∞ 1 ǫ ,andρ(=1/φ)istherootofthecharacteristicpolynomial1−φz.
j=0ρj t−j
P
ArgimiroArratia,AlejandraCaban˜aandEnriqueM.Caban˜a 271
p
Moreover, the AR(p) process X = φ X +ǫ ( or φ(B)X =ǫ ), where φ(z)=
t j t−j t t t
j=1
X
p p
1− φ zj=∏(1−z/ρ )hasrootsρ =eλj,j=1,...,p,canbeobtainedbyapplying
j j j
j=1 j=1
thecXompositionofthemovingaveragesMA tothenoise,thatis:
1/ρj
p
X =∏MA ǫ
t 1/ρj t
j=1
NowconsidertheoperatorMAe−λ thatmapsǫt onto
MAe−λǫt = e−λ(t−l)ǫl
l≤t,integer
X
AcontinuousversionofthisoperatorisOU thatmapsy(t),t ∈Ronto
λ
t
OU y(t)= e−λ(t−s)dy(s), (6)
λ
Z−∞
whenever the integral can be defined. The definition of OU is extended to include
λ
complexprocesses,byreplacingλbyκ=λ+iµ,λ>0,µ∈Rin(6).Thesetofcomplex
numberswith positiverealpartisdenotedbyC+, andthe conjugateofκis denotedby
κ¯.
For p≥1andparametersκκκ=(κ ,...,κ ),thepreviousargumentsuggeststodefine
1 p
thefollowingprocessobtainedasrepeatedcompositionsofoperatorsOU , j=1,...,p:
κj
p
OU y(t):=OU OU ···OU y(t)=∏OU y(t) (7)
κκκ κ1 κ2 κp κj
j=1
ThisiscalledOrnstein-Uhlenbeckprocessoforderpwithparametersκκκ=(κ ,...,κ )∈
1 p
(C+)p. The composition ∏p OU is unambiguouslydefined becausethe application
j=1 κj
ofOU operatorsiscommutativeasshowninTheorem1(i)below.
κj
The particular case of interest where the underlying noise is a second order Le´vy
processΛ,namely,
p
OU Λ(t):=OU OU ···OU Λ(t)=∏OU Λ(t) (8)
κκκ κ1 κ2 κp κj
j=1
is called the Le´vy-driven Ornstein-Uhlenbeck process of order p with parametersκκκ=
(κ ,...,κ )∈(C+)p.
1 p
272 Aconstructionofcontinuous-timeARMAmodelsbyiterationsofOrnstein-Uhlenbeckprocesses
Fortechnicalreasons,itisconvenienttointroducetheOrnstein-Uhlenbeckoperator
OU(h) ofdegreehwithparameterκthatmapsyonto
κ
t (−κ(t−s))h
OU(h)y(t)= e−κ(t−s) dy(s) (9)
κ Z−∞ h!
andΛonto
t (−κ(t−s))h
ξ(h)(t)= e−κ(t−s) dΛ(s) (10)
κ Z−∞ h!
We call the process (10) generalized Ornstein-Uhlenbeck process of order 1 and
degree h. For the remainder of the paper we restrict the underlying noise to a second
orderLe´vyΛ,butnotethatthegeneralpropertiesoftheOU operatorthatwearegoing
κ
toshowholdforanyrandomfunctiony(t)forwhichtheintegral(6)isdefined.
3.1. Properties of the operatorOOOUUU
κκκ
Thefollowingstatementssummarizesomepropertiesofproducts(compositions)ofthe
operatorsdefinedby(7)and(9),andcorrespondingly,ofthestationarycentredprocesses
ξ(h), h ≥ 0. In particular, the Ornstein-Uhlenbeck processes of order 1 and degree 0,
κ
ξ(0)=ξ aretheordinaryOrnstein-Uhlenbeckprocesses(1).
κ κ
Theorem1
(i) Whenκ 6=κ ,theproductOU OU canbecomputedas
1 2 κ2 κ1
κ κ
1 OU + 2 OU
κ −κ κ1 κ −κ κ2
1 2 2 1
andisthereforecommutative.
(ii) Thecomposition∏p OU constructedwithpairwisedifferentκ ,...,κ isequal
j=1 κj 1 p
tothelinearcombination
p p
∏OU = K (κ ,...,κ )OU , (11)
κj j 1 p κj
j=1 j=1
X
withcoefficients
1
K (κ ,...,κ )= . (12)
j 1 p
∏ (1−κ/κ )
κl6=κj l j
ArgimiroArratia,AlejandraCaban˜aandEnriqueM.Caban˜a 273
(iii) Fori=1,2,...,OU OU(i)=OU(i)−κOU(i+1).
κ κ κ κ
(iv) Foranypositiveinteger pthe p-thpoweroftheOrnstein-Uhlenbeckoperatorhas
theexpansion
p−1
p−1
OUp = OU(j). (13)
κ j κ
j=0(cid:18) (cid:19)
X
(v) Letκ ,...,κ bepairwisedifferentcomplexnumberswithpositiverealparts,and
1 q
p ,...,p positiveintegers,andletusdenotebyκκκacomplexvectorin(C+)pwith
1 q
componentsκ repeated p times, p ≥1, h=1,...,q, q p = p. Then, with
h h h h=1 h
K (κκκ)definedby(12),
h
P
q q 1 q
∏OUph = OUph = K (κκκ)OUph.
h=1 κh h=1 ∏l6=h(1−κl/κh)pl κh h=1 h κh
X X
An immediate consequenceis that the operatorOU with p-vectorparameterκκκ canbe
κκκ
written as a linear combination of p operators OU or OU(h) for suitable scalar values
κ κ
κ and non-negative integer h. Therefore, the process OU Λ can be written as a linear
κκκ
combinationofOUprocessesdrivenbythesameLe´vyprocess,asstatedinthefollowing
Corollary.
Corollary1
q
(i) TheprocessOU (Λ)=∏OUph(Λ)canbeexpressedasthelinearcombination
κκκ κh
h=1
q ph−1
OU (Λ)= K (κκκ) ph−1 ξ(j) (14)
κκκ h j κh
h=1 j=0
X X(cid:0) (cid:1)
ofthe pprocesses{ξ(j):h=1,...,q,j=0...,p −1}(see(10)).
κh h
(ii) Consequently,
q ph−1
OUκκκΛ(t)= Kh(κκκ) ph−j 1 −t∞e−κh(t−s)(−κh(jt!−s))jdΛ(s)
h=1 j=0
X X(cid:0) (cid:1)R
Corollary2 For real λ,µ, with λ>0, the product OU OU is real, that is, ap-
λ+iµ λ−iµ
pliedtoarealprocessproducesarealimage.
TheproofsofTheorem1andcorollariesareinAppendixA.
274 Aconstructionofcontinuous-timeARMAmodelsbyiterationsofOrnstein-Uhlenbeckprocesses
3.2. Computingthe covariances
Therepresentation
q ph−1 p −1
x:=OU (Λ)= K (κ) h OU(j)(Λ)
κκκ h j κh
h=1 j=0 (cid:18) (cid:19)
X X
of x as a linear combination of the processes ξ(i) = OU(i)(Λ) allows a direct compu-
κh κh
tation of the covariances γ(t) = Ex(t)x¯(0) through a closed formula, in terms of the
covariancesγ(i1,i2)(t)=Eξ(i1)(t)ξ¯(i2)(0):
κ1,κ2 κ1 κ2
q ph′−1 q ph′′−1 p −1 p −1
γ(t)= K (κκκ)K¯ (κκκ) h′ h′′ γ(i′,i′′) (t) (15)
h′ h′′ i′ i′′ κh′,κh′′
h′=1 i′=0 h′′=1 i′′=0 (cid:18) (cid:19)(cid:18) (cid:19)
X X X X
withv2=VarΛ(1),
0 (t−s)i1 (−s)i2
γ(i1,i2)(t)=v2(−κ )i1(−κ¯ )i2 e−κ1(t−s) e−κ¯2(−s) ds
κ1,κ2 1 2 Z−∞ i1! i2!
i1 i tj 0
=v2(−κ )i1(−κ¯ )i2e−κ1t 1 e(κ1+κ¯2)s(−s)i1+i2−jds
1 2
j=0(cid:18)j(cid:19)i1!i2!Z−∞
X
v2(−κ )i1(−κ¯ )i2e−κ1t i1 tj(i +i − j)!
1 2 1 2
= (16)
i2! j!(i1− j)!(κ1+κ¯2)(i1+i2−j+1)
j=0
X
Arealexpressionforthecovariancewhentheimaginaryparametersappearasconjugate
pairscanbeobtainedbutitismuchmoreinvolvedthanthisone.
4. The OU(ppp) process as a state-space model
Theorem 1 and its corollaries lead to express the OU(p) process by means of linear
state-space models. The state-space modelling provides a unified methodology for the
analysisoftimeseries(seeDurbinandKoopman,2001).
In the simplest case, where the elements of κκκ are all different, the process x(t) =
OU Λ(t)isalinearcombinationofthestatevectorξξξ (t)=(ξ (t),ξ (t),...,ξ (t))T,
κκκ κκκ κ1 κ2 κp
whereξ =OU (Λ).
κj κj
ArgimiroArratia,AlejandraCaban˜aandEnriqueM.Caban˜a 275
Moreprecisely,thevectorialprocess
ξξξ (t)=(ξ (t),ξ (t),...,ξ (t))T, ξ =OU (Λ)
κκκ κ1 κ2 κp κj κj
andx(t)=OU Λ(t)satisfythelinearequations
κκκ
ξξξ (t)=diag(e−κ1τ,e−κ2τ,...,e−κpτ)ξξξ (t−τ)+ηηη (t) (17)
κκκ κκκ κκκ,τττ
and x(t)=KKKT(κκκ)ξξξ(t), (18)
t
ηηη (t)=(η (t),η (t),...,η (t))T, η (t)= e−κj(t−s)dΛ(s),
κκκ,τττ κ1,τ κ2,τ κp,τ κj,τ
Zt−τ
t 1−e−(κj+κ¯l)τ
Var(ηηη (t))=v2((v )), v = e−(κj+κ¯l)(t−s)ds= (19)
κκκ,τττ j,l j,l
κ +κ¯
Zt−τ j l
andthecoefficientsfrom(12),KKKT(κκκ)=(K (κκκ),K (κκκ),...,K (κκκ)).
1 2 p
The initial value ξξξ(0) is estimated by means of its conditional expectation ξξξˆ(0)=
KKKT(κκκ)Vx(0) 1
E(ξ(0)|x(0))= ,withV =Var(ξξξ(0))= .
KKKT(κκκ)VKKK (cid:18)(cid:18)κj+κ¯l(cid:19)(cid:19)
AnapplicationofKalmanfiltertothisstate-spacemodelleadstocomputethelikeli-
hoodofxxx=(x(0),x(τ),...,x(nτ)).SomeKalman filterprograms includedin software
packages require the processes in the state-space to be real. That condition is not ful-
filled by the modeldescribedby equations(17) and (18). An equivalentdescriptionby
meansofrealprocessescanbeobtainedbyorderingtheparametersκκκwiththeimaginary
componentspairedwith theirconjugatesinsuchawaythatκ =κ¯ ,h=1,2,...,c
2h 2h−1
andtheimaginarycomponentℑ(κ )=0ifandonlyif2c< j≤ p.
j
Then the matrix M=((M )) with all elements equal to zero except M =
j,k 2h−1,2h−1
M =1,−M =M =i, h=1,2,...,candM =1, 2c< j≤ p,induces
2h−1,2h 2h,2h−1 2h,2h j,j
thelineartransformationξξξ 7→Mξξξ thatleadstothenewstate-spacedescription
Mξξξ(t)=Mdiag(e−κ1τ,e−κ2τ,...,e−κpτ)M−1Mξξξ(t−τ)+Mηηη(t), (20)
x(t)=KKKTM−1Mξξξ(t), (21)
wheretheprocessesMξξξ arereal.
276 Aconstructionofcontinuous-timeARMAmodelsbyiterationsofOrnstein-Uhlenbeckprocesses
Observethatthereisnolossofgeneralityinchoosingthespacingτ betweenobser-
vations as unity for the derivation of the state-spaceequations. Hence, we set τ =1 in
thesequeland,inaddition,τ willbeomittedfromthenotation.
When κ ,...,κ are all different, p ,...,p are positive integers, q p = p and
1 q 1 q h=1 h
κκκ is a p-vector with p repeated components equal to κ , the OU(p) process x(t) =
h h
P
OU Λ(t)isalinearfunctionofthestate-spacevector
κκκ
ξ(0),ξ(1),...,ξ(p1−1),...,ξ(0),ξ(1),...,ξ(pq−1)
κ1 κ1 κ1 κq κq κq
(cid:16) (cid:17)
where the components are given by (10), and the transition equation is no longer ex-
pressedbyadiagonalmatrix.Inthiscasethestate-spacemodelhasthefollowingform
ξξξ(t)=Aξξξ(t−1)+ηηη(t)
x(t)=KKKTξξξ(t) (22)
Weleavethetechnicaldetailsofthis derivationto AppendixB. Thetermsξξξ(t),A,ηηη(t)
and KKK are precisely defined in (36). The real version of (22), when the processξξξ has
imaginary components is obtained by multiplying both equations by a block-diagonal
matrix C (which is defined precisely in the Appendix), giving us the real state-space
model
Cξξξ(t)=(CAC−1)(Cξξξ(t−1))+Cηηη(t), (23)
x(t)=(KKKTC−1)(Cξξξ(t)). (24)
5. The OU(ppp) as an ARMA(ppp, ppp−−−111)
Thestudiesofpropertiesoflineartransformationsandaggregationsofsimilarprocesses
haveproducedagreatamountofworkstemmingfromtheseminalpaperbyGrangerand
Morris (1976) on the invariance of MA and ARMA processes under these operations.
These results and extensions to vector autoregressive moving average (VARMA) pro-
cessesarecompiledinthetextbookbyLu¨tkepohl(2005).
The description of the OU(p) process x = OU (Λ) with parameters κκκ as a linear
κκκ
state-space model, given in the previous section, will allow us to show that the series
x(0), x(1), ..., x(n) satisfies an ARMA(p,q) model with q smaller than p. We refer
the reader to (Lu¨tkepohl, 2005, Ch. 11) for a presentation on VARMA processes and,
in particular, to the following result on the invariance property of VARMA processes
underlineartransformations,whichwequotewithaminorchangeofnotation: