Estimation of weak ARMA models with regime changes

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Yacouba Boubacar Maïnassara, Landy Rabehasaina

To cite this version:

Yacouba Boubacar Maïnassara, Landy Rabehasaina. Estimation of weak ARMA models with regime changes. Statistical Inference for Stochastic Processes, Springer Verlag, 2020, 23 (1), pp.1-52. �hal- 01691099v3�

hanges

Yaouba Boubaar Maïnassara and Landy Rabehasaina

Université BourgogneFranhe-Comté,

Laboratoire demathématiques deBesançon,

UMRCNRS6623,

16routede Gray,

25030Besançon,Frane.

e-mail:yaouba.boubaar_mainassarauniv-fomte.fr;lrabehasuniv-fomte.fr

Abstrat: Inthis paper we derive the asymptotiproperties ofthe leastsquares es-

timator(LSE)of autoregressive moving-average (ARMA)models withregimehanges

undertheassumptionthattheerrorsareunorrelatedbutnotneessarilyindependent.

Relaxingtheindependeneassumptiononsiderablyextendstherangeofappliationof

thelassofARMAmodelswithregimehanges.Conditionsaregivenfortheonsisteny

andasymptotinormalityoftheLSE.Apartiularattentionisgiventotheestimation

oftheasymptotiovarianematrix,whihmaybeverydierentfromthatobtainedin

thestandardframework.ThetheoretialresultsareillustratedbymeansofMonteCarlo

experiments.

AMS2000subjetlassiations:Primary62M10,62F03,62F05;seondary91B84,

62P05.

Keywordsand phrases: Leastsquareestimation,Randomoeients,weakARMA

models.

1. Introdution

Sine the works of Hamilton (1988, 1989) and NihollsandQuinn (1982), the time series

models with time-varyingoeients have beome inreasingly popular. In statistial appli-

ations, a largepart ofthe literature isdevotedto the non-stationary autoregressive moving-

average (ARMA) models with time-varying parameters (see Azrak andMélard (1998, 2006);

Bibi andFranq(2003);Dahlhaus(1997)),seealsothelassofARMAmodelswithperiodio-

eients(forinstane Andersonand Meershaert(1997); Basawaand Lund (2001)). Butthe

mostpopularlassdealswiththetreatmentofregimeshiftsandnon-linearmodelingstrategies.

For instane, aMarkov-swithing modelis anon-linear speiation in whih dierent states

of the world aet the evolution of a time series (see, for examples, Franq and Roussignol

(1997);Hamilton(1990);Hamilton andSusmel(1994)).TheasymptotipropertiesofMarkov-

swithing ARMA models are well known in the literature (see, for instane, Billioet al.

(1999);Franq and Roussignol(1998);Franq andZakoïan(2001,2002);Kim andKim(2015)

or Hamilton (1994)).

The fatthathangesin regimes maybe veryimportant for the evolution ofinterest rates

hasbeenemphasizedin anumber ofreent studies. Ourattentionhereisfousedonthelass

ofARMAmodelswithregimehanges(ARMARCforshort);forinstane,ARMAmodelswith

reurrent but non neessarily periodi hanges in regime. We onsider a time series (X_t)_t∈Z

exhibitinghangesinregimeatknowndatesandwesupposethatwehaveniteregimes.Con-

trarilytothefamousMarkov-swithingapproah,weassumethattherealizationoftheregimes

is observed. Suh a situation may be realisti,and would orrespond e.g. to time series with

periods of harsh and mild weather whih are observed in pratie. Thismodel ould also be

applied toeonomitimeserieswhosebehaviourdependsonworked daysandpubliholidays,

whih are known in advane. Another motivating example would be nanial times series,

where regimes orresponding to typial known major events leading to highand quiet(low)

volatilitysubperiods areobserved,seee.g.Figure1.2p.7inFranq and Zakoïan(2010)where

the highvolatilitylusters orrespondstolargely famousevents suhasSeptember 11th2001

or the2008 nanialrisis.Anotherexample anbefoundfor instaneinFranq and Gautier

(2004b).

Forsuhmodels,Franq andGautier(2004a,b)gavegeneralonditionsensuringonsisteny

and asymptoti normality of least squares (LS) and quasi-generalized least-squares (QGLS)

estimators under the assumption thatthe innovationproessesisindependent.Thisindepen-

dene assumption is often onsidered too restritive by pratitioners. Relaxing the indepen-

deneassumptiononsiderablyextendstherangeofappliationsoftheARMARCmodels,and

allows to over general nonlinear proesses. Indeed suh nonlinearitiesmay arisefor instane

whenthe errorproessfollowsanautoregressiveonditionalheterosedastiity(ARCH)intro-

dued byEngleEngle(1982)and extendedto thegeneralized ARCH(GARCH)byBollerslev

(1986), all-pass(seeAndrews et al.(2006))or othermodels displaying a seondorderdepen-

dene (see Amendolaand Franq (2009)). Other situations where the errors are dependent

anbefoundinFranq and Zakoïan(2005),seealsoRomanoand Thombs(1996).Thispaper

isdevotedtothe problemofestimatingARMARCrepresentationsunderthe assumptionthat

the errorsareunorrelated butnot neessarilyindependent. ThesearealledweakARMARC

modelsin ontrast to the strong ARMARCmodels above-ited, in whih the error termsare

supposedtobeindependentandidentiallydistributed(iid).Thus,themaingoalofourpaper

is to omplete the above-mentioned results onerning the statistial analysis of ARMARC

models, by onsidering the estimation problem under general error terms. We establish the

asymptotidistributionoftheLSestimatorofweakARMARCmodels,under stronglymixing

assumptions.

Thepaperisorganizedasfollows.Setion2presentstheARMARCmodelsthatweonsider

here. In Setion 3, we established the strit stationarity ondition and it is shown that the

LSestimator(LSE)isasymptotiallynormallydistributedwhenlinearinnovationproess(ǫ_t)

satisesmildmixingassumptions.TheasymptotiovarianeoftheLSEmaybeverydierent

in theweakandstrongases.Partiularattentionisgiventothe estimationofthisovariane

matrix. Modied version of the Wald test is proposed for testing linear restritions on the

parameters. In Setion 4, we present two examples of weak ARMARC(1,0) ^{models with iid}

and orrelated realization of the regimes. Numerial experiments are presented in Setion 5.

The proofsofthe main results areolleted in the appendix.

2. Model and assumptions

Let (∆_t)_t∈Z ^{be a stationary ergodi observed proess with values in a nite set} S ^{of size}

Card(S) =K^{. Weonsider the ARMARC}(p, q) ^proess (X_t)_t∈Z ^denedby

Xt−

p

X

i=1

a⁰_i(∆t)Xt−i =ǫt−

q

X

j=1

b⁰_j(∆t)ǫt−j ⁽¹⁾

wherethelinearinnovationproessǫ:= (ǫ_t)_t∈Z^{isassumedtobeastationarysequenesatises}

E(ǫ_t) = 0, E(ǫ_tǫ_t^′) =σ²¹_[t=t^′_]^{. Underthe above} assumptions, the proess ǫ^{is alled a weak}

white noise.

AnimportantexampleofaweakwhitenoiseistheGARCHmodel(seeFranqand Zakoïan

(2010)).Inthemodelingofnanialtime seriestheGARCHassumptionontheerrorsisoften

usedtoapturethe onditionalheterosedastiity. However,the multipliative noisestruture

of this GARCH model is often too restritive in pratial situations. This is one motivation

of this paper, whih onsidersan even more general weaknoise, where the error issubjet to

unknownonditional heterosedastiity.

This representation issaid tobe aweak ARMARC(p, q) representation under the assump- tionthatǫ^{isaweakwhitenoise.ForthestatistialinfereneofARMAmodels,theweakwhite}

noise assumption is oftenreplaed bythe strong whitenoise assumption,i.e. the assumption

that ǫ ^{is an iid sequene of random variables with mean 0 and ommon variane. Obviously}

the strong whitenoise assumption ismore restritive than the weakwhite noise assumption,

beause independeneentailsunorrelatedness. Consequently weakARMARCrepresentation

is more general than the strongone.

Theunknownparameterofinterestdenotedθ₀:= (a⁰_i(s), b⁰_j(s), i= 1, . . . , p, j= 1, . . . , q, s∈ S)^{lies in aompat set ofthe form}

Θ⊂n

(a_i(s), b_j(s), i= 1, . . . , p, j = 1, . . . , q, s∈ S)∈R^(p+q)×Ko ,

with nonemptyinterior,withinwhihwesupposethatθ0 ^{lies.Theparameter}σ^2isonsidered

asanuisane parameter.Inordertoestimateθ₀^{,wethushaveatourdisposaltheobservations} (X_t,∆_t)^, t = 1, . . . , n^{, from whih we aim to build a strongly onsistent and} asymptotially normal estimator

θˆn^{. We now introdue, the strong mixing oeients} (αZ(h))_h∈Z ^{of a sta-}

tionary proess (Z_t)_t∈Z ^{dened by}

αZ(h) := sup

A∈F−∞^t , B∈F_t+h^∞ |P(A∩B)−P(A)·P(B)|, ⁽²⁾

measuring the temporaldependene ofthe proess andwhere F−∞^{t , and} F_t+h^{∞ bethe} σ^-elds

generated by {Z_u, u ≤t} ^and {Z_u, u≥ t+h}^, respetively. We will make an integrability assumption on the moment of the noise and a summability ondition on the strong mixing

oeients (α_Z(h))_h≥0^{. Letus supposethe following} assumptions.

(A1) ^{The proesses}(ǫ_t)_t∈Z ^and (∆_t)_t∈Z ^{areergodisequenes, stritly}stationary, independent fromeah other.

(A2) ^{For some} ν >0, ^{the proesses} (ǫ_t)_t∈Z ^and(∆_t)_t∈Z ^satisfy P_∞

h=0α_ǫ(h)^ν+2ν <+∞

and

P_∞

h=0α_∆(h)^ν+2ν <+∞. (A3) ^{The proess}(ǫ_t)_t∈Z ^{also satises}E[|ǫ_t|^2ν+4]<+∞. (A4) ^{We have} θ₀∈Θ^◦, ^where Θ^{◦ denotes the interiorof} Θ.

Notethatthestrongwhitenoiseassumptionentailstheergodiityonditionfor(ǫ_t)_t∈Z^.Thisis

not the aseifweimposethe weakwhitenoise assumptiononly, henethe assumption (A1).

Likewise, the ergodiity onditionon (∆t)t∈Z ^{is imposedin that} assumption. For example,if

(∆_t)_t∈Z ^{is a nite Markov hain, then a neessary and suient ondition for ergodiity is}

that itisirreduible,whih ensuresitspositivereurrene (seeTheorem1.10.2 p.53in Norris

(1998)), seefor instane the examplein Setion 4.

Weintroduethefollowingnotationssoastoemphasizedependeneofunknownparameter

θ₀ ^{in (1). For all} θ = (a_i(s), b_j(s), i = 1, . . . , p, j = 1, . . . , q, s ∈ S) ∈ Θ^{, we let} a_i :=

(a_i(s), s∈ S)^, i= 1, . . . , p^and b_j := (b_j(s), s∈ S)^, j = 1, . . . , q^{. Let}e(s) ^{be the rowvetor of}

size 1×K ^{suh thatthe} i^{thomponent is1}_[s=i]^{. Thenwe notie that}∀t∈Z

a_i(∆_t) =<e(∆_t), a_i>:=g^a_i(∆_t, θ), b_j(∆_t) =<e(∆_t), a_j >:=g_j^b(∆_t, θ), i= 1, . . . , p, j = 1, . . . , q,

where <·,·>^{denotes thesalarprodut between vetorsof}appropriatedimension.Thus (1) reads

X_t−

p

X

i=1

g^a_i(∆_t, θ₀)X_t−i=ǫ_t−

q

X

j=1

g^b_j(∆_t, θ₀)ǫ_t−j. ⁽³⁾

Let us furthermore note that for all i^, j ^and s^, g_i^a(s, θ) ^and g_j^b(s, θ) ^{are linear in} θ^{. We thus}

introdue the following ompanion matries

A(s) :=











g^a₁(s, θ₀) · · · g_p^a(s, θ₀) 0 Ip−1

.

.

.

0











, B(s, θ) :=











g₁^b(s, θ) · · · g^b_q(s, θ) 0 Iq−1

.

.

.

0











for all s∈ S^, θ ∈ Θ^{. A remark that will prove useful later on is that} θ 7→ B(s, θ) ^{is, for all} s∈ S^{, an anefuntion.}

Wenextintroduetheresidualsorrespondingtoparameterθ∈Θ^{asthestationaryproess} (ǫ_t(θ))_t∈Z ^satisfying

ǫ_t(θ)−

q

X

j=1

g^b_j(∆_t, θ)ǫ_t−j(θ) =X_t−

p

X

i=1

g_i^a(∆_t, θ)X_t−i, ∀t∈Z. ⁽⁴⁾

ThisproessisuniqueinL^{2,asexplainedin}Proposition3.1.Inpartiular,wehave(ǫ_t(θ₀))_t∈Z = (ǫ_t)_t∈Z^{, the initial white noise. We next dene the} approximating residuals as the proess

(e_t(θ))_t∈Z ^verifying

e_t(θ)−

q

X

j=1

g^b_j(∆_t, θ)e_t−j(θ) = ˜X_t−

p

X

i=1

g_i^a(∆_t, θ) ˜X_t−i, ∀t∈Z, ⁽⁵⁾

where valuesorrespondingtonegativeindiesaresettozero,i.e.theproesses(e_t(θ))_t∈Z ^and

( ˜X)_t∈Z ^verify

e_t(θ) = 0, t≤0,

X˜_t = X_t¹_[t≥1], ∀t∈Z.

The basi idea behind denition of (e_t(θ))_t∈Z ^{is that, given a} realization X₁, X₂, . . . , X_n ^of

length n^, ǫt(θ)^is approximated,for 0< t≤n^{, by}et(θ)^{. Next,we dene the ost funtion} Q_n(θ) = 1

2n

n

X

t=1

e²_t(θ). ⁽⁶⁾

Finally, we letfor alln∈Nthe randomvariableθˆ_n ^{the leastsquaredestimatorthatsatises,}

almost surely,

Q_n(ˆθ_n) = min

θ∈ΘQ_n(θ), ⁽⁷⁾

We nishthis setionbygiving some notations. Inthe following,||.||^{will denotethe norm of}

matries or vetors of appropriate size, depending on the ontext, whereas ||.||p ^{will denote}

the L^{p norm dened by} ||X||p = [E(|X|^p)]^{1/p for all random variable} X ^{admitting a} p−^th

order moment, p ≥ 1^{. For all matrix} M^, M^{′ will denote its transpose. For all three times}

dierentiable funtion f : Θ −→ R, we will let ∇f(θ) =

∂

∂θkf(θ)

k=1,...,(p+q)K

, ∇²f(θ) = ∂²

∂θ_i∂θ_jf(θ)

i,j=1,...,(p+q)K

and ∇³f(θ) =

∂³

∂θ_ℓ∂θ_i∂θ_jf(θ)

ℓ,i,j=1,...,(p+q)K

respetively the rst,

seond and thirdorder derivativeswith respet tothe variableθ^.

3. Case of general orrelated proess (∆t)t∈Z

In this setion,wedisplay our mainresults.

3.1. Weak stationarity

Arststeponsistsingivingsuientonditionssuhthattheproesses(X_t)_t∈Z^and(ǫ_t(θ))_t∈Z

dened in (1) and(4) arestritlystationaryandadmitsmomentsofsuientlyhighorderso

astoobtainonsistenyand asymptotinormalityresults.Thisapproahisstandard,seee.g.

(Franq andZakoïan, 2001, Theorem 1 and Setion 3) and (Stelzer, 2009, Theorems 2.1 and

4.1). Let||.||^{beanynormonthe setof matries,and letusintrodue thefollowing notations} w₁ := (1,0, . . . ,0)∈R^p+q,

w_p+1 := (w_p+1,i)i=1,...,p+q, w_p+1,1= 1, w_p+1,i=¹_[i=p+1], i= 2, . . . , p+q, M := (m_ij)i,j=1,...,p+q, m_i,j =¹_[i=q+1,j=1or i=1,j=1],

Φ(s, θ) :=



























g^a₁(s, θ) · · · g_p^a(s, θ)

B(s, θ) 0

0 · · · 0

0 I_p−1 ^...

0



























, s∈ S, θ∈Θ,

Ψ(s) :=



























g^b₁(s, θ₀) · · · g_q^b(s, θ₀)

A(s) 0

0 · · · 0

0 I_q−1 ^...

0



























, s∈ S.

Let us note that the matries Φ(s, θ) ^and Ψ(s) ^{are, like} B(s, θ) ^and A(s)^{, reminisent of}

ompanion matries. Asfor B(s, θ)^{, we also notie in partiular that} θ7→Φ(s, θ) ^{isan ane}

funtion for all s∈ S^{. Wehave the following result.}

Proposition 3.1. Let us suppose that

(A5a) lim sup

t→∞

1 tlnE



sup

θ∈Θ

t

Y

i=1

Φ(∆i, θ)

8

<0, lim sup

t→∞

1 t lnE





t

Y

i=1

Ψ(∆i)

8

<0,

then for all t∈Zand θ∈Θ^{, the unique stationary solutionto (4)is given by} ǫ_t(θ) =

∞

X

i=0

c_i(θ,∆_t, . . . ,∆_t−i+1)ǫ_t−i, ^{where (8)}

c_i(θ,∆_t, . . . ,∆_t−i+1) =

i

X

k=0

w₁

k−1

Y

j=0

Φ(∆_t−j, θ)M

i−1

Y

j^′=k

Ψ(∆_t−j^′)w^′_p+1, ⁽⁹⁾

withtheusualonvention

Qj

i = 1^ifi > j^.Furthermore,foreaht∈Z,(c_i(θ,∆_t, . . . ,∆_t−i+1))_i∈N

is the unique sequene in the set of sequenes of random variables

H:=

(

(d_i)_i∈N independent from (ǫ_t)_t∈Z ^s.t. E

∞

X

i=0

d²_i

!

<+∞ )

satisfying the deomposition (8).

Theuniqueness propertyin thispropositionanbeseenasanidentiabilityproperty. Suh

a property is guaranteed in a similar ontext by Assumption A6 page 56 in Gautier (2004)

(see also Franq and Gautier (2003)) in the ase of strong ARMA proesses modulated by

a Markov hain. Note also that the deomposition (8) is a slight generalization of the Wold

deomposition of stationaryproesses whih aresquaredintegrable, seeTheorem 5.7.1 p.187

of Brokwell andDavis (1991). Remark that the stability ondition (A5a) is reminisent of

the one in (Franq andZakoïan, 2001, Theorem 1) and (Stelzer, 2009, Theorem 2.1) (see

also Brandt (1986)); itishowever stronger asweneed integrability onditionsfor the proess

(ǫ_t(θ))_t∈Z ^{(aswell asonits} derivatives),uniformlyon θ∈Θ^{. Morepreisely, we notethatthe}

right inequalityondition in (A5a)is equivalent to (Stelzer, 2009, Remark 4.1(a)).

Corollary 3.2. The proess (e_t(θ))_t∈Z ^{dened by (5) has the following deomposition}

e_t(θ) =

∞

X

i=0

c^e_i(t, θ,∆_t, . . . ,∆_t−i+1)ǫ_t−i, t≥p+ 1, ^{where (10)}

c^e_i(t, θ,∆_t, . . . ,∆_t−i+1) =

min(t−1,i)

X

k=0

w₁

k−1

Y

j=0

Φ(∆_t−j, θ)M

i−1

Y

j^′=k

Ψ(∆_t−j^′)w^′_p+1, ⁽¹¹⁾

where the matrix M ^andvetors w₁^, w_p+1^{,are dened atthe beginning of the setion.}

Lemma 3.3. The random oeients ci(θ,∆t, . . . ,∆t−i+1)^, i∈Z, t∈Z,verifythe following

• θ7→c_i(θ,∆_t, . . . ,∆_t−i+1)^,θ7→ ∇[c_i(θ,∆_t, . . . ,∆_t−i+1)]^{2 and}θ7→ ∇²[c_i(θ,∆_t, . . . ,∆_t−i+1)]²

are a.s. polynomialfuntions,

• ^{Letus assume,instead of (A5a), that the stronger assumption} (A5b) lim sup

t→∞

1 tlnE



sup

θ∈Θ

t

Y

i=1

Φ(∆_i, θ)

4ν+8

<0, lim sup

t→∞

1 t lnE





t

Y

i=1

Ψ(∆_i)

4ν+8

<0

holds. Thenwe have

lim sup_i→∞¹_ilnE sup_θ∈Θ[c_i(θ,∆_i, . . . ,∆₁)]^2ν+4

< 0, lim sup_i→∞¹_i lnE

sup_θ∈Θ

∇^j[c_i(θ,∆_i, . . . ,∆₁)]

2ν+4

< 0, j= 2,3. ⁽¹²⁾

Furthemore, the oeients c^e_i(t, θ,∆_t−1, . . . ,∆_t−i)^, i∈Z, t≥0^{, satisfy} lim sup_i→∞¹_i ln sup_t≥0E sup_θ∈Θ[c^e_i(t, θ,∆_t, . . . ,∆_t−i+1)]^2ν+4

< 0, lim sup_i→∞ ¹_i ln sup_t≥0E

sup_θ∈Θ

∇^j[c^e_i(t, θ,∆_t, . . . ,∆_t−i+1)]

2ν+4

< 0, j= 2,3.

(13)

Notethatone ofthe diereneswithGautier (2004);Franq andGautier (2003)(apart for

the obviousonewhere thenoiseisweakhere)isthat(A5b)leadstothe exponentialderease

(12) for the oeient [ci(θ,∆i, . . . ,∆1)]^{2ν+4 (uniformlyin} θ^{) aswell asits} derivatives. This isto beompared with ConditionA8 page56ofGautier(2004) (seealsoFranq and Gautier

(2003)), where the exponent is 4 ^insteadof 2ν+ 4^{. This} ν > 0 ^{is what makes the dierene}

between weakandstrongnoise,asthisisthe parameterthatmeasuresthedependeneamong

the random variables in the (non iid) sequene (ǫ_t)^{. Alsonote that(12) and (13) are akin to}

Conditions (A2) ^and(A8) ^{in Franq and Gautier(2004a).}

3.2. Preliminary results

We dene the ostfuntion

O_n(θ) = 1 2n

n

X

t=1

ǫ²_t(θ). ⁽¹⁴⁾

Similarly to

θˆ_n^{, let us introdue} θˇ_n ^{the least squared estimators} orresponding to the ost funtion O_n(θ)^:

O_n(ˇθ_n) = min

θ∈ΘO_n(θ). ⁽¹⁵⁾

The following results are neessary in order to prove the asymptoti properties for the esti-

mators

θˆn ^and θˇn ^{dened in (15) and (7) . We rst justify that} et(θ) asymptotially behaves asǫ_t(θ) ^ast→ ∞ ^{for all}θ ^asfollows:

Lemma 3.4. Letus supposethat (A1)andthatstationarityondition(A5a)hold.Sequenes

(ǫ_t(θ))_t∈Z ^and(e_t(θ))_t∈Z ^satisfy

1. ||sup_θ∈Θ|ǫ₀(θ)|||₄ <+∞ ^andsup_t≥0||sup_θ∈Θ|e_t(θ)|||₄ <+∞^,

2. ||sup_θ∈Θ|ǫ_t(θ)−e_t(θ)|||₂ ^{tends to}0 exponentially fast as t→ ∞^,

3. For all α >0^, t^αsup_θ∈Θ|ǫ_t(θ)−e_t(θ)| −→0 ^{a.s. as}t→ ∞^,