Monitoring procedure for parameter change in causal time series

Transcription

1 Moitorig procedure for parameter chage i causal time series Jea-Marc Bardet, William Chakry Kege To cite this versio: Jea-Marc Bardet, William Chakry Kege. Moitorig procedure for parameter chage i causal time series. 22. <hal-7342v> HAL Id: hal Submitted o 2 Sep 22 (v), last revised 2 Ja 23 (v2) HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Moitorig procedure for parameter chage i causal time series Jea-Marc BARDET ad William KENGNE September 2, 22 SAMM, Uiversité Paris Pathéo-Sorboe, 9 rue de Tolbiac Paris Cedex 3, Frace. Jea-Marc.Bardet@uiv-paris.fr ; William-Charky.Kege@malix.uiv-paris.fr Abstract : We propose a ew sequetial procedure to detect chage i the parameters of a process X = (X t ) t Z belogig to a large class of causal models (such as AR( ), ARCH( ), TARCH( ), ARMA-GARCH processes). The procedure is based o a differece betwee the historical parameter estimator ad the updated parameter estimator, where both these estimators are based o the quasi-likelihood of the model. Ulike classical recursive fluctuatio test, the updated estimator is computed without the historical observatios. The asymptotic behavior of the test is studied ad the cosistecy i power as well as a upper boud of the detectio delay are obtaied. Some simulatio results are reported with comparisos to some other existig procedures exhibitig the accuracy of our ew procedure. The procedure is also applied to the daily closig values of the Nikkei 225, S&P 5 ad FTSE stock idex. We show i this real-data applicatios how the procedure ca be used to solve off-lie multiple breaks detectio. Keywords: Sequetial chage detectio; Chage-poit; Causal processes; Quasi-imum likelihood estimator; Weak covergece. Itroductio I statistical iferece, may authors have poited out the dager of omittig the existece of chages i the data. May papers have bee devoted to the problem of test for parameter chages i time series models whe all data are available, see for istace Horváth [5], Icla ad Tiao [7], Kokoszka ad Leipus [2], Kim et al. [9], Aue et al. [4], Bardet et al. [5], Kege [8]. These papers cosider retrospective (off-lie) chages i.e. chages i parameters whe all data are available. Aother poit of view is the chage detectio whe ew data arrive; this is sequetial chage-poit problem. A importat turig o this topic was made i 996 with the paper of Chu et al.. They cosidered the sequetial chage i regressio model ad poited out the effects of repeatig retrospective test everytime whe ew dataare observed; this ca icreasethe probability oftype errorofthe test. They successfully applied a fluctuatio test to solve the sequetial chage-poit problem. Two procedures are developed based o cumulative sum (CUSUM) of residuals ad recursive parameter fluctuatios. Their idea has bee geeralized ad several procedures are ow based o this approach. Leisch et al. [23] itroduced the geeralized fluctuatio test based o the recursive movig estimator which cotais the test of Chu et al. [2] as a special case. Horváth et al. [6] itroduced residual CUSUM moitorig procedure where the recursive parameter is based o the historical data. This procedure has bee geeralized by Aue et al. [2] to the class of liear model with depedet errors. Berkes et al. [6] cosidered sequetial chages i the parameters of GARCH process. Accordig to the fact that the fuctioal limit theorem assumed by Chu et al. [2] is ot satisfied by the squares of residuals of GARCH process, they developed a procedure based o quasi-likelihood scores. Na et al. [25] developed a moitorig procedure for Supported by Uiversité Paris ad AUF (Agece Uiversitaire de la Fracophoie).

3 2 the detectio of parameter chages i geeral time series models. They show that uder the ull hypothesis of o chage, their detector statistic coverges weakly to a kow distributio. However, the asymptotic behavior of this detector is ukow uder the alterative of parameter chages. I this ew cotributio, we cosider a large class of causal time series ad ivestigate the asymptotic behavior uder the ull hypothesis of o chage but also uder the alterative hypothesis of chage. More precisely, let M,f : IR IN IR be measurable fuctios, (ξ t ) t Z be a sequece of cetered idepedet ad idetically distributed (iid) radom variables satisfyig var(ξ ) = σ 2 ad let Θ be a fixed compact subset of IR d. Let T Z, ad for ay θ Θ, defie Class M T (M θ,f θ ): The process X = (X t ) t Z belogs to M T (M θ,f θ ) if it satisfies the relatio: X t+ = M θ ( (Xt i ) i IN ) ξt +f θ ( (Xt i ) i IN ) for all t T. () The existece ad properties of this geeral class of causal ad affie processes were studied i Bardet ad Witeberger [4]. Numerous classical time series (such as AR( ), ARCH( ), TARCH( ), ARMA-GARCH or biliear processes) are icluded i M Z (M,f). The off-lie chage detectio for such a class of models has already bee studied i Bardet et al. [5] ad Kege [8]. Suppose ow that we have observed X,,X which are available historical data. We assume that the historical data depeds o oe parameter i.e. there exists θ Θ such as (X,,X ) belogs to M {,,} (M θ,f θ ). The, we observe ew data X +,X +2,X k, : the moitorig starts. For each ew observatio, we would like to kow if a chage occurs i the parameter θ. More precisely, we cosider the followig test : H : θ is costat over the observatio X,,X,X +, i.e. the observatios X,,X,X +, belog to M IN (M θ,f θ ); H : there exist k >, θ Θ such that X,,X,X +,,X k,x k +, belogs to M {,,k }(M θ,f θ ) M {k +, }(M θ,f θ ). Whe ew data arrive, Chu et al. [2] proposed to compute a estimator of the parameter based o all the observatios ad to compare it to a estimator based o historical data. A large distace betwee both these estimators meas that ew data come from a model with differet parameters. The the ull hypothesis H is rejected ad the moitorig stops; otherwise, the moitorig cotiues. I their procedure, Leisch et al. [23] suggested to compute the recursive estimators o a movig widow with a fixed width. They fixed a moitorig horizo so that, the procedure will stop after a fixed umber of steps eve if o chage is detected. As Chu et al. [2], the recursive estimators computed by Na et al. [25] are based o all the observatios. As we will see i the ext sectios, their procedure caot be effective i terms of detectio delay or if a small chage i the parameter occurs. For ay k, l,l {,,k} (with l l ) let θ(x l,,x l ) be the quasi-imum likelihood estimator (QMLE i the sequel) of the parameter computed o {l,,l } (see its defiitio i (6)). Whe ew data arrive at time k, we explore the segmet {l,l+,,k} with l { v, v + k v } (where (v ) IN is a fixed sequece of iteger umbers) that the distace betwee θ(x l,,x k ) ad θ(x,,x ) is the largest. If the orm θ(x l,,x k ) θ(x,,x ) is greater tha a suitable critical value, the H is rejected ad the moitorig stops; otherwise, the moitorig cotiues. More precisely, we costruct a detector that takes ito accout the distace betwee θ(x l,,x k ) ad θ(x,,x ). It is show that this detector is almost surely fiite uder the ull hypothesis ad almost surely diverges to ifiity uder the alterative. Hece, the cosistecy of our procedure follows. Simulatios results compared to the procedure of Horváth et al. [6] (see also Aue et al. [2]) ad Na et al. [25] show that our procedure outperforms i terms of test power ad detectio delay.

4 3 I the forthcomig Sectio 2 the assumptios ad the defiitio of the quasi-likelihood estimator are provided. I Sectio 3 we preset the moitorig procedure ad the asymptotic results. Sectio 4 is devoted to a simulatio study for AR() ad GARCH(,) processes. I Sectio 5 we apply our procedure to Nikkei 225, S&P 5 ad FTSE stock idex. After coclusios i Sectio 6, the proofs of the mai results are provided i Sectio 7. 2 Assumptios ad defiitio of the quasi-likelihood estimator 2. Assumptios o the class of models M Z (f θ,m θ ) Let θ IR d ad M θ ad f θ be umerical fuctios such that for all (x i ) i IN IR IN, M θ ( (xi ) i IN ) ad ( ) f θ (xi ) i IN IR. Deote hθ := Mθ 2. We will use the followig classical otatios:. applied to a vector deotes the Euclidea orm of the vector; 2. for ay compact set K IR d ad for ay g : K IR d, g K = θ K ( g(θ) ); 3. for ay set K IR d, K deotes the iterior of K. Throughout the sequel, we will assume that the fuctios θ M θ ad θ f θ are twice cotiuously differetiable o Θ. Let Ψ θ = f θ, M θ ad i =,, 2, the for ay compact set K Θ defie Assumptio A i (Ψ θ,k): Assume that i Ψ θ ()/ θ i Θ < ad there exists a sequece of o-egative real umbers (α (i) j (Ψ θ,k)) j such that α (i) j (Ψ θ, ) < ad i Ψ θ (x) θ i j= i Ψ θ (y) K θ i j= α (i) j (Ψ θ,k) x j y j for all x,y IR IN. I the sequel we refer to the particular case called ARCH-type process if f θ = ad if the followig assumptio holds with h θ = M 2 θ : Assumptio A i (h θ,k): Assume that i h θ ()/ θ i Θ < ad there exists a sequece of o-egative real umbers (α (i) j (h θ,k)) j such as α (i) j (h θ,k) < ad j= i h θ (x) θ i i h θ (y) K θ i j= α (i) j (h θ,k) x 2 j y 2 j for all x,y IR IN. The Lipschitz-type hypothesis A i (Ψ θ,k) are classical whe studyig the existece of solutios of the geeral model (see for istace [3]). Usig a result of [4], for each model M Z (M θ,f θ ) it is iterestig to defie the followig set: Θ(r) := { θ Θ, A (f θ,{θ}) ada (M θ,{θ})hold with α () j (f θ,{θ})+(e ξ r ) /r } α () j (M θ,{θ}) < j j { θ Θ, f θ = ad A (h θ,{θ})holds with(e ξ r ) 2/r } α () j (h θ,{θ}) <. j The, if θ Θ(r) the existece of a uique causal, statioary ad ergodic solutio X = (X t ) t Z M Z (f θ,m θ ) is esured (see more details i [4]). The subset Θ(r) is defied as a reuio to cosider accurately geeral causal models ad ARCH-type models simultaeously. Here there are assumptios required for studyig QMLE asymptotic properties: Assumptio D(Θ): h > such that if θ Θ ( h θ(x) ) h for all x IR IN.

5 4 Assumptio Id(Θ): For all (θ,θ ) Θ 2, ( ) f θ (X,X, ) = f θ (X,X, ) ad h θ (X,X, ) = h θ (X,X, ) a.s. θ = θ. AssumptioVar(Θ): Forallθ Θ,oeofthefamilies ( f θ θ i(x,x, ) ) i d is a.s. liearly idepedet. or ( h θ θ i (X,X, ) ) i d Assumptio K(f θ,m θ,θ): for i=,, 2, A i (f θ,θ) ad A i (M θ,θ) (or A i (h θ,θ)) hold ad there exists l > 2 such that α (i) j (f θ,θ)+α (i) j (M θ,θ)+α (i) j (h θ,θ) = O(j l ) for j IN. Note that i this last assumptio, as i [4], we use the covetio that if A i (M θ,θ) holds the α (i) l (h θ,θ) = ad if A i (h θ,θ) holds the α (i) l (M θ,θ) =. 2.2 Two first examples. ARMA(p, q) processes. Cosider the ARMA(p, q) process defied by: X t + p a i X t i = i= q b j ξ t j, t Z (2) with b, θ = (a,,a p,b,,b q) Θ IR p+q+ ad (ξ t ) a white oise such as E(ξ 2 ) =. Whe q j= b j Xj ad + p i= a i Xi for all X, this process ca be also writte as: X t = b ξ t + j= φ j (θ )X t i, t Z j= where θ Θ φ j (θ) are fuctios oly depedig o θ ad decreasig expoetially fast to (j ). The process (2) belogs to the class M Z (M θ,f θ ) where f θ (x, ) = j φ j(θ)x j ad M θ b for all θ Θ. The Assumptios D(Θ), A (f θ,θ), A (M θ,θ) hold with h = b > ad α() j (f θ,θ) = φ j (θ) Θ while α () j (M θ,θ) = for j IN. Assumptio K(f θ,m θ,θ) holds sice there exists c > ad C > such as φ j Ce cj for j IN. Moreover, if (ξ t ) is a sequece of o-degeerate radom variables (i.e. ξ t are ot equal to a costat), Assumptios Id(Θ) ad Var(Θ) hold. Fially, for ay r such that E ξ r <, the Θ(r) = {θ IR p+q+, φ j (θ) < }. Note that if θ Θ(r) with r the the previous coditios of statioarity q j= b jx j ad + p i= a ix i for all X are satisfied. j 2. GARCH(p, q) processes. Cosider the GARCH(p, q) process defied by: p q X t = σ t ξ t, σt 2 = α + α j X2 t j + βj σ2 t j, t Z (3) j= with E(ξ 2) = ad θ := (α,,α p,β,,β q ) Θ where Θ is a compact subset of ], [ [, [p+q such that p j= α j + q j= β j < for all θ Θ. The there exists (see Bollerslev [] or Nelso ad Cao [26]) a oegative sequece (ψ j (θ)) j such that σt 2 = ψ (θ)+ j ψ j(θ)x t j 2 with ψ (θ) = α /( q j= β j ). This process belogs to the class M Z (M θ,f θ ) where f θ ad M θ (x, ) = ψ (θ)+ j ψ j(θ)x 2 j for all θ Θ. Assumptio D(Θ) holds with h = if θ Θ (ψ (θ)) >. If there exists < ρ < such that for ay j=

6 5 q θ Θ, j= α j + p j= β j ρ the the sequeces ( ψ j (θ) Θ ) j, ( ψ j (θ) Θ) j ad ( ψ j (θ) Θ) j decay expoetially fast (see Berkes et al. [7]) ad Assumptio K(f θ,m θ,θ) holds. Moreover, (ξt) 2 is a sequece of o-degeerate radom variables (i.e. ξt 2 are ot equal to a costat), Assumptios Id(Θ) ad Var(Θ) hold. Fially for r 2 we obtai Θ(r) = { θ Θ ; (E ξ r ) 2/r φ j (θ) < }. 2.3 The quasi-imum likelihood estimator Let k 2, if (X,,X k ) M {,,k} (M θ,f θ ), the for T {,,k}, the coditioal quasi-(log)likelihood computed o T is give by: L(T,θ) := 2 t T j= q t (θ) with q t (θ) = (X t f t θ )2 h t θ +log(h t θ ) (4) where f t θ = f θ( Xt,X t 2... ), M t θ = M θ( Xt,X t 2... ) ad h t θ = Mt θ2. The classical approximatio of this coditioal log-likelihood is give by: L(T,θ) := ( Xt q t (θ) where q t (θ) := f t 2 θ) 2 t T ĥ t θ +log ( ĥ t ) θ with f t θ = f θ( Xt,...,X,,, ), M t θ = M θ( Xt,...,X,,, ) ad ĥt θ = ( M t θ )2. For T {,,k}, defie the quasi imum-likelihood estimator (QMLE) of θ computed o T by (5) θ(t) := arg( L(T,θ)). (6) θ Θ IBardetadWiteberger[4]itwasestablishedthatif(X,,X )isaobservedtrajectoryofx M Z (f θ,m θ ) with θ Θ(4) ad if Θ is a compact set such as Assumptios A i (f θ,m θ,θ) (or A i (h θ,θ)) hold for i =,,2 ad uder Assumptios D(Θ), Id(Θ), Var(Θ), K(f θ,m θ,θ), the ( θ(t, ) θ ) D N (, F G F ), (7) with [ q (θ G := E ) q (θ ) ] [ 2 q (θ ad F := E ) ] θ θ θ θ, (8) where deotes the traspose ad with q defied i (4). Note that uder assumptios D(Θ) ad Var(Θ), G is symmetric positive defiite (see [8]) ad F is o-sigular (see [4]). Also defie the matrix Ĝ(T) := Card(T) ( qt ( θ(t)) )( qt ( θ(t)) θ θ t T ) 2 ( 2 Lm (T, θ(t)) ) ad F(T) := Card(T) θ θ. (9) Uder the previous assumptios, Ĝ(T,) ad F(T, ) coverge almost surely to G ad F respectively. Hece, Ĝ(T, ) /2 F(T, ) ( θ(t, ) θ ) D N (, I d ) with I d the idetity matrix. This result will be the startig poit of the followig moitorig procedure. () 3 The moitorig procedure ad asymptotic results 3. The moitorig procedure Ithesequel,(X,,X )isposedtobethehistoricalavailableobservatiosbelogigtotheclassm {,,} (f θ,m θ ). For l l, deote T l,l := {l,l+,,l }.

7 6 At a moitorig istat k, our idea is to evaluate the differece betwee θ(t l,k ) ad θ(t, ) for ay l =,,k. More precisely, from (), for ay k > defie the statistic (called the detector) Ĉ k,l := k l k Ĝ(T, ) /2 F(T, ) ( θ(tl,k ) θ(t, ) ) for l =,,k. Sice the matrix Ĝ(T,) is asymptotically symmetric ad positive defiite (see [8]), Ĝ(T,) /2 exists for large eough ad Ĉk,l is well defied. At the begiig of the moitorig ad whe l is close to k, the legth of T l,k is too small, therefore the umerical algorithm used to compute θ(t l,k ) caot coverge. This ca itroduce a large distortio i the procedure. To avoid this, we itroduce a sequece of iteger umbers (v ) IN with v << ad compute Ĉk,l for l { v, v +,,k v }. Thus, for ay k > deote For techical reasos, assume that, Π,k := { v, v +,,k v }. v ad v / ( ). Accordig to Remark of [8], we ca choose v = [(log) δ ] with δ >. Note that, if chage does ot occur at time k >, for ay l Π,k, the two estimators θ(t l,k ) ad θ(t, ) are close ad the detector Ĉk,l is ot large eough. Hece, the moitorig stops ad reject H at the first time k > where there exists l Π,k satisfyig Ĉk,l > c for a fixed costat c >. To be more geeral, we will use a b : (, ) (, ), called a boudary fuctio satisfyig: Assumptio B: b : (, ) (, ) is a o-icreasig ad cotiuous fuctio such as If b(t) >. <t< The the moitorig procedure stops at the first time k > such as there exists l Π,k satisfyig Ĉk,l > b((k l)/). Hece defie the stoppig time: Therefore, we have τ() := If{k > / l Π,k, Ĉ k,l > b((k l)/)} = If { k > / { P{τ() < } = P Ĉ } k,l l Π,k b((k l)/) > for some k > { = P k> l Π,k The challege is to choose a suitable boudary fuctio b( ) such as for some give α (,) ad lim P H {τ() < } = α lim P H {τ() < } = where the hypothesis H ad H are specified i Sectio. Ĉ k,l b((k l)/) > }. Ĉ } k,l l Π,k b((k l)/) >. () I the case where b( ) is a costat positive value, b c with c >, these coditios lead to compute a threshold c = c α depedig o α. If chage is detected uder H i.e. τ() < ad τ() > k, the the detectio delay is defied by d = τ() k. Usig the previous otatios, Na et al. [25] used the followig detector D k := Ĝ(T, ) /2 F(T, ) ( θ(t,k ) θ(t, ) ). At the step k of the moitorig, their recursive estimator is based o X,,X,,X k. Oe ca see that this estimator is highly iflueced by the historical data. Assume that a chage occurs at time k k, i the sequel of

8 7 the procedure, the recursiveestimator cotets the observatiosx,,x,,x k which depeds o θ. The, oe must wait loger before the differece betwee θ(x,,x ) ad θ(x,,x,,x k ) becomes sigificat at a step k > k. Therefore, their procedure caot be effective i terms of detectio delay. Moreover, if teds to ifiity, it is ot almost sure that this chage will be detected. These are cofirmed by the results of simulatios (see Sectio 4). Berkes et al. (24) cosidered a estimator based o historical data to compute the quasi-likelihood scores. They used the fact that the partialderivatives applied to a vectoruis equal to if ad oly if u is the true parameter of the model. So, whe chage occurs, their detector growths asymptotically to ifiity. Therefore, their procedure is cosistet. They proved this result for GARCH(p,q) models. 3.2 Asymptotic behaviour uder the ull hypothesis Uder H, the parameter θ does ot chage over the ew observatios. Thus we have the result Theorem 3. Assume D(Θ), Id(Θ), Var(Θ), K(f θ,m θ,θ), B ad θ Θ(4). Uder the ull hypothesis H, the { lim P{τ() < } = P where W d is a d-dimesioal stadard Browia motio. t> <s<t W d (s) sw d ()) t b(s) I the simulatios, we will use the most atural boudary fuctio b( ) = c with c a positive costat sice it satisfies the above assumptios imposed to b( ). I such case, the forthcomig corollary idicates that the asymptotic distributio of Theorem 3. ca be easily computed: } > Corollary 3. Assume b(t) = c > for t. Uder the assumptios of Theorem 3., { lim P{τ() < } = P t> <s<t where U d = <u< f(u) W d (u) with f(u) = } t W d(s) sw d ()) > c = P{U d > c} 9 u+ u ( 9 u+3 u Remark 3. By the law of the iterated logarithm, it comes that <s<t t W d(s) sw d ()) a.s. W d()). t 2 3 u+ (9 u)( u) So, the two distributios <s<t t W d(s) sw d ()) as t (resp. t ) ad f(u) W d (u) as u (resp. u ) are equal. It is easy to show (see proof of Corollary 3.) that t> <s<t t W d(s) sw d ()) D = <u< f(u) W d (u). Therefore, at a omial level α (,), take c = c(α) be the ( α)-quatile of the distributio of U d = <u< f(u) W d (u) which ca be computed through Mote-Carlo simulatios. Table shows the ( α)- quatile of this distributio for α =.,.5,. ad d =,,5. d = d = 2 d = 3 d = 4 d = 5 α = α = α = Table : Empirical ( α)-quatile of the distributio of U d, for d =,,5. ) /2.

9 8 3.3 Asymptotic behaviour uder the alterative hypothesis Uder the alterative H, the parameter chages from θ to θ at k >, where θ Θ ad θ θ. The Theorem 3.2 Assume D(Θ), Id(Θ),Var(Θ), K(f θ,m θ,θ) ad B. Uder the alterative H, if θ θ ad θ,θ Θ(4) the for k = k () such as lim k ()/ < ad k = k ()+ δ with δ (/2,), l Π,k Ĉ k,l b((k l)/) a.s. The forthcomig Corollary 3.2 ca be immediately deduced from the relatio (). Corollary 3.2 Uder assumptios of Theorem 3.2, lim P{τ() < } =. Remark 3.2 We kow that the moitorig is stopped ad rejects H at the first time k where. l Π,k Ĉ k,l b((k l)/) >. Therefore, it follows from Theorem 3.2 that uder the hypothesis H, the detectio delay d of the procedure ca be bouded by O P ( /2+ε ) for ay ε > (or eve by O P ( (log) a ) with a > usig the same kid of proof). 3.4 Examples 3.4. AR( ) processes Cosider the geeralizatio of ARMA(p, q) processes defied i (2) i.e. a AR( ) processes defied by: X t = φ (θ)+ j (θ)x j φ t j +ξ t, t Z (2) with θ Θ, where we ca chose Θ as a compact subset of Θ(4) IR d where Θ(4) = { θ IR d ; φ j (θ) < }. This process belogs to the class M Z (M θ,f θ ) where f θ (x, ) = j φ j(θ)x j ad M θ φ (θ) for all θ Θ ad therefore α () j (f θ,θ) = φ j (θ) Θ ad α () j (M θ,θ) = for j IN. The Assumptio D(Θ) holds if h = if θ Θ ( φ (θ) ) > ; j Assumptio K(f θ,m θ,θ) holds if there exists l > 2 ad ad if θ φ j (θ) are twice differetiable fuctios satisfyig ( ψ j (θ) Θ, φ j (θ) Θ, φ j (θ) Θ) = O(j l ) for j IN. if (ξ t ) is a sequece of o-degeerate radom variables (i.e. ξ t are ot equal to a costat), Assumptios Id(Θ) ad Var(Θ) hold. Case of AR(p) process Assume that p X t = φ + φ jx t j +ξ t with p IN. j= The true parameter of the model is deoted by θ = (φ,φ,,φ p ) Θ where Θ = {θ = (φ,φ,,φ p ) IR p+ / p φ j < }. The, Θ(r) = Θ for ay r. Assume that a trajectory (X,,X k ) has bee observed, j=

10 9 for ay t =,,k ad θ Θ we have, q t (θ) = ( X t φ p ) 2, q t (θ) φ j X t j = 2 ( X t φ p ) φ j X t j j= θ j= 2 q t (θ) 2 q t (θ) (,X t,x t 2,,X t p ). Moreover, = 2, for j =,,p, = 2X t j ad for i,j p, φ φ φ φ j 2 q t (θ) = 2X t i X t j. φ i φ j ARCH( ) processes Cosider the geeralizatio GARCH(p, q) processes defied i (3) i.e. a ARCH( ) processes defied by: X t = σ t ξ t ad σt 2 = ψ (θ )+ ψ j (θ )X2 t j, t Z (3) j= with θ Θ, where we ca chose Θ as a compact subset of Θ(4) IR d where Θ(4) = { θ IR d ; (E ξ 4 ) /2 φ j (θ) < }. Thisprocess,itroducedbyRobiso[28], belogstotheclassm Z (f θ,m θ,)wheref θ (x, ) admθ 2(x, ) = ψ (θ) + j ψ j(θ)x 2 j for all θ Θ ad therefore α() j (f θ,θ) = ad α () j (h θ,θ) = φ j (θ) Θ for j IN (X is of course a ARCH-type process). The Assumptio D(Θ) holds if h = if θ Θ (ψ (θ)) > ; Assumptio K(f θ,m θ,θ) holds if there exists l > 2 ad ad if θ φ j (θ) are twice differetiable fuctios satisfyig ( ψ j (θ) Θ, ψ j (θ) Θ, ψ j (θ) Θ) = O(j l ) for j IN. if (ξ 2 t) is a sequece of o-degeerate radom variables (i.e. ξ 2 t are ot equal to a costat), Assumptios Id(Θ) ad Var(Θ) hold. j= Case of GARCH(, ) process Assume that X t = σ t ξ t with σ 2 t = α +α X 2 t +β σ 2 t with θ = (α,α,β) Θ ], [ [, [ 2 ad satisfyig α + β <. The ARCH( ) represetatio is σt 2 = α /( β )+α (β )j Xt j 2. If a trajectory (X,,X k ) has bee observed, for ay t =,,k ad θ Θ we have, j Thus, it follows that q t(θ) θ ĥ t θ = α /( β )+α X 2 t +α = ĥ t θ ( )( X2 t ĥ t θ ĥ t θ t t j=2 ), ĥt θ, ĥt θ α α β ad ĥt θ = α /( β ) 2 +α Xt 2 2 +α (j )β j 2 Xt k 2 β. j=3 Let θ = (α,α,β ) = (θ,θ 2,θ 3 ) Θ, for i,j 3, we have with ĥt θ β 2 2ĥt θ α 2 =, 2ĥt θ α α =, 2 q t (θ) = ( 2X 2 t θ i θ j (ĥt θ )2 ĥ t θ 2ĥt θ α 2 = 2α /( β ) 3 +2α X 2 t 3 +α =, β j X 2 t j ad q t (θ) = X 2 t / ĥt θ +log(ĥt θ ). with ĥt θ α = X 2 t + t j=2 ĥ ) t θ ĥt θ + ( ) X2 t 2 ĥ t θ θ i θ j ĥ t θ ĥ t θ θ i θ j 2ĥt θ = Xt 2 2 α β + t t (j )(j 2)β j 3 Xt j 2. j=4 j=3 β j X 2 t j ĥt θ α = /( β ), (j )β j 2 Xt j 2, 2ĥt θ = /( β ) 2 ad α β

11 3.4.3 TARCH( ) processes The process X is called Threshold ARCH( ) (TARCH( ) i the sequel) if it satisfies X t = σ t ξ t ad σ t = b (θ )+ [ ] b + j (θ )(X t j,) b j (θ )mi(x t j,), t Z (4) j= where the parameters b (θ), b + j (θ) ad b j (θ) are assumed to be o egative real umbers ad θ Θ where Θ is a compact subset of Θ(4) where Θ(4) = {θ / IR d ( E ξ 4) /4 ( } b j (θ),b+ j (θ)) < sice α () j (M,{θ}) = ( b j (θ),b+ j (θ)). This class of processes is a geeralizatio of the class of TGARCH(p,q) processes (itroduced by Rabemaajara ad Zakoïa [27]). The, Assumptio D(Θ) holds if h = if θ Θ b (θ) > ; AssumptioK(f θ,m θ,θ)holdsifthereexistsl > 2adadifθ b j (θ) adθ b+ j (θ) aretwicedifferetiable fuctios satisfyig j= ( b j (θ) Θ, b + j (θ) Θ, θ b j (θ) Θ, θ b+ j (θ) Θ, 2 θ 2b j (θ) Θ, 2 θ 2b+ j (θ) ) Θ = O(j l ) for j IN. Ufortuately, for TARCH( ) it is ot possible to provide simple coditios for obtaiig Assumptios Id(Θ) ad Var(Θ) as for AR( ) or ARCH( ) processes. 4 Some simulatio ad umerical experimets First remark that, at a time k >, we eed to compute Ĉk,l for all l Π,k to test whether chage occurs or ot. O ca see that, the computatioal time is very log ad icreases with k. To reduce it, we itroduce a iteger sequece (u ) (satisfyig u / as ; typically u = [l()]) ad compute Ĉk,l oly for l Π,k := { v, v +u, v +2u,,k v }. We have Π,k Π,k ad for ay t = l with l Π,k, we ca fid a iteger j l such that v + j l u Π,k ad v +j l u l v +(j l +)u. This implies that v+j lu t v+j lu + u. Thus, we have asymptotically (as ), t v+j lu. It shows that the previous asymptotic results still hold by computig Ĉ k,l for l Π,k. The coditio u / esures that the Theorem 3.2 still holds by choosig k = k ()+ δ with δ (/2,). I practice, the use of Π,k ca itroduce a distortio i the detectio delay. But, the ew detectio delay must be betwee d ad d +u (where d is the detectio delay obtaied by usig Π,k ). I the sequel, we use u = [l()]. Moreover, if b c > is a costat fuctio, accordig to (), we have { P{τ() < } = P Thus, deote k> Ĉ k,l > c l Π,k Ĉ k = Ĉ k,l for ay k >. l Π,k }. (5) The procedure is moitored from k = + to k = +5. The set {+,,+5} is called moitorig period. Accordig to the Remark of [8], v = [(log) δ ] (with δ 3) is chose. We evaluated the performace of the procedure with v = [log], [(log) 3/2 ], [(log) 2 ], [(log) 3 ] ad we recommed to use v = [(log) 3/2 ] for liear model ad v = [(log) 2 ] for ARCH-type model. The omial level used i the sequel is α =.5.

12 a ) C k for GARCH(,) without chage b ) C k for GARCH(,) with oe chage at k*= k k Figure : Typical realizatio of the statistics Ĉk for = 5 ad k = 5,,. a-) The parameter θ = (.,.3,.2) is costat ; b-) the parameter θ = (.,.3,.2) chages to θ = (.5,.5,.2) at k = 75. The horizotal solid lie represets the limit of the critical regio, the vertical dotted lie idicates where the chage occurs ad the vertical solid lie idicates the time where the moitorig procedure detectig the chage. 4. A illustratio We cosider GARCH(,) process : X t = σ t ξ t with σt 2 = α +α X2 t +β σ2 t. Thus, the parameter of model is θ = (α,α,β ). The historical available data are X,,X 5 (therefore = 5) ad the moitorig period is {5,,}. At the omial level α =.5, the critical values of the procedure is C α = The Figure is a typical realizatio of the statistic (Ĉk) 5<k. We cosider a sceario without chage (Figure a-)) ad a sceario with chage at k = +25 = 75 (Figure b-)). Figure a-) shows that, the detector Ĉk is bellow uder the horizotal lie which represets the limit of the critical regio. O Figure b-) we ca see that, before chage occurs, Ĉk is bellow uder the horizotal lie ad icreases with a high speed after chage. Such growth over a log period idicates that somethig happeig i the model. 4.2 Moitorig mea shift i times series Let (X,,X ) be a (historical) observatio of a process X = (X t ) t Z. We assume that X satisfy { Xt = µ +ǫ t for t k X t = µ +ǫ t for t > k with k >, µ µ ad (ǫ t ) a zero mea statioary time series belogs to a class M Z (f θ,m θ ). Uder H, k =. The moitorig procedure start at k = + ad the aim is to test mea shift over the ew observatio X +,X +2,. This problem ca be see as moitorig chages i liear model (see Horváth et al. [6], Aue et al. [2]) with costat regressor. The empirical mea X = i= X i

13 2 is a cosistet estimator of µ. The recursive residual is defied by ǫ k = X k X ; for k >. Horváth et al. [6] ad Aue et al. [2] proposed the detector Q k = σ c ( k )( k )γ k i=+ where σ 2 is a cosistet estimator of the log-ru variace σ 2 = lim Var( ǫ i ). ǫ i k > ; c > ; γ < /2 If the process (ǫ t ) are ucorrelated (for istace GARCH-type model), empirical variace of the historical data ca be used as estimator of σ 2. If (ǫ t ) are correlated, the popular Bartlett estimator (see [8]) ca be used. Uder some regular coditios, it hold that (see [6] ad [2]) { lim P{τ() < } = P i= <s< W (s) s γ } > c. Hece, at a omial level α =.5, the critical value of the test is the ( α)-quatile of the distributio of W (s) /s γ. Whe γ =, these quatiles are kow (see Table of [25] for values obtaied through a Mote <s< Carlo simulatio). We compare our procedure to the previous residuals CUSUM oe (with γ = ) i two situatios. (ǫ t ) is a AR() process; ǫ t = φ ǫ t +ξ t with φ =.2; 2. (ǫ t ) is a GARCH(,) process; ǫ t = σ t ξ t with σt 2 = α +α ǫ2 t +β σ2 t ad (α,α,β ) = (.,.3,.2). The historical sample size are = 5 ad =. These procedure are evaluated at times k = +, + 2,+3,+4,+ 5, while the chage occurs at k = + 5 or k = Tables 2 ad 3 idicate the empirical levels ad the empirical powers based of 2 replicatios. The elemetary statistics of the empirical detectio delay are reported i Tables 4. The results of Table 2 ad Table 3 show that both procedure based o detectors Ĉk ad Q k are more coservative. Oe ca also see that, icreasig legth of historical data reduces the size distortio of these procedures. This is due to the fact that the legth of moitorig period is fixed ad ot icreasig with. Uder H, the chage has bee detected before the moitorig time k = +5. But, as we metioed above, the challege of this problem is to miimize the detectio delay. For this criteria, it is see i Table 4 that for the mea shift i AR process, our procedure works well as Horváth et al. s procedure whe the chage occurs at the begiig of the moitorig (k = +5) ad it is little better whe the chage occurs log time after the begiig of the moitorig (k = +25). For the mea shift i GARCH process, our test procedure outperforms the Horváth et al. s test i terms of mea, Q, media ad Q 3 of detectio delay. 4.3 Moitorig parameter chage i AR() ad GARCH(,) models I this subsectio, we preset some simulatios results for moitorig parameter chage i AR() ad GARCH(,) models ad compare our procedure to that proposed by Na et al. [25]. First recall that if the boudary fuctio b c > is costat, this procedure is based o the relatio { P{τ() < } = P k> } D k > c

14 3 k Empirical levels = 5 Ĉ k Q k = Ĉ k Q k Empirical powers = 5 ; k = +5 Ĉ k.3 Q k.335 = 5 ; k = +25 Ĉ k...9 Q k = ; k = +5 Ĉ k.75 Q k.95 = ; k = +25 Ĉ k...35 Q k Table 2: Empirical levels ad powers for moitorig meas shift i AR() with φ =.2. The empirical levels are computed whe µ = ad the empirical powers are computed whe the mea µ = chages to µ =.2. k Empirical levels = 5 Ĉ k Q k = Ĉ k Q k..... Empirical powers = 5 ; k = +5 Ĉ k Q k = 5 ; k = +25 Ĉ k..5 Q k...92 = ; k = +5 Ĉ k.995 Q k.985 = ; k = +25 Ĉ k...98 Q k Table 3: Empirical levels ad powers for moitorig meas shift i GARCH(,) with (α,α,β ) = (.,.3,.2). The empirical levels are computed whe µ = ad the empirical powers are computed whe the mea µ = chages to µ =.3.

15 4 d Mea SD Mi Q Med Q 3 Max AR() = 5 ; k = +5 Ĉ k Q k = 5 ; k = +25 Ĉ k Q k = ; k = +5 Ĉ k Q k = ; k = +25 Ĉ k Q k GARCH(,) = 5 ; k = +5 Ĉ k Q k = 5 ; k = +25 Ĉ k Q k = ; k = +5 Ĉ k Q k = ; k = +25 Ĉ k Q k Table 4: Elemetary statistics of the empirical detectio delay for moitorig mea shift i AR() ad GARCH(,).

16 5 where D k := Ĝ(T, ) /2 F(T, ) ( θ(t,k ) θ(t, ) ). Na et al. show that uder H, { lim P{τ() < } = lim P k> } { } D k > c = P W d (s) > c. <s< Hece, at a omial level α, the critical value of their procedure is the ( α)-quatile of the distributio of W d (s) which ca be foud i Table of [25]. <s< The comparisos are made i the followigs situatios.. For AR() model : X t = φ X t +ξ t. Uder H, θ = φ =.2. Uder H, θ chages to θ =.5 at k. 2. For GARCH(,) model : X t = σ t ξ t with σ 2 t = α +α X2 t +β σ2 t. Uder H, θ = (α,α,β ) = (.,.3,.2), while uder H, θ = (.,.3,.2) chages to θ = (.5,.5,.2) at k. The historical sample size are = 5 ad =. These procedure are evaluated at times k = +, + 2,+3,+4,+5,, while the chage occurs at k = +5 or k = +25. Tables 5 ad 6 idicate the empirical levels ad the empirical powers based of 2 replicatios. The elemetary statistics of the empirical detectio delay are reported i Tables 7. k Empirical levels = 5 Ĉ k D k = Ĉ k D k Empirical powers = 5 ; k = +5 Ĉ k.335 D k = 5 ; k = +25 Ĉ k D k = ; k = +5 Ĉ k D k = ; k = +25 Ĉ k D k Table 5: Empirical levels ad powers for moitorig parameter chage i AR() process. The empirical levels are computed whe θ = φ =.2 is costat ad the empirical powers are computed whe θ =.2 chages to θ =.5. The cosidered AR ad GARCH processes have zero mea. Cotrary to the mea shift studied above, this mea is ot estimated. For AR model, it appears i Table 5 that both procedures based o detector Ĉk ad D k are coservative. This is ot the case for GARCH model (Table 6). The high size distortios whe = 5 is due to the difficulty to estimate the parameter of GARCH model. This size distortio decreases whe icreases ad Corollary 3. esures that with ifiite moitorig period, the empirical level teds to the omial oe as. For both the cases of AR ad GARCH processes, the procedure based o detector Ĉk detect the chage after the moitorig time k = + 5. Ulike Na et al. [25], we cosider a sceario of GARCH model with moderate chage i parameter, it is see i Table 6 that the procedure based o detector D k provides usatisfactory results. At the moitorig time k = +5, it is ot sure that the chage must be detected eve whe k = +5. This

17 6 k Empirical levels = 5 Ĉ k D k = Ĉ k D k Empirical powers = 5 ; k = +5 Ĉ k.89 D k = 5 ; k = +25 Ĉ k D k = ; k = +5 Ĉ k.835 D k = ; k = +25 Ĉ k D k Table 6: Empirical levels ad powers for moitorig parameter chage i GARCH(,) process. The empirical levels are computed whe θ = (α,α,β ) = (.,.3,.2) is costat (hypothesis H ) ad the empirical powers are computed whe θ = (.,.3,.2) chages to θ = (.5,.5,.2) (hypothesis H ). d Mea SD Mi Q Med Q 3 Max AR() = 5 ; k = +5 Ĉ k D k = 5 ; k = +25 Ĉ k D k = ; k = +5 Ĉ k D k = ; k = +25 Ĉ k D k GARCH(,) = 5 ; k = +5 Ĉ k D k = 5 ; k = +25 Ĉ k D k = ; k = +5 Ĉ k D k = ; k = +25 Ĉ k D k Table 7: Elemetary statistics of the empirical detectio delay for moitorig parameter chage i AR() ad GARCH(,).

18 7 is ot surprisig accordig to the commet of subsectio 3.. Table 7 idicates the distributio of the detectio delay d. We ca see i Table 7 (eve i Table 4) that for our procedure, the relatio d /5 d 5 is globally satisfied (from Theorem 3.2, we deduced that d = O P ( /2 log ) whe is large eough). It is also see that, mea, Q, media ad Q 3 of our test are shorter tha Na et al. s oe. The results of Table 5, 6 ad 7 show that, our test is uiformly better ad the procedure based o detector Ĉk is highly recommeded. 5 Real-Data Applicatios We cosider the returs of the daily closig values of the Nikkei 225 stock idex (from Jauary 2, 995 to October 9, 998), S&P 5 ad FTSE (from Jauary 2, 24 to Jue, 22). These data are available o Yahoo! Fiace at They are represeted o Figure 2 ad Figure 6. These series are kow to represet ARCH effect ad GARCH(,2) (resp. GARCH(,)) ca be used to capture it i returs of Nikkei 225 (resp. S&P 5 ad FTSE ), see the book of Fracq ad Zakoïa 2. Let us take the observatios goig from Jauary 2, 995 to December 3, 996 (resp. Jauary 2, 24 to December 3, 25) as a historical data for the Nikkei 225 (resp. S&P 5 ad FTSE ) stock idex. These periods are kow to be stable i fiacial commuity. To verify it, we apply three procedures to test for parameter chage i the historical observatios. The ull hypothesis is that the parameter is costat over the observatios agaist the parameter chages alterative. The first test is proposed by Kege [8]. Defie the asymptotic covariace matrix (which take ito accout the chage possibility) of the estimator θ (T, ) by Σ,k := k F (T,k )Ĝ(T,k ) F (T,k ) det( Ĝ (T,k )) + k F (T k, )Ĝ(T k, ) F (T k, ) det( Ĝ (T k, )). The test is based o the statistic Q := ( Q() ) (2), Q where Q () := Q() v,k with k v Q (),k k2 := ( θ (T,k ) θ (T ) ) Σ,k ( θ (T k ) θ (T ) ), Q (2) := Q(2),k with v k v Q (2),k This test is applied with v = (log) δ where 2 δ 5/2. The secod test (see Lee ad Sog [22]) is based o the statistic with v = (log) 2. ( k)2 := ( θ (T k, ) θ (T, ) ) Σ,k ( θ (T k, ) θ (T k, ) ). ( k Q () 2 := ( θ (T,k ) v k v θ (T, ) ) Σ, ( θ (T,k ) θ (T, ) )) The third procedure is the CUSUM test see Kulperger ad Yu [2]. At a omial level α (,), each of these procedure rejects ull hypothesis if the test statistic is greater tha a critical value C α. Table 8 provides the results of these tests to the historical data that we have chose. Q Q CUSUM Nikkei (3.98) 2.3 (3.45).98 (.36) S&P (3.47) 2. (3.6).93 (.36) FTSE.95 (3.47) 2.3 (3.6).3 (.36) Table 8: Results of test for parameter chages i the historical data of Nikkei 225 (from Jauary 2, 995 to December 3, 996), S&P 5 ad FTSE (from Jauary 2, 24 to December 3, 25). Figures i brackets the critical values of the procedure at the omial level α =.5.

19 8 Note that, these series are very closed to a ostatioary process, i the sese that q j= α j + p j= β j (see (3)). Therefore, it would be difficult to compute the estimator θ (T k,l ) ( k < l ). For the statistics Q ad Q (), we cosider oly the time k that the computatio of θ (T,k ) ad θ (T k, ) coverges. This certaily itroduces distortios o these tests. O the other had, the CUSUM procedure eeds to compute oly the estimator θ (T, ) which covergece is obtaied. Accordig to these results, we coclude that the parameter does ot chage over these historical observatios. For Nikkei 225 data, moitorig startig at Jauary 2, 997. Figure 2 shows the realizatio of the sequece (Ĉk) for k goig from Jauary 2, 997 to October 9, 998. Moitorig procedure stops at October 27, 997. Recall that, the moitorig scheme ca be used as a alarm system. Whe it triggered, we eed to apply retrospective test to estimate the breakpoit. Accordig to Kege [8], the test based o Q () ad Q are more powerful tha the CUSUM test. Thus, i the retrospective procedure, we applied these two tests ad cosidered the oe which provides more sigificat result (i terms of p-value). Retrospective procedure is applied to the observatios goig from Jauary 2, 995 to October 27, 997 ad break is detected at t N September 7, 997; see Figure 2. This chage correspodsto the Asia fiacial crisis ( ) where the turmoil period started at July 997. We are goig to see for S&P 5 ad FTSE data how multiple chages ca be moitored. For these series, moitorig starts at Jauary 2, 26. Figure 3 represets the sequece (Ĉk) for k goig from Jauary 2, 26 to December 3, 28. Accordig to the Figure 3, the moitorig stops at November 6, 27 ad September 4, 27 for S&P 5 ad FTSE data respectively. Retrospective procedure is applied to the series goig from Jauary 2, 26 to November 6, 27 (for S&P 5 data) ad the series goig from Jauary 2, 26 to September 4, 27 (for FTSE data). Breaks are detected at t S, Jue 8, 27 (for S&P 5 data) ad t F, July 6, 27 (for FTSE data); see Figure 6. These breaks correspod to the begiig of the Subprime Crisis i US. After moitored the first chage, we eed to update the procedure. The ew historical data are the series goig from t S, to November 6, 27 (for S&P 5 data) ad the series goig from t F, to September 4, 27 (for FTSE data). Therefore, moitorig cotiues at November 9, 27 (for S&P 5 data) ad at September 5, 27 (for FTSE data). Figure 4 shows the curve of the sequece (Ĉk). The moitorig stops at October 7, 28 ad November, 28 for S&P 5 ad FTSE data respectively. The retrospective test is applied ad the break poit estimatio are t S,2 August 4, 28 ad t F,2 September 7, 28 respectively for these two series; see Figure 6. These breaks correspod to the Lehma Brothers Bakruptcy which affects the worldwide fiacial system. After that, the procedure is updated ad moitorig cotiues at October 2, 28 ad November, 28 for S&P 5 ad FTSE data. Figure 5 shows the sequece (Ĉk). The moitorig stopped at March 7, 29 (S&P 5) ad March 9, 29 (FTSE ) ad retrospective test detected chage at t S,3 Jauary 5, 29 (i S&P 5) ad t F,3 December 29, 28 (i FTSE data); see Figure 6. These breaks correspod to the worldwide govermets itervetio to solve the fiacial crisis. The procedure cotiues util Jue 22, other breaks are detected at t S,4 26 Jue 29, t S,5 5 April 2, t S,6 27 September 2, t S,7 9 July 2, t S,8 Jauary 22 (for S&P 5 data) ad t F,4 3 Jue 29, t F,5 27 July 2, t F,6 2 December 2 (for FTSE data). They are represeted o Figure 6. These breaks correspod to the turmoils periods i the Greece ad Europea debt crisis. Summary of the real-data applicatios Both moitorig procedure (based o detector Ĉk) ad retrospective test have bee applied to detect breaks i the Nikkei 225, S&P 5 ad FTSE stock idex. The followig results are obtaied :. For the Nikkei 225, from Jauary 2, 995 to October 9, 998 ; break is detected at t N 7 September 997 which correspod to the turmoil period of the Asia fiacial crisis ( ).

20 9 See also Figure For the S&P 5 ad FTSE, from Jauary 2, 24 to Jue, 22 ; break are detected at ( t S,i ad t F,i are referred to the breakpoit i the S&P 5 ad FTSE respectively) t S, 8 Jue 27 ad t F, 6 July 27 which correspod to the begiig of the Subprime Crisis i US; t S,2 4 August 28 ad t F,2 7 September 28 which correspod to the Lehma Brothers Bakruptcy; t S,3 5Jauary29ad t F,3 29December28whichcorrespodworldwidegovermetsitervetio to solve the fiacial crisis; t S,4 26 Jue 29, t S,5 5 April 2, t S,6 27 September 2, t S,7 9 July 2, t S,8 See also Figure 6. Jauary 22 ad t F,4 3 Jue 29, t F,5 27 July 2, t F,6 2 December 2. These breaks idicates the turmoils periods i the Greece ad Europea debt crisis. 6 Coclusios This paper is devoted to the sequetial chage-poit detectio i the parameters of a large class of causals models. We costruct a moitorig scheme based o a differece betwee the historical parameter estimate ad the updated parameter estimate computed without the historical observatios. The simulatio results for the moitorig mea shift i AR() ad GARCH(,) show that our procedure works well (i terms of power ad detectio delay) as Horváth et al. s procedure whe the chage occurs at the begiig of the moitorig ad it is preferable whe the chage occurs log time after the begiig of the moitorig. The simulatio results for the moitorig parameter chage i AR() ad GARCH(,) show that our procedure outperforms (i terms of power ad detectio delay) the Na et al. s oe. The real-data study shows the applicability of the procedure o the daily closig values of the Nikkei 225, S&P 5 ad FTSE stock idex. It is show how this moitorig scheme with retrospective test ca be used to off-lie multiple breaks detectio.

21 2 C k for the Nikkei Jul Oct Time Breaks detectio i the Nikkei 225 Returs Nikkei t N 998 Time Figure 2: The top figure is a realizatio of the statistics Ĉk with k goig from Jauary 2, 995 to October 9, 998 for Nikkei 225 data; the historical data cosidered are the series goig from Jauary 2, 995 to December 3, 996. The horizotal solid lie represets the limit of the critical regio, the vertical dotted lie idicates the date of the begiig of the Asia fiacial crisis ( ) ad the vertical solid lie idicates the time where the moitorig procedure will stop. The bottom figure is the returs of Nikkei 225 data from Jauary 2, 995 to October 9, 998; the vertical solid lie idicates the date where break have bee detected usig retrospective test after the moitorig stops.

22 2 C k for the S&P Jul Nov Time C k for the FTSE Jul Sep Time Figure 3: Realizatio of the statistics Ĉk with k goig from Jauary 2, 26 to December 3, 28 for S&P 5 ad FTSE data; the historical data cosidered are the series goig from Jauary 2, 24 to December 3, 25. The horizotal solid lie represets the limit of the critical regio, the vertical dotted lie idicates the date of the begiig of the Subprime Crisis i US ad the vertical solid lie idicates the time where the moitorig procedure stopped.

23 22 C k for the S&P Sep Oct 29 Time C k for the FTSE Sep Nov 29 Time Figure 4: Realizatio of the statistics Ĉk for S&P 5 ad FTSE data; the historical data are the series goig from Jue 8, 27 to November 6, 27 (for S&P 5 data) ad July 6, 27 to September 4, 27 (for FTSE data). The horizotal solid lie represets the limit of the critical regio, the vertical dotted lie idicates the date of the Lehma Brothers Crisis ad the vertical solid lie idicates the time where the moitorig procedure stopped.

24 23 C k for the S&P 5 C k for the FTSE Oct Mar 29 May 29 Time Nov Mar 29 Jul 29 Time Figure 5: Realizatio of the statistics Ĉk for S&P 5 ad FTSE data; the historical data are the series goig from August 4, 28 to October 7, 28 (S&P 5) ad September 7, 28 to November, 28 (FTSE ). The horizotal solid lie represets the limit of the critical regio, the vertical dotted lie idicates the date of the begiig of stabilizatio i fiacial system due to the worldwide govermets itervetio ad the vertical solid lie idicates the time where the moitorig procedure stopped. 7 Proofs of the mai results Let us prove first some useful lemmas. I the sequel, for ay x IR, [x] deotes the iteger part of x. Let (ψ ) ad (r ) be sequeces of radom variables. Throughout this sectio, we use the otatio ψ = o P (r ) to mea : for all ε >, P( ψ ε r ) as. Write ψ = O P (r ) to mea : for all ε >, there exists C > such that P( ψ C r ) ε for large eough. Recall that (X,,X ) is a observed trajectory of a process M Z (M θ,f θ ). Let k 2 ad T, = {,,}, T l,k = {l,l+,,k} with l Π,k = {v,v +,,k v }, ad defie with θ defied i (6). Lemma 7. Uder assumptios of Theorem 3., k> C k,l := k l G /2 F ( θ(tl,k ) k θ(t, ) ), Ĉk,l C k,l = op () as. l Π,k b((k l)/) Proof. For ay, we have k> l Π,k b((k l)/) Ĉ k,l C k,l = Now, proceed similarly as i the proof of Lemma 5.3 of [8]. if s> b(s) k> Ĉk,l C k,l. l Π,k

25 24 Breaks detectio i the S&P 5 Returs S&P t S, t S,2 t S,3 t S,4 t S,5 t S,6 t S,7 t S, Time Breaks detectio i the FTSE Returs FTSE t F, t F,2 t F,3 t F,4 t F,5 t F, Time Figure 6: Break detectio i the returs of S&P 5 ad FTSE data usig moitorig procedure based o Ĉk. The verticals lies idicate the dates where breaks have bee detected.

26 25 Lemma 7.2 Uder assumptios of Theorem 3. k> l Π,k b((k l)/) (k l)f ( θ(tl,k ) k θ(t, ) ) 2 ( θ L(T l,k,θ) k l θ L(T,,θ) ) = o P () as. Proof. Let k ad T {,,k}. By applyig the Taylor expasio to the coordiates of L(T, )/ θ, ad usig the fact that L(T, θ(t))/ θ = we have 2 Card(T) θ L(T,θ ) = F(T) ( θ(t) θ ) where F(T) = 2 ( Card(T) for some θ i (T) betwee θ(t) ad θ. Hece for ay l Π,k F ( θ(t l,k ) θ) = 2 k l θ L(T l,k,θ)+ ( F F(T l,k ) )( θ(tl,k ) θ ) ad 2 L(T, θi (T)) θ θ i ) i d + 2 ( l,k,θ k l θ L(T ) θ L(T l,k,θ) ). F ( θ(t, ) θ) = 2 θ L(T,,θ)+ ( F F(T, ) )( θ(t, ) θ ) 2( +,,θ θ L(T ) θ L(T,,θ) ). Therefore, for ay l Π,k k ( (k l)f ( θ(tl,k ) θ(t, ) ) 2 ( θ L(T l,k,θ) k l = k l k k l k For k > ad with some l k Π,k, we have θ L(T,,θ) )) ( F F(Tl,k ) )( θ(tl,k ) θ) ( +2 l,k,θ k θ L(T ) θ L(T l,k,θ )) ( F F(T, ) )( θ(t, ) θ) k l ( 2,,θ k θ L(T ) θ L(T,,θ )). (6) k l l Π,k b((k l)/) k F F(T l,k ) θ(t l,k ) θ k If b(s) lk F F(T lk,k) θ(t lk,k) θ. s> Accordig to [4] ad [5], F F(T lk,k) = o P () ad θ(t lk,k) θ = O P (/ k l k ) as k l k. Hece k> Similar argumets imply that k> k l l Π,k b((k l)/) k F F(T l,k ) θ(t l,k ) θ = o P() as. (7) k l l Π,k b((k l)/) k F F(T, ) θ(t, ) θ = o P() as. (8) For k > ad for some l k Π,k, we have l Π,k b((k l)/) k θ L(T l,k,θ) θ L(T l,k,θ) Accordig to [4], If b(s) lk,k,θ k lk θ L(T ) θ L(T l k,k,θ ). s> k lk θ L(T lk,k, ) θ L(T l k,k, ) Θ = o P () as k l k. Hece k> l Π,k b((k l)/) k θ L(T l,k,θ ) θ L(T l,k,θ ) = o P() as. (9)

27 26 Similar argumets show that k> l Π,k b((k l)/) k l k Thus, Lemma 7.2 follows from (6), (7), (8), (9) ad (2).,,θ θ L(T ) θ L(T,,θ) = o P () as. (2) Lemma 7.3 Uder assumptios of Theorem 3. k> k l l Π,k b((k l)/) k F ( θ(tl,k ) θ(t, ) ) D t> <s<t W G (s) sw G () t b(s) where W G is a d-dimesioal Gaussia cetered process with covariace matrix E(W G (s)w G (τ) ) = mi(s,τ)g. Proof. We are goig to apply Lemma 7.2 for specifyig the asymptotic behaviour of θ(t l,k ) θ(t, ). For k > ad l Π,k, we have 2 k ( θ L(T l,k,θ ) k l Now we are goig to proceed i two steps. Step. Let T >. We have <k<t l Π,k b((k l)/) = <k<t l Π,k b((k l)/) = 2 k k k θ L(T,,θ )) = k θ L(T l,k,θ ) k l i=l q i (θ ) θ t {,+,,T} s { v,2 v,,t v } k l b(([t] [s])/) i= ( k i=l+ θ L(T,,θ) q i (θ ) θ ( [t] q i (θ ) θ [t] i=[s]+ k l q i (θ ) θ i= q i (θ ) ). θ [t] [s] i= q i (θ) ). θ Defie the set S := {(t,s) [,T] [,T]/ s < t}. Accordig to [4], ( q i (θ ) ) is a statioary ergodic martigale θ t Z differece sequece with covariace matrix G. By Cramér-Wold device (see [9] p. 26), it holds that [t] i=[s]+ q i (θ ) θ D(S) W G(t s). with D(S) meas the weak covergece o the Skorohod space D(S). Hece ( [t] i=[s]+ q i (θ ) θ [t] [s] i= q i (θ ) θ ) D(S) W G(t s) (t s)w G (). Therefore 2 <k<t l Π,k b((k l)/) k θ L(T l,k,θ k l ) D D <t<t <t<t <s<t <s<t θ L(T,,θ ) W G (t s) (t s)w G () t b(t s) W G (s) sw G (). (2) t b(s) Step 2. We will show that the limit distributio (as,t ) of k>t 2 l Π,k b((k l)/) k θ L(T l,k,θ ) k l θ L(T,,θ )