Detecting changes in cross-sectional dependence in multivariate time series

Transcription

1 Detectig chages i cross-sectioal depedece i multivariate time series arxiv: v4 [math.st] 2 May 204 Axel Bücher Istitut für Agewadte Mathematik Uiversität Heidelberg Im Neueheimer Feld 294, 6920 Heidelberg, Germay axel.buecher@rub.de Iva Kojadiovic, Tom Rohmer Laboratoire de mathématiques et applicatios, UMR CNRS 542 Uiversité de Pau et des Pays de l Adour B.P. 55, 6403 Pau Cedex, Frace {iva.kojadiovic,tom.rohmer}@uiv-pau.fr Joha Segers Istitut de statistique, biostatistique et scieces actuarielles Uiversité catholique de Louvai Voie du Roma Pays 20, B-348 Louvai-la-Neuve, Belgium joha.segers@uclouvai.be May 22, 204 Abstract Classical ad more recet tests for detectig distributioal chages i multivariate time series ofte lack power agaist alteratives that ivolve chages i the cross-sectioal depedece structure. To be able to detect such chages better, a test is itroduced based o a recetly studied variat of the sequetial empirical copula process. I cotrast to earlier attempts, raks are computed with respect to relevat subsamples, with beeficial cosequeces for the sesitivity of the test. For the computatio of p-values we propose a multiplier resamplig scheme that takes the serial depedece ito accout. The large-sample theory for the test statistic ad the resamplig scheme is developed. The fiite-sample performace of the procedure is assessed by Mote Carlo simulatios. Two case studies ivolvig time series of fiacial returs are preseted as well.

2 Itroductio Give a sequece X,...,X of d-dimesioal observatios, chage-poit detectio aims at testig H 0 : F such that X,...,X have c.d.f. F (.) agaistalterativesivolvigtheocostacyofthec.d.f. UderH 0 adtheassumptio that X,...,X have cotiuous margial c.d.f.s F,...,F d, we have from the work of Sklar (959) that the commo multivariate c.d.f. F ca be writte i a uique way as F(x) = C{F (x ),...,F d (x d )}, x R d, where the fuctio C : [0,] d [0,] is a copula ad ca be regarded as capturig the depedece betwee the compoets of X,...,X. It follows that H 0 ca be rewritte as H 0,m H 0,c, where H 0,m : F,...,F d such that X,...,X have margial c.d.f.s F,...,F d, (.2) H 0,c : C such that X,...,X have copula C. (.3) Classical oparametric tests for H 0 are based o sequetial empirical processes; see e.g. Bai (994), Csörgő ad Horváth (997, Sectio 2.6) ad Ioue (200). For moderate sample sizes, however, such tests appear to have little power agaist alterative hypotheses that leave the margis uchaged but that ivolve a chage i the copula, i.e., whe H 0,m ( H 0,c ) holds. Empirical evidece of the latter fact ca be foud i Holmes et al. (203, Sectio 4). For that reaso, oparametric tests for chage-poit detectio particularly sesitive to chages i the depedece structure are of practical iterest. Several tests desiged to capture chages i cross-sectioal depedece structure were proposed i the literature. Tests based o Kedall s tau were ivestigated by Gombay ad Horváth (999) (see also Gombay ad Horváth, 2002), Quessy et al. (203) ad Dehlig et al. (203). Although these have good power whe the copula chages i such a way that Kedall s tau chages as well, they are obviously useless whe the copula chages but Kedall s tau does ot chage or oly very little. Tests based o fuctioals of sequetial empirical copula processes were cosidered i Rémillard (200), Bücher ad Ruppert(203),va Kampe ad Wied(203)adWied et al.(203). However, the power of such tests is ofte disappoitig; see Sectio 5 for some umerical evidece. It is our aim to costruct a ew test for H 0 that is more powerful tha its predecessors agaist alteratives that ivolve a chage i the copula. The test is based o sequetial empirical copula processes as well, but the crucial differece lies i the computatio of the raks. Whereas i Rémillard (200) ad subsequet papers, raks are always computed with respect to the full sample, we propose to compute the raks with respect to the relevat subsamples; see Sectio 2 for details. The ituitio is that i this way, the copulas of those subsamples are estimated more accurately, so that differeces betwee copulas of disjoit subsamples are detected more quickly. The pheomeo is aki to the oe observed i Geest ad Segers (200) that the empirical copula, which is based o 2

3 pseudo-observatios, is ofte a better estimator of a copula tha the empirical distributio fuctio based o observatios from the copula itself. For aother illustratio i the cotext of tail depedece fuctios, see Bücher (203a). The paper is orgaized as follows. The test statistic is preseted i Sectio 2, ad its asymptotic distributio uder the ull hypothesis is foud i Sectio 3. Next, Sectio 4 cotais a detailed descriptio of the multiplier resamplig scheme ad its asymptotic validity uder the ull hypothesis. The results of a large-scale Mote Carlo simulatio study are reported i Sectio 5, ad two brief case studies are give i Sectio 6. Sectio 7 cocludes. Proofs ad details regardig the simulatio study are deferred to the Appedices. I the rest of the paper, the arrow deotes weak covergece i the sese of Defiitio.3.3 i va der Vaart ad Weller (2000). Give a set T, let l (T) deote the space of all bouded real-valued fuctios o T equipped with the uiform metric. 2 Test statistic We ow describe our test statistic ad highlight the differece with the oe i Rémillard (200) ad Bücher ad Ruppert (203). Let X,...,X be radom vectors. For itegers k l, let C k:l be the empirical copula of the sample X k,...,x l. Specifically, for u [0,] d, where Û k:l i = C k:l (u) = l k + (Rk:l l k + l i=k (Ûk:l i u), (2.) i,...,rk:l id ), i {k,...,l}, (2.2) with R k:l ij = l t=k (X tj X ij ) the (maximal) rak of X ij amog X kj,...,x lj. (Because of serial depedece, there ca be ties, eve if the margial distributio is cotiuous; thik for istace of a movig maximum process.) A importat poit is that the raks are computed withi the subsample X k,...,x l ad ot withi the whole sample X,...,X. As we cotiue, we adopt the covetio that C k:l = 0 if k > l. Write = {(s,t) [0,] 2 : s t}. Let λ (s,t) = ( t s )/ for (s,t). Our test statistic is based o the differece process, D, defied by D (s,u) = λ (0,s)λ (s,){c : s (u) C s +: (u)} (2.3) for (s,u) [0,] d+. For every s [0,], it gives a weighted differece betwee the empirical copulas at u of the first s ad the last s poits of the sample. Large absolute differeces poit i the directio of a chage i the copula. To aggregate over u, we cosider the Cramér vo Mises statistic S,k = {D (k/,u)} 2 dc : (u), k {,..., }. [0,] d 3

4 The test statistic for detectig chages i cross-sectioal depedece is the S = max S,k = {D (s,u)} 2 dc : (u). (2.4) k s [0,] [0,] d Other aggregatig fuctios ca be thought of too, leadig for istace to Kolmogorov Smirov ad Kuiper statistics. I umerical experimets, the resultig tests were foud to be less powerful tha the oe based o the Cramér vo Mises statistic ad hece are ot cosidered further i this paper. The ull hypothesis of a costat distributio is rejected whe S is large. The p- values are determied by the ull distributio of S, whose large-sample limit is derived i Sectio 3. To estimate the p-values from the data, a multiplier bootstrap method is proposed i Sectio 4. Fially, if H 0 is rejected, there could be oe or several abrupt or smooth chages i the joit distributio. Moreover, the chage(s) could cocer oe or more margial distributios, the copula, or both. I the case where there is just a sigle (abrupt) chage-poit k {,..., }, oe ca for istace estimate it by k = argmax S,k. (2.5) k We do ot pursue the issue of sigle or multiple chage-poit estimatio or the diagosis of the ature of the chage-poit. Our test statistic S differs from the oe cosidered i Rémillard (200, Sectio 5.2) ad Bücher ad Ruppert (203, Sectio 3.2) i the way the copulas of the subsamples X k,...,x l are estimated. Rather tha the empirical copula C k:l, these authors propose to use C k:l, (u) = l k + l i=k (Û: i u), u [0,] d, (2.6) with the covetio that C k:l, = 0 if k > l. I compariso with C k:l i (2.), the raks for the subsample X k,...,x l are computed relative to the complete sample X,...,X. The estimators C k:l, yield the differece process D R (s,u) = λ (0,s)λ (s,){c : s, (u) C s +:, (u)} (2.7) for (s,u) [0,] d+. The process D R is to be compared with the process D i (2.3). The differece lies i the use of C k:l, rather tha C k:l. From the process D R, oe obtais the test statistic { S R = D R (s,u) } 2 dc: (u), (2.8) s [0,] [0,] d which is the aalogue of S i (2.4). I the Mote Carlo simulatio experimets (Sectio 5), we will see that S is usually more powerful tha S R for detectig chages i the cross-sectioal copula. Ituitively, the reaso is that the empirical copula C k:l i (2.) is ofte a better copula estimator tha C k:l, i (2.6). Note that C k:l is ot oly the empirical copula of X k,...,x l, it 4

5 is also equal to the empirical copula of Ûk :,...,Û: l, of which C k:l, is the empirical distributio fuctio. I Geest ad Segers (200), situatios are idetified where the empirical copula of a idepedet radom sample draw from a give bivariate copula has a lower asymptotic variace tha the empirical distributio fuctio of that sample. Of course, the situatio here is differet from the oe i the cited paper: multivariate rather tha bivariate, serial depedece rather tha idepedece. But still, we suspect the same mechaisms to be active. 3 Large-sample distributio The asymptotic distributio uder H 0 of our test statistic S i (2.4) ca be obtaied by writig it as a fuctioal of the two-sided sequetial empirical copula process studied i Bücher ad Kojadiovic (203). Let X,X 2,... be a strictly statioary d-variate time series with statioary c.d.f. F havig cotiuous margis F,...,F d ad copula C. Recall C k:l i (2.) ad Ûk:l i i (2.2). The two-sided sequetial empirical copula process, C, is defied by C (s,t,u) = λ (s,t){c s +: t (u) C(u)} (3.) = t i= s + { (Û s +: t i } u) C(u), (3.2) for (s,t,u) [0,] d. The lik of C to our test statistic S i (2.4) is that, uder H 0, the differece process D i (2.3) ca be writte as for (s,u) [0,] d+. D (s,u) = λ (s,)c (0,s,u) λ (0,s)C (s,,u), (3.3) Before focusig o the weak limit of the process D uder H 0, let us briefly recall the otio of strogly mixig sequece. For a sequece of d-dimesioal radom vectors (Y i ) i Z, the σ-field geerated by (Y i ) a i b, a,b Z {,+ }, is deoted by F b a. The strog mixig coefficiets correspodig to the sequece (Y i ) i Z are defied by α r = p Z A F p,b F+ p+r P(A B) P(A)P(B) for positive iteger r. The sequece (Y i ) i Z is said to be strogly mixig if α r 0 as r. The weak limit of the two-sided empirical copula process C defied i (3.2) uder strog mixig was established i Bücher ad Kojadiovic (203) uder the followig two coditios: Coditio 3.. With probability oe, there are o ties i each of the d compoet series X j,x 2j,..., where j {,...,d}. 5

6 Coditio 3.2. For ay j {,...,d}, the partial derivatives Ċj = C/ u j exist ad are cotiuous o V j = {u [0,] d : u j (0,)}. Coditio 3. was cosidered i Bücher ad Kojadiovic (203) as cotiuity of the margial distributios is ot sufficiet to guaratee the absece of ties whe the observatios are serially depedet (see e.g. Bücher ad Segers, 203, Example 4.2). Oe of the cotributios of this work is to show that it actually ca be dispesed with. Coditio 3.2 was proposed i Segers (202) ad is orestrictive i the sese that it is ecessary for the cadidate weak limit of C to exist poitwise ad have cotiuous trajectories. As we cotiue, for ay j {,...,d}, we defie Ċj to be zero o the set {u [0,] d : u j {0,}} (see also Segers, 202; Bücher ad Volgushev, 203). Also, for ay j {,...,d} ad ay u [0,] d, u (j) is the vector of [0,] d defied by u (j) i = u j if i = j ad otherwise. The weak covergece of the process C defied i (3.2) actually follows from that of the process B (s,t,u) = t i= s + {(U i u) C(u)}, (s,t,u) [0,] d, (3.4) where U,...,U is the uobservable sample obtaied from X,...,X by the probability itegral trasforms U ij = F j (X ij ), i {,...,}, j {,...,d}, ad with the covetio that B (s,t, ) = 0 if t s = 0. If U,...,U isdrawfromastrictly statioarysequece (U i ) i Z whosestrog mixig coefficiets satisfy α r = O(r a ) with a >, we have from Bücher (203b) that B (0,, ) coverges weakly i l ([0,] d+ ) to a tight cetered Gaussia process Z C with covariace fuctio cov{z C (s,u),z C (t,v)} = mi(s,t) k Z cov{(u 0 u),(u k v)}. The latter is actually a cosequece of Lemma 2 i Bücher (203b) statig that B (0,, ) is asymptotically uiformly equicotiuous i probability, which i tur implies that Z C has cotiuous trajectories with probability oe. As a cosequece of the cotiuous mappig theorem, B B C i l ( [0,] d ), where B C (s,t,u) = Z C (t,u) Z C (s,u), (s,t,u) [0,] d. (3.5) The followig result is a cosequece of Theorem 3.4 of Bücher ad Kojadiovic (203) ad the argumets used i the proof of Lemma A.2 of Bücher ad Segers (203). Its proof is give i Appedix A. Propositio 3.3. Let X,...,X be draw from a strictly statioary sequece (X i ) i Z with cotiuous margis ad whose strog mixig coefficiets satisfy α r = O(r a ), a >. The, provided Coditio 3.2 holds, C (s,t,u) C (s,t,u) P 0, (3.6) (s,t,u) [0,] d 6

7 where C (s,t,u) = B (s,t,u) d Ċ j (u)b (s,t,u (j) ). (3.7) Cosequetly, C C C i l ( [0,] d ), where, for (s,t,u) [0,] d, C C (s,t,u) = B C (s,t,u) j= d Ċ j (u)b C (s,t,u (j) ). (3.8) I view of (3.3), the weak limit of D uder H 0 is a mere corollary of Propositio 3.3 ad the cotiuous mappig theorem. Corollary 3.4. Uder the coditios of Propositio 3.3, D D C i l ([0,] d+ ), where, for ay (s,u) [0,] d+, j= D C (s,u) = C C (0,s,u) sc C (0,,u), (3.9) with C C defied i (3.8). As a cosequece, S S = {D C (s,u)} 2 dc(u). (3.0) s [0,] [0,] d The covariace fuctio of D C ca be expressed i terms of the oe of C C by cov{d C (s,u),d C (t,v)} = {mi(s,t) st} cov{c C (0,,u),C C (0,,v)}. 4 Resamplig I order to compute p-values for S based o (3.0), we propose to use resamplig methods. Tracig back the defiitio of S via D to C i (3.) ad usig the approximatio via C i (3.7), we fid that it suffices to costruct a resamplig scheme for B defied i (3.4) ad to estimate the first-order partial derivatives, Ċ j, of C. 4. Multiplier sequeces I the case of i.i.d. observatios, Scaillet (2005) proposed to use a multiplier approach i the spirit of va der Vaart ad Weller (2000, Chapter 2.9) to resample B. Whe the first-order partial derivatives of C are estimated by fiite-differecig as i Rémillard ad Scaillet (2009), the resultig resamplig scheme for C is frequetly referred to as a multiplier bootstrap. I a osequetial settig based o idepedet observatios, Bücher ad Dette (200) compared the fiite-sample behavior of the various resamplig techiques proposed i the literature ad cocluded that the multiplier bootstrap of Rémillard ad Scaillet (2009) has, overall, the best fiite-sample properties. This techique was revisited theoretically by Segers (202) who showed its asymptotic validity uder Coditio 3.2. A sequetial geeralizatio of the latter result will be stated later i this sectio. I the case of idepedet observatios, the multiplier bootstrap is based o i.i.d. multiplier sequeces. We say that a sequece of radom variables (ξ i, ) i Z is a i.i.d. multiplier sequece if: 7

8 (M0) (ξ i, ) i Z is i.i.d., idepedet of X,...,X, with distributio ot chagig with, havig mea 0, variace, ad beig such that 0 {P( ξ 0, > x)} /2 dx <. Startig from the semial work of Bühlma(993, Sectio 3.3), Bücher ad Ruppert (203) ad Bücher ad Kojadiovic(203) have studied a depedet multiplier bootstrap for C which exteds the multiplier bootstrap of Rémillard ad Scaillet (2009) to the sequetial ad strogly mixig settig. The key idea i Bühlma (993) is to replace i.i.d. multipliers by suitably serially depedet multipliers that will capture the serial depedece i the data. I the rest of the paper, we say that a sequece of radom variables (ξ i, ) i Z is a depedet multiplier sequece if: (M) The sequece (ξ i, ) i Z is strictly statioary with E(ξ 0, ) = 0, E(ξ 2 0,) = ad E( ξ 0, ν ) < for all ν, ad is idepedet of the available sample X,...,X. (M2) There exists a sequece l of strictly positive costats such that l = o() ad the sequece (ξ i, ) i Z is l -depedet, i.e., ξ i, is idepedet of ξ i+h, for all h > l ad i N. (M3) There exists a fuctio ϕ : R [0,], symmetric aroud 0, cotiuous at 0, satisfyig ϕ(0) = ad ϕ(x) = 0 for all x > such that E(ξ 0, ξ h, ) = ϕ(h/l ) for all h Z. Ways to geerate depedet multiplier sequeces are metioed i Sectio 5 ad Appedix C. 4.2 Computig p-values via resamplig Let M be a large iteger ad let (ξ () i, ) i Z,...,(ξ (M) i, ) i Z be M idepedet copies of the same multiplier sequece. We will defie two multiplier resamplig schemes for the process B i (3.4). These will lead to two resamplig schemes for the test statistic S i (2.4), o the basis of which approximate p-values ca be computed. Recall that C k:l i (2.) is the empirical copula of X k,...,x l, which is the empirical distributio of the vectors of rescaled raks Ûk:l i i (2.2). For ay m {,...,M} ad (s,t,u) [0,] d, let ˆB (s,t,u) = t i= s + ξ i, {(Û: i u) C : (u)}, (4.) ad ˇB (s,t,u) = t i= s + = t i= s + ξ i, {(Û s +: t i u) C s +: t (u)} (ξ i, ξ s +: t )(Û s +: t i u), (4.2) 8

9 where ξ k:l is the arithmetic mea of ξ i, for i {k,...,l}. By covetio, the sums are zero if s = t. Note that the raks are computed relative to the complete sample X,...,X for ˆB (s,t, ), whereas they are computed relative to the subsample X s +,...,X t for ˇB (s,t, ). I order to get to resamplig versios of C i (3.7), we eed estimators of the firstorder partial derivatives of C. A simple estimator based o X k,...,x l cosists of fiite differecig at a badwidth of h h(k,l) = mi{(l k +) /2,/2}. Varyig slightly upo the defiitio i Rémillard ad Scaillet (2009) ad followig Kojadiovic et al. (20a, Sectio 3), we put Ċ j,k:l (u) = C k:l(u+he j ) C k:l (u he j ) mi(u j +h,) max(u j h,0) foru [0,] d, where e j isthejthcaoical uit vector ir d. Notethat ifh u j h, the deomiator is just 2h. The more geeral form of the deomiator corrects for boudary effects (u j close to 0 or ). Proceedig for istace as i Kojadiovic et al. (20a, proof of Propositio 2), we fid that the previous estimator is uiformly bouded. ˆB ˇB The resamplig versios ad versios for C : for (s,t,u) [0,] d, d Ĉ (s,t,u) = ˆB (s,t,u) Ċ j,: (u) j= d Č (s,t,u) = ˇB (s,t,u) Ċ j, s +: t (u) j= of B the lead to the followig resamplig ˆB (s,t,u(j) ), ˇB (s,t,u(j) ). (4.3) Recall that λ (s,t) = ( t s )/. The differece process D is to be resampled by oe of the followig two methods: ˆD Ď (s,u) = λ(s,)ĉ (0,s,u) λ(0,s)ĉ (s,,u) = Ĉ (0,s,u) λ (0,s)Ĉ (0,,u), (s,u) = λ(s,)č (0,s,u) λ(0,s)č (s,,u). For resamplig the test statistic, oe has the choice betwee Ŝ = {ˆD (s,u)}2 dc : (u), (4.4) s [0,] [0,] d Š = {Ď (s,u)}2 dc : (u). (4.5) s [0,] [0,] d Fially, approximate p-values of the observed test statistic S ca be computed via either M M m= (Ŝ ) S or 9 M M m= ( Š S ). (4.6)

10 The ull hypothesis is rejected if the estimated p-value is smaller tha the desired sigificace level. By compariso, ote that for the test statistic S R i (2.8) based o the process DR i (2.7), a approximate p-value ca be computed usig the multiplier processes ˆB D R, (s,u) = ˆB (0,s,u) λ (0,s) ˆB (0,,u), (4.7) where is defied i(4.); see also Rémillard(200, Sectio 5.2) ad Bücher ad Ruppert (203, Sectio 3.2). 4.3 Asymptotic validity of the resamplig scheme We establish the asymptotic validity of the multiplier resamplig schemes described above uder the ull hypothesis. First, we eed to impose coditios o the data geeratig process X,...,X ad the multiplier sequeces (ξ i, ) i Z for m {,...,M}. Coditio 4.. Oe of the followig two coditios holds: (i) The radom vectors X,...,X are i.i.d. ad (ξ () i, ) i Z,...,(ξ (M) i, ) i Z are idepedet copies of a multiplier sequece satisfyig (M0). (ii) The radom vectors X,...,X are draw from a strictly statioary sequece (X i ) i Z whose strog mixig coefficiets satisfy α r = O(r a ) for some a > 3+3d/2, ad (ξ () i, ) i Z,...,(ξ (M) i, ) i Z are idepedet copies of a depedet multiplier sequece satisfyig (M) (M3) with l = O( /2 γ ) for some 0 < γ < /2. I both cases, the statioary distributio of X i has cotiuous margis ad a copula C satisfyig Coditio 3.2. If the radom vectors X,...,X are i.i.d., they ca also be cosidered to be draw from a strogly mixig, strictly statioary sequece. Hece, for the multiplier sequeces (ξ i, ) i Z, oe could either assume (M0) or (M) (M3): both should work. However, as discussed i Bücher ad Kojadiovic (203, Sectio 2), the use of depedet multipliers i the case of idepedet observatios is likely to result i a efficiecy loss. This is illustrated i the Mote Carlo simulatios reported i Bücher ad Ruppert (203, Sectio 3) ad carried out for the test based o the statistic S R defied i (2.8) which is resampled usig multiplier processes asymptotically equivalet to those give i (4.7): the use of depedet multipliers i the case of serially idepedet data usually results i a loss of power ad i a slightly more coservative test. Thus, i fiite samples, if there is o evidece agaist serial idepedece, it appears more sesible to work uder (M0). We ca ow state the asymptotic distributios of the multiplier resamplig schemes uder the ull hypothesis of a costat distributio. We provide two propositios, oe for the resamplig scheme based o ˆB i (4.) ad aother oe for the scheme based o ˇB i (4.2). 0

11 Propositio 4.2. If Coditio 4. holds, the ( ) ( ) C,Ĉ(),...,Ĉ(M) C C,C () C,...,C(M) C i {l ( [0,] d )} M+, where C C is defied i (3.8), ad C () C,...,C(M) C copies of C C. As a cosequece, also ( ) ( ) () (M) D, ˆD,..., ˆD D C,D () C,...,D(M) C, i {l ([0,] (d+) )} M+, where D C is defied i (3.9) ad D () C,...,D(M) C copies of D C. Fially, ( ) S,Ŝ(),...,Ŝ(M) ( S,S (),...,S (M)) are idepedet are idepedet where S is defied i (3.0) ad S (),...,S (M) are idepedet copies of S. Uder Coditio 4.(i), the above result ca be easily proved by startig from Theorem of Holmes et al. (203) ad adaptig the argumets used i Segers (202, proof of Propositio 3.2). Uder Coditios 4.(ii) ad 3., the result was obtaied i Bücher ad Kojadiovic (203, Propositio 4.2). The additioal argumets allowig to avoid Coditio 3. will be give i the proof of the ext result. Propositio 4.3. If Coditio 4. holds, the the coclusios of Propositio 4.2 also hold with Ĉ replaced by Č, ˆD replaced by Ď, ad Ŝ replaced by Š. TheproofofPropositio4.3issomewhativolvedadisgiveidetailiAppedixB. Combiig the last claims of Propositios 4.2 ad 4.3 with Propositio F. i Bücher ad Kojadiovic (203), we obtai that a test based o S whose p-value is computed usig oe of the two approaches i (4.6) will hold its level asymptotically as followed by M. 5 Simulatio study Large-scale Mote Carlo experimets were carried out i order to study the fiite-sample performace of the derived tests for detectig chages i cross-sectioal depedece. The mai questios addressed by the study are the followig: (i) How well do the tests hold their size uder the ull hypothesis H 0 i (.) of o chage? (ii) Whatisthepower ofthetests agaistthealterativeh,c ofasiglechageicrosssectioal depedece at costat margis? Specifically, the alterative hypothesis is H,c H 0,m with H 0,m i (.2) ad H,c defied by H,c : distict C ad C 2, ad k {,..., } such that X,...,X k have copula C ad X k +,...,X have copula C 2. (5.)

12 (iii) What happes if the chage i distributio is oly due to a chage i the margis, thecopula remaiig costat? Specifically, thealterative hypothesis ish,m H 0,c with H 0,c give i (.3) ad H,m defied by H,m : distict F,,F,2 as well as F 2,...,F d ad k {,..., } such that X,...,X k have margial c.d.f.s F,,F 2,...,F d ad X k +,...,X have margial c.d.f.s F,2,F 2,...,F d. (5.2) I additio to the three questios above, may others ca be formulated, ivolvig other alterative hypotheses for istace. The problem is complex ad there are coutless ways of combiig factors i the experimetal desig. I our study, the settigs were chose to represet a wide ad hopefully represetative variety of situatios, i fuctio of the three questios above. The mai factors of our experimets are summarized below: Test statistics: Our statistic S i (2.4) with p-values computed via resamplig usig Ŝ or Š i (4.4) ad (4.5), respectively. As we cotiue, we shall simply talk about the test based o Ŝ or Š, respectively, to distiguish betwee these two situatios. The statistic S R i (2.8) of Bücher ad Ruppert (203), with p-values computed accordig to the resamplig method for D R i (4.7). Sample size: {50,00,200}. Number of samples per settig: 000. Cross-sectioal dimesio: d {2, 3}. Sigificace level: α = 5%. Serial depedece: The data were geerated either as beig serially idepedet or via two time-series models, a autoregressive process ad a multivariate versio of the expoetial autoregressive model cosidered i Auestad ad Tjøstheim (990) ad Paparoditis ad Politis (200, Sectio 3.3). Idepedet stadard ormals were used as multipliers for idepedet observatios, while for the serially depedet datasets, the depedet multiplier sequeces were geerated from iitial idepedet stadard ormal sequeces usig the movig average approach proposed iitially i Bühlma(993) ad revisited i some detail i Bücher ad Kojadiovic (203, Sectio 6.). The value of the badwidth parameter l defied i Coditio(M2) was chose automatically usig the approach described i Bücher ad Kojadiovic (203, Sectio 5). See Appedix D for details. Margis: i all but oe settig, the margis were kept costat, i.e., H 0,m i (.2) was assumed. I oe case (see Table 5), a break as i H,m i (5.2) was assumed, the margial distributio of the first compoet chagig from the N(0, ) to the N(µ, ) distributio. Copulas: Clayto, Gumbel Hougaard, Normal, Frak, with positive or egative(isofar possible) associatio, as well as asymmetric versios obtaied via Khoudraji s device (Khoudraji, 995; Geest et al., 998; Liebscher, 2008). Alterative hypotheses ivolvig a sigle chage-poit occurrig at time k = t with t {0.,0.25,0.5,0.75}: H 0,m H,c with a chage of the parameter withi a copula family. H 0,m H,c with a chage of the copula family at costat Kedall s tau. 2

13 H,m H 0,c, i.e., a chage of oe of the margis rather tha of the copula. For the serially depedet case, a chage i the copula of the iovatios, leadig to a gradual chage of the copula of the margial distributios of the observables. The experimets were carried out i the R statistical system(r Developmet Core Team, 203) usig the copula package(hofert et al., 203). To allow us to reuse previously writte code, the rescaled raks i (2.2) were computed by dividig the raks by l k + 2 istead of l k+. Because (4.4) oly ivolves rescaled raks computed from the etire sample, the test based o Ŝ ca be implemeted to be substatially faster tha the oe based o Š for larger sample sizes. The correspodig routies are available i the R package pcp (Kojadiovic, 204). For the sake of brevity, oly a represetative subset of the results is reported here. Specifically, the followig tables are provided i Appedix D: Size of the tests uder the ull hypothesis H 0 : Table : Percetage of false rejectios whe data are serially idepedet. Table 2: Percetage of false rejectios whe data are serially depedet. Power of the tests agaist specific alteratives: Table 3: Power agaist H 0,m H,c ivolvig a chage of the copula parameter withi a copula family ad at serial idepedece. Table 4: Power agaist H 0,m H,c ivolvig a chage of copula family at a costat value of Kedall s tau ad at serial idepedece. Table 5: Power agaist H,m H 0,c ivolvig a chage i oe of the margis ad at serial idepedece.. Table 6: Power agaist H 0 whe data are serially depedet ad the chage occurs i the copula of the iovatios. Besides fidigs of a more aecdotical ature, the followig coclusios may be draw from the results: All tests hold their level reasoably well i the case of serial idepedece (Table ), with mior fluctuatios depedig o sample size, test statistic, copula parameter ad copula family. I case of serial depedece, the test based o Ŝ is too coservative for the sample sizes uder cosideratio (Table 2). I lie with this observatio, the test based o Š appears to be more powerful tha the oe based o Ŝ (Table 6). For alterative hypotheses ivolvig a chage i the copula, the tests based o Ŝ ad Š have a higher power tha S R (Tables 3 ad 4). Whe the copula chages i such a way that Kedall s tau remais costat, the power of S R is especially low. With respect to that last settig, ote that distiguishig copulas o the basis of lowamouts ofdataiskow to bedifficult (Geest et al.,2009; Kojadiovic et al., 20b). The fact that the chage-poit is ukow makes the problem eve harder. 3

14 For alterative hypotheses ivolvig a chage i oe of the margis, it is the test statistic S R that is substatially more powerful tha S (Table 5). The weak power of S ca be explaied by the fact that it is desiged for detectig chages i the copula. Aother tetative readig of the results is that the test based o S, regarded as a procedure for testig H 0,c, is relatively robust agaist small chages i oe margi. I cotrast, the test based o S R behaves as a all-purpose test for the hypothesis of a costat distributio rather tha as a test for a costat copula. 6 Case studies As a illustratio, we first applied the test based o Š to bivariate fiacial data cosistig of daily logreturs computed from the DAX ad the Stadard ad Poor 500 idices. Followig Dehlig et al. (203, Sectio 7), attetio was restricted to the years The correspodig closig quotes were obtaied from usig the get.hist.quote fuctio of the tseries R package (Trapletti ad Horik, 203), which resulted i = 993 bivariate logreturs. Depedet multiplier sequeces were geerated as explaied i Appedix C. A approximate p-value of 0.04 was obtaied, providig some evidece agaist H 0. The coclusio is i lie with the results reported i Dehlig et al. (203). Of course, as discussed earlier, it is oly uder the assumptio that H 0,m i (.2) holds that it would be fully justified to decide to reject H 0,c i (5.) o the basis of the previous approximate p-value. The value of the chage- poit estimator k i (2.5) is 529, correspodig to February 22d, As a secod illustratio, we followed agai Dehlig et al. (203) ad cosidered = 504 bivariate logreturs computed from closig daily quotes of the Dow Joes Idustrial Average ad the Nasdaq Composite for the years 987 ad 988. The former quotes, ot beig available o aymore, were take from the R package QRM (Pfaff ad McNeil, 203). This two-year period is of iterest because it cotais October 9th, 987, kow as black Moday (see Dehlig et al., 203, Figure 4). A approximate p-value of 0.59 was obtaied. Hece, despite the extreme evets that occurred durig the period uder cosideratio, the test based o Š detects o evidece agaist H 0 i the data, which is i lie with the results reported i Dehlig et al. (203). 7 Coclusio We have demostrated that the sesitivity of rak-based tests for the ull hypothesis of a costat distributio agaist chages i cross-sectioal depedece ca be improved if raks are computed with respect to relevat subsamples. I this way, the test we propose achieves i may cases a higher power tha the oe proposed i Bücher ad Ruppert (203). The limit distributio of the test statistic uder the ull hypothesis is uwieldy, but approximate p-values ca still be computed via a multiplier resamplig scheme. To deal with potetial serial depedece, we make use of depedet multiplier sequeces, a idea goig back to Bühlma (993) ad revisited i Bücher ad Kojadiovic (203). 4

15 Here are some potetial aveues for further research: Oce the ull hypothesis has bee rejected, the ature of the ostatioary eeds to be ivestigated further: is there a sigle chage-poit or is there more tha oe? Or maybe the chage is gradual rather tha sudde? Ad does the chage cocer the margis or the copula? Ca oe detect a chage i the copula without the hypothesis that the margis are costat? The procedure is computatioally itesive because the raks have to be recomputed for every k {,..., }. Efficiet algorithms for reutilizig calculatios from oe value of k to the ext oe might speed up the computatios. Ackowledgmets The authors are grateful to Mark Holmes, Jea-Fraçois Quessy ad Marti Ruppert for fruitful discussios, ad to a aoymous referee for poitig out that Coditio 3. might be dispesed with. The research by A. Bücher has bee ported i parts by the Collaborative Research Ceter Statistical modelig of oliear dyamic processes (SFB 823, Project A7) of the Germa Research Foudatio (DFG), which is gratefully ackowledged. J. Segers gratefully ackowledges fudig by cotract Projet d Actios de Recherche Cocertées No. 2/7-045 of the Commuauté fraçaise de Belgique ad by IAP research etwork Grat P7/06 of the Belgia govermet (Belgia Sciece Policy). Refereces B. Auestad ad D. Tjøstheim. Idetificatio of oliear time series: First order characterizatio ad order determiatio. Biometrika, 77: , 990. J. Bai. Weak covergece of the sequetial empirical processes of residuals i ARMA models. The Aals of Statistics, 22(4): , 994. A. Bücher. A ote o oparametric estimatio of bivariate tail depedece. Statistics ad Risk Modelig, page i press, 203a. URL A. Bücher. A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig. Joural of Theoretical Probability, page i press, 203b. A. Bücher ad H. Dette. A ote o bootstrap approximatios for the empirical copula process. Statistics ad Probability Letters, 80(23 24): , 200. A. Bücher ad I. Kojadiovic. A depedet multiplier bootstrap for the sequetial empirical copula process uder strog mixig. arxiv: ,

16 A. Bücher ad M. Ruppert. Cosistet testig for a costat copula uder strog mixig based o the tapered block multiplier techique. Joural of Multivariate Aalysis, 6: , 203. A. Bücher ad J. Segers. Extreme value copula estimatio based o block maxima of a multivariate statioary time series. arxiv:3.3060, 203. A. Bücher ad S. Volgushev. Empirical ad sequetial empirical copula processes uder serial depedece. Joural of Multivariate Aalysis, 9:6 70, 203. P. Bühlma. The blockwise bootstrap i time series ad empirical processes. PhD thesis, ETH Zürich, 993. Diss. ETH No M. Csörgő ad L. Horváth. Limit theorems i chage-poit aalysis. Wiley Series i Probability ad Statistics. Joh Wiley & Sos, Chichester, UK, 997. H. Dehlig, D. Vogel, M. Wedler, ad D. Wied. A efficiet ad robust test for a chage-poit i correlatio. arxiv: , 203. C. Geest ad J. Segers. O the covariace of the asymptotic empirical copula process. Joural of Multivariate Aalysis, 0: , 200. C. Geest, K. Ghoudi, ad L.-P. Rivest. Discussio of Uderstadig relatioships usig copulas, by E. Frees ad E. Valdez. North America Actuarial Joural, 3:43 49, 998. C. Geest, B. Rémillard, ad D. Beaudoi. Goodess-of-fit tests for copulas: A review ad a power study. Isurace: Mathematics ad Ecoomics, 44:99 23, E. Gombay ad L. Horváth. Chage-poits ad bootstrap. Evirometrics, 0(6), 999. E. Gombay ad L. Horváth. Rates of covergece for U-statistic processes ad their bootstrapped versios. Joural of Statistical Plaig ad Iferece, 02: , M. Hofert, I. Kojadiovic, M. Mächler, ad J. Ya. copula: Multivariate depedece with copulas, 203. URL R package versio M. Holmes, I. Kojadiovic, ad J-F. Quessy. Noparametric tests for chage-poit detectio à la Gombay ad Horváth. Joural of Multivariate Aalysis, 5:6 32, 203. A. Ioue. Testig for distributioal chage i time series. Ecoometric Theory, 7(): 56 87, 200. A. Khoudraji. Cotributios à l étude des copules et à la modélisatio des valeurs extrêmes bivariées. PhD thesis, Uiversité Laval, Québec, Caada, 995. I. Kojadiovic. pcp: Some oparametric tests for chage-poit detectio i (multivariate) observatios, 204. URL R package versio

17 I. Kojadiovic, J. Segers, ad J. Ya. Large-sample tests of extreme-value depedece for multivariate copulas. The Caadia Joural of Statistics, 39(4): , 20a. I. Kojadiovic, J. Ya, ad M. Holmes. Fast large-sample goodess-of-fit for copulas. Statistica Siica, 2(2):84 87, 20b. E. Liebscher. Costructio of asymmetric multivariate copulas. Joural of Multivariate Aalysis, 99: , E. Paparoditis ad D.N. Politis. Tapered block bootstrap. Biometrika, 88(4):05 9, 200. B. Pfaff ad A. McNeil. QRM: Provides R-laguage Code to Examie Quatitative Risk Maagemet Cocepts, 203. URL R package versio J.-F. Quessy, M. Saïd, ad A.-C. Favre. Multivariate Kedall s tau for chage-poit detectio i copulas. The Caadia Joural of Statistics, 4:65 82, 203. R Developmet Core Team. R: A Laguage ad Eviromet for Statistical Computig. R Foudatio for Statistical Computig, Viea, Austria, 203. URL ISBN B. Rémillard. Goodess-of-fit tests for copulas of multivariate time series. Social Sciece Research Network, : 32, 200. URL B. Rémillard ad O. Scaillet. Testig for equality betwee two copulas. Joural of Multivariate Aalysis, 00(3): , J.P. Romao ad M. Wolf. A more geeral cetral limit theorem for m-depedet radom variables with ubouded m. Statistics ad Probability Letters, 47:5 24, O. Scaillet. A Kolmogorov-Smirov type test for positive quadrat depedece. Caadia Joural of Statistics, 33:45 427, J. Segers. Asymptotics of empirical copula processes uder orestrictive smoothess assumptios. Beroulli, 8: , 202. A. Sklar. Foctios de répartitio à dimesios et leurs marges. Publicatios de l Istitut de Statistique de l Uiversité de Paris, 8:229 23, 959. A. Trapletti ad K. Horik. tseries: Time series aalysis ad computatioal fiace, 203. URL R package versio A.W. va der Vaart ad J.A. Weller. Weak covergece ad empirical processes. Spriger, New York, Secod editio. M. va Kampe ad D. Wied. A oparametric costacy test for copulas uder mixig coditios. workig paper, 203. URL 7

18 D. Wied, H. Dehlig, M. va Kampe, ad D. Vogel. A fluctuatio test for costat Spearma s rho with uisace-free limit distributio. Computatioal Statistics ad Data Aalysis, page i press, 203. A Proof of Propositio 3.3 Let us first itroduce additioal otatio. For itegers k l, let H k:l deote the empirical c.d.f. of the uobservable sample U k,...,u l ad let H k:l,j, for j {,...,d}, deote its margis. The empirical quatile fuctios are H k:l,j (u) = if{v [0,] : H k:l,j(v) u}, u [0,], which are collected i a vector via H k:l (u) = ( H k:l, (u ),...,H k:l,d (u d) ), u [0,] d. By covetio, the previously defied quatities are all take equal to zero if k > l. From the proof of Theorem 3.4 i Bücher ad Kojadiovic (203), we have that (3.6) holds with C replaced by C alt, where C alt (s,t,u) = t i= s + To show (3.6), it remais therefore to prove that [{U i H s +: t (u)} C(u)], (s,t,u) [0,]d. C alt (s,t,u) C (s,t,u) P 0. (s,t,u) [0,] d (A.) To do so, we adapt the argumets used i Lemma A.2 of Bücher ad Segers (203). Fix k l ad u [0,] d. For i {k,...,l}, the d compoets of Ûi k:l defied i (2.2) ca be expressed as Ûk:l ij = H k:l,j (U ij ), j {,...,d}. Next, otice that { U ij H k:l,j (u j) } ( ) { Ûij k:l u j = Uij < H k:l,j (u j) } + { U ij = H k:l,j (u j) } ( ) Ûij k:l (Ûk:l ) < u j ij = u j = { U ij = H k:l,j (u j) } ( Û k:l ij = u j ), as x < H (u) if ad oly H(x) < u for ay distributio fuctio H. Sice Ûk:l H k:l,j (U ij ) = u j implies U ij = H k:l,j (u j), we obtai that 0 { U ij H k:l,j (u j) } ( Û k:l ij u j ) { Uij = H k:l,j (u j) }. Combiig the previous iequality with the decompositio {U i H k:l (u)} (Ûk:l i u) [ d = {U ij H k:l,j (u j)} p= j p j p p<j d (Ûk:l ij u j ) {U ij H k:l,j (u j)} p <j d (Ûk:l ij u j ) ij = ], 8

19 we obtai that 0 {U i H k:l (u)} (Ûk:l i u) d j= { U ij = H k:l,j (u j) }. (A.2) It follows that the remum i (A.) is smaller tha d j= (s,t,u) [0,] t i= s + { U ij = H s +: t,j (u)} d j= u [0,] (U ij = u). Usig the fact that ( U ij = u ) ( U ij u ) ( U ij u / ), the latter is smaller d B (0,,u) B (0,,v) +d u,v [0,] d u v where B is defied i (3.4). Usig the asymptotic uiform equicotiuity i probability of B established i Lemma 2 of Bücher (203b), we fially obtai (A.), which completes the proof. /2, i= B Proof of Propositio 4.3 We shall oly prove the result i the case of strogly mixig observatios, that is, whe Coditio 4.(ii) is assumed. The proof is similar but simpler whe Coditio 4.(i) is assumed istead. Š It is sufficiet to show the statemet ivolvig Č. The statemets for Ď the follow from the cotiuous mappig theorem. For ay m {,...,M} ad (s,t,u) [0,] d, put ad B (s,t,u) = t C (s,t,u) = B i= s + (s,t,u) d ξ i, {(U i u) C(u)}, j= Ċ j (u)b (s,t,u(j) ). (B.) [Recall that u (j) = (,...,,u j,,...,) [0,] d, with u j appearig at the j-th coordiate.] From Theorem 2. i Bücher ad Kojadiovic (203), we have that ( ) ( B,B (),...,B(M) BC,B () ) C,...,B(M) C i {l ( [0,] d )} M+, where B () C,...,B(M) C are idepedet copies of B C i (3.5), ad thus, from the cotiuous mappig theorem ad (3.6), we fid that ( ) ( C,C (),...,C(M) CC,C () ) C,...,C(M) C 9

20 i {l ( [0,] d )} M+. It is therefore sufficiet to show that ( Č C )(s,t,u) P 0 (s,t,u) [0,] d (B.2) for every m {,...,M}. Below, we will show the followig two assertios: first, (ˇB B )(s,t,u) P 0, (s,t,u) [0,] d ad secod, for every δ (0,/2) ad every ε (0,), u [0,] d δ u j δ (s,t) [0,] 2 t s ε Ċj, s +: t (u) Ċj(u) P 0. (B.3) (B.4) I view of the structure of Č we show ext. Clearly, ( Č C )(s,t,u) (ˇB B i (4.3), the assertios (B.3) ad (B.4) imply (B.2), as d )(s,t,u) + Ċj, s +: t (u) (ˇB + j= B )(s,t,u(j) ) d Ċj, s +: t (u) Ċj(u) B (s,t,u(j) ). (B.5) j= Takig rema over (s,t,u) [0,] d, thefirst adthe secod termothe right-had side of (B.5) coverge to zero i probability because of assertio(b.3) ad uiform boudedess of Ċ j,k:l (see Kojadiovic et al., 20a, proof of Propositio 2). The third term o the right-had side of (B.5) coverges to zero i probability because of assertio (B.4) ad the fact that (s,t,u) B (s,t,u (j) ) vaishes as soo as s = t or u j {0,}, ad is asymptotically uiformly equicotiuous i probability as a cosequece of Lemma A.3 i Bücher ad Kojadiovic (203). It remais to show (B.3) ad (B.4). The proof of the latter assertio is simplest ad is give first. Proof of (B.4). Observe that C s +: t (u) = C(u)+ λ (s,t) C (s,t,u). Fix δ (0,/2) ad ε (0,). Without loss of geerality, assume that is large eough so that the badwidth h = h (s,t) = / t s is less tha δ wheever t s ε. The, for t s ε ad u [0,] d with δ u j δ, we have Ċ j, s +: t (u) = 2h {C(u+he j) C(u he j )} + 2h λ (s,t) {C (s,t,u+he j ) C (s,t,u he j )}. 20

21 By the assumptio of existece ad cotiuity of Ċ j o V j (see Coditio 3.2), ad sice 0 Ċj, it follows from the mea-value theorem that 2h {C(u+he j) C(u he j )} Ċj(u) 0, h 0. u [0,] d δ u j δ Usig (3.6)adthefactthatB isasymptoticallyuiformlyequicotiuous iprobability, it ca be verified that C is asymptotically uiformly equicotiuous i probability as well. It follows that u [0,] d δ u j δ (s,t) [0,] 2 t s ε C (s,t,u+he j ) C (s,t,u he j ) P 0. Fially, 2h λ (s,t) = 2 λ (s,t) 2 ε /. Combie the four previous displays to arrive at the desired coclusio. The proof of (B.3) is more complicated. Usig the otatio itroduced i Appedix A, let us defie the auto-cetered versio of the process B i (B.) as B (s,t,u) = t i= s + = t i= s + ξ i, {(U i u) H s +: t (u)} (ξ i, with the usual covetio that empty sums are zero. ξ s +: t )(U i u), Proof of (B.3). Cosider the decompositio (ˇB B )(s,t,u) ˇB (s,t,u) B (s,t,h s +: t (u)) + B (s,t,h s +: t (u)) B (s,t,h s +: t (u)) + B (s,t,h s +: t (u)) B (s,t,u). B Write out the defiitios of the processes B, ad respectively, ad take rema over (s,t,u) [0,] d to obtai (ˇB B )(s,t,u) (s,t,u) [0,] d ˇB (s,t,u) [0,] d + (s,t,u) [0,] d (B.6) ˇB i (B.), (B.6) ad (4.2), (s,t,u) B (s,t,h s +: t (u)) t Each term requires a differet treatmet. i= s + ξ i, H s +: t (u) C(u) + (s,t,u) [0,] d B (s,t,h s +: t (u)) B (s,t,u). 2 (B.7) (B.8) (B.9)

22 - The term (B.8): Let δ (0,/2), to be specified later. We split the remum ito two parts, accordig to whether t s is smaller or larger tha a = /2 δ : A, = A,2 = (s,t,u) [0,] d t s a (s,t,u) [0,] d t s a t i= s + t i= s + ξ i, ξ i, H s +: t (u) C(u), H s +: t (u) C(u). We will show that both A, ad A,2 coverge to zero i probability. (a) Sice both H k:l ad C take values i [0,], a crude boud for A, is A, (a +) max i ξ i, 2 δ max i ξ i,. Usig the fact that, from (M), for ay ν, E[ ξ, ν ] <, we have that, for every α > 0 ad ν such that ν > /α, ( ) P max i ξ i, α P ( ξ, α) να E[ ξ, ν ] 0. Apply the previous display with α (0,δ) to fid that A, coverges to zero i probability. (b) Recall B i (3.4). Observe that We have A,2 = B (s,t,u) = t s (s,t,u) [0,] d t s a s < t t + s a max k l l k a t s l k + k l l k b t s l i=k ξ i, {H s +: t (u) C(u)}. t i= s + t i= s + ξ i, ξ i, B (s,t,u) B (s,t,u). (s,t,u) [0,] d B (s,t,u) (s,t,u) [0,] d By weak covergece B B C i l ( [0,] d ), the remum at the ed of the previous display is bouded i probability. Writig b = a, it is sufficiet to show that max l ξ i, l k + P 0,. i=k 22

23 Fix η > 0. The probability that the previous maximum exceeds η is bouded by P k l l k b [ l k + l i=k ξ i, ]. > η (B.0) Fix ν 2, to be specified later. By statioarity ad Markov s iequality, the previous expressio is bouded by η k l l k b [ l k+ ν (l k +) ν E i= ξ i, ν] η ν b r [ r r ν E i= ξ i, ν]. Recall that the sequece (ξ i, ) i Z is l -depedet from (M2) ad assume that is sufficietly largesothat 2l +b. The, bycorollarya.iromao ad Wolf (2000), there exists a costat C ν, depedig oly o ν, such that [ r E i= ξ i, ν] C ν ν (4l r) ν/2 E[ ξ, ν ]. Usig the fact that, from (M), E[ ξ, ν ] <, ad up to a multiplicative costat, the expressio i (B.0) is bouded by b r r ν (l r) ν/2 2 b ν/2 l ν/2 = O ( 2 (/2 δ)ν/2+(/2 γ)ν/2) = O ( 2+(δ γ)ν/2). The right-had side coverges to zero if we choose δ = γ/2 ad the ν > 8/γ. 2- The term (B.9): We have to show that, for every η,λ > 0, [ P B {s,t,h s +: t (u)} B (s,t,u) ] > λ η, (s,t,u) [0,] d for all sufficietly large. Fix η,λ > 0. Sice (B.9) is smaller tha 2 (s,t,u) [0,] d B (s,t,u), ad usig the fact that B vaishes o the diagoal s = t ad is asymptotically uiformly equicotiuous i probability, there exists ε (0, ) such that, for all sufficietly large, P (s,t,u) [0,] d t s<ε B {s,t,h s +: t (u)} B (s,t,u) > λ η/2. 23

24 Settig ζ = (s,t,u) [0,] d H s +: t (u) u, we shall ow show that, for all t s ε sufficietly large, A = P (s,t,u,v) [0,] 2d t s ε, u v ζ (s,t,u) B (s,t,v) > λ η/2, B which will complete the proof. Usig agai the asymptotic uiformly equicotiuity i probability of B, there exists µ (0,) such that, for all sufficietly large, A, = P B (s,t,u) B (s,t,v) > λ η/4. (s,t,u,v) [0,] 2d t s ε, u v µ WetheboudA bya, +A,2, wherea,2 = P(ζ > µ). Fromtheweakcovergece of B to B C i l ( [0,] d ), we have that (s,t,u) [0,] d t s ε H s +: t (u) C(u) (s,t,u) [0,] d B (s,t,u) /2 (s,t) t s ε {λ (s,t)} P 0. Usig the fact that u [0,] H s +: t,j (u) u = u [0,] H s +: t,j (u) u for j {,...,d} (for istace, by symmetry argumets o the graphs of H s +: t,j ad H s +: t,j ), we immediately obtai that ζ P 0, which implies that, for all sufficietly large, A,2 η/4, ad thus that, for all sufficietly large, A η/2. 3- The term (B.7): For the followig argumets, it is sufficiet to assume that the sequece (ξ i, ) i Z appearig i ˇB ad B satisfies oly (M) with E[{ξ 0, } 2 ] > 0 ot ecessarily equal to oe. Let K > 0beacostat adlet usfirst pose that, foray adi {,...,}, ξ i, K. With (A.2) i mid, the term (B.7) is smaller tha A, +A,2, where A, = ad t (s,t,u) [0,] d i= s + K + ξ A,2 = (s,t,u) [0,] d (ξ s +: t [ i, +K) {U i H t i= s + ] i u) s +: t (u)} (Û s +: t [ ] {U i H s +: t (u)} (Û s +: t i u). Let us first show that A, P 0. Pluggig (A.2) ito the expressio of A,, we boud A, by A,, + +A,,d, where A,,j = (s,t,u) [0,] t i= s + 24 (ξ i, +K)( U ij = u ).

25 To prove that A P, 0, we shall ow show that A P,,j 0 for all j {,...,d}. Fix j {,...,d}. Usig the fact that (U ij = u) (U ij u) (U ij u /), we obtai that A,,j is smaller tha A,,j +A,,j +A A,,j = A,,j = K A,,j = (s,t,u) [0,] (s,t,u) [0,] (s,t,u) [0,] t i= s + t ξ i= s + s +: t (ξ i,,,j, where ξ s +: t ){ ( U ij u ) ( U ij u / )}, { ( Uij u ) ( U ij u / )}, t i= s + { ( Uij u ) ( U ij u / )}. From Lemma A.3 of Bücher ad Kojadiovic (203), we kow that B is asymptotically uiformly equicotiuous i probability uder the weaker coditios o the sequece (ξ i, ) i Z cosidered above. The treatmet of the term (B.8) carried out previously remais valid uder these coditios ad esures that (s,t,u) [0,] d B (s,t,u) B (s,t,u) P 0, B which implies that is asymptotically uiformly equicotiuous i probability as well uder the same weaker coditios o the sequece (ξ i, ) i Z. The latter immediately implies that A P,,j 0. For A,,j, we have A,,j K /2 B (0,,u) B (0,,v) +K P 0, u,v [0,] d u v by asymptotic uiform equicotiuity i probability of B. The fact that A,,j P 0 ca be show by proceedig as for the term (B.8). Hece, we have that A,,j P 0, which implies that A, P 0. The fact that A P,2 0, follows from (A.2) which implies that A,2 is smaller tha d j= (A,,j +A,,j ). This completes the proof uder the coditio ξ i, K. To show that this coditio is ot ecessary, we proceed as at the ed of the proof of Lemma A.3 of Bücher ad Kojadiovic (203). Let Z + i, = max(ξ i,,0), Z i, = max( ξ i,,0), K+ = E(Z 0,) + ad K = E(Z0,). Furthermore, defie ξ,+ i, = Z + i, K + ad ξ, i, = Z i, K. The, usig the fact that K + K = 0, we ca write Let B,+ ξ i, = Z+ i, Z i, = Z+ i, K+ (Z i, K ) = ξ,+ i, ξ, i,. ad B, be the aalogues of B ad (ξ, i, ) i Z, respectively, ad similarly for 25 defied from the sequeces (ξ,+ i, ) i Z ad. The case treated B,+ B,

26 above yields ˇB,+ (s,t,u) [0,] d ˇB, (s,t,u) [0,] d (s,t,u) B,+ (s,t,h s +: t (u)) P 0, (s,t,u) B, (s,t,h s +: t (u)) P 0. The desired result fially follows from the fact that B B,+ B,. = B,+ B, ad B = C O the set-up of the simulatio experimets For the umerical experimets ivolvig serially depedet observatios, we restricted ourselves to the bivariate case ad oly focused o the tests based o Š ad Ŝ. Give a bivariate copula C, two models were used to geerate serially depedet observatios uder H 0 defied i (.). The first oe is a simple autoregressive model of order oe, AR(). Let U i, i { 00,...,0,...,}, be a bivariate i.i.d. sample from a copula C. The, set ǫ i = (Φ (U i ),Φ (U i2 )), where Φ is the c.d.f. of the stadard ormal distributio, ad X 00 = ǫ 00. Fially, for ay j {,2} ad i { 99,...,0,...,}, compute recursively X ij = 0.5X i,j +ǫ ij. (AR) The secod model is a bivariate versio of the expoetial autoregressive (EXPAR) model cosidered i Auestad ad Tjøstheim(990) ad Paparoditis ad Politis(200, Sectio 3.3) (see also Bücher ad Kojadiovic, 203). The sample X,...,X is geerated as previously with (AR) replaced by X ij = {0.8.exp( 50X 2 i,j)}x i,j +0.ǫ ij. (EXPAR) Data uder ( H 0 ) H 0,m, with H 0,m as i (.2), were geerated usig the procedures described above except that the bivariate radom vectors U i, i { 00,...,0,...,} are idepedet such that U i, i { 00,...,0,...,k } are i.i.d. from a copula C ad U i, i {k +,...,} are i.i.d. from a copula C 2, where C C 2 ad k = t for some t (0,). The resultig samples X,...,X are therefore ot samples uder H,c H 0,m sice thechageithedepedece isgradualby (AR)or(EXPAR). The copulasc ad C 2 were take to be both either bivariate Clayto, Gumbel Hougaard, Normal or Frak copulas such that C has a Kedall s tau of 0.2 ad C 2 a Kedall s tau of τ {0.4,0.6}. The parameter t defiig k was chose i {0.25,0.5}. The depedet multiplier sequeces ecessary to carry out the tests were geerated usig the movig average approach proposed iitially i Bühlma (993, Sectio 6.2) ad revisited i some detail i Bücher ad Kojadiovic (203, Sectio 6.). A stadard 26