Common persistence in conditional variance: A reconsideration. chang-shuai Li

Transcription

1 Common perssence n condonal varance: A reconsderaon chang-shua L College of Managemen, Unversy of Shangha for Scence and Technology, Shangha, 00093, Chna E-mal:chshua865@63.com Ths paper demonsraes he flaws of co-perssence heory proposed by Bollerslev and Engle (993) whch cause he heory can hardly be appled. Wh he nroducon of he half-lfe of decay coeffcen as he measure of he perssence, and boh he weak defnon of perssence and co-perssence n varance, hs sudy aemps o solve he problems by usng exhausve search algorhm for obanng co-perssen vecor. In addon, hs mehod s llusraed o research he co-perssence of sock reurn volaly n 0 European counres. Keywords: Perssence n varance; Co-perssence n varance; Long memory Inegraed-GARCH (IGARCH); Vecor GARCH (VGARCH); Decay coeffcen JEL classfcaon Numbers: C0, C3.

2 INTROCUTION The perssence of fnancal volaly mples ha he shocks of he curren condonal varance affecs he condonal varances of all fuure horzons permanenly,.e., he curren condonal varance would never dmnsh. The perssence n he auoregressve condonal heeroscedascy process means here exss a un roo, whch s he so-called IGARCH model proposed by Engle and Bollerslev (986). The exsence of perssence ncreases he uncerany of fuure nvesmen. Forunaely n ceran suaons nvesors can elmnae perssence by allocang porfolo o be co-perssen. Accordng o porfolo heory, dversfcaon of asses can reduce rsk bu meanwhle, ends o ncrease he volaly perssence of porfolo whch s dscussed by Karnasos e al (999). Therefore s necessary o research volaly co-perssence. Bollerslev and Engle (993) exend he analyss of perssence o he mulvarae vecor GARCH process and nae he research feld of he heory of co-perssence n condonal varances. There are some sudes focusng on he heorecal expanson of co-perssence n recen years. L and Zhang (00) defne he perssence and common perssence of vecor GARCH process from he vew of he negraon and gve he properes and he error correcon model of vecor GARCH process under he condon of he co-perssence. L and Zhang (00) sudy he perssence and co-perssence n SV model nsead of GARCH model and presen he co-perssence heorem. LIU Dan-hong, XU Zheng-guo and ZHANG Sh-yng (00) gve ou he defnon of he nonlnear common perssence and make use of he wavele neural nework o approach he nonlnear common perssence hen esfy he nonlnear common perssence of shangha and Shenzhen sock marke. Du and Zhang (003) generalze he concep o paral-co-perssence and blocked-co-perssence, hey also show he no-exsence of co-perssence under wo condons, and exsence of co-perssence when wo varables exs lnear relaon. L and Zhang (00) dscuss he negraon and perssence of he BEKK model developed by Engle and Kroner (995) and sugges he suffcen and necessary condon of co-perssence n varance of he BEKK model. Based on mpulse response analyss, Xu and Zhang (005) pus forward he defnon of volaly perssence and common perssence n fraconal dmenson, and nvesgaes he perssence of FIGARCH process. I s regreable ha few papers abou applcaon sudy of co-perssence appeared snce Bollerslev and Engle (993) had nroduced he co-perssence heory, one of he reasons accoun for hs phenomenon s ha he vecor GARCH model for analyzng co-perssence has some unavodable flaws. Ths arcle demonsraes four flaws of analyzng co-perssence by means of vecor GARCH. As a resul, he co-perssence heory can hardly be appled n he fnancal feld. Wh he nroducon of he half-lfe of decay coeffcen as he measure of he perssence, and he weak defnon of boh perssence and co-perssence n varance, hs sudy herefore aemps o solve he problem of obanng co-perssence vecor wh exhausve search algorhm. In addon, hs mehod s llusraed o research he co-perssence of sock reurn volaly n 0 European counres n hree fnancal regons. The resul shows ha here are seven counres socks are co-perssen wh ohers, besdes, he socks n he Germanc area and he Scandnavan area are volaly co-perssen, bu he socks n he French area are no co-perssen. The res of he paper s organzed as follows: The nex secon revews he relave defnons and heorems nroduced by Bollerslev and Engle (993). Secon 3 explores he four flaws of co-perssence heory. Secon 4 akes he half-lfe of decay

3 coeffcen as he measure of he perssence hen pu forward he weak defnon of boh perssence and co-perssence n varance, fnally proposes exhausve search algorhm for obanng co-perssen vecor. Secon 5 presens daa and emprcal analyss of he flaws of he vecor GARCH mehod and he advanages of our mehod. Secon 6 concludes. REVIEW OF THE CO-PERSISTENCE THOERY Ths secon revews he relave defnons and he correspondng heorems nroduced by Bollerslev and Engle (993). Le {y } denoe he N vecor sochasc process wh he condonal mean and varance funcons: E [ ] y = M (.) var [ y ] = H =,,... (.) E [ ] and var [ ] denoe he condonal expecaon and condonal varance based on he avalable nformaon se a me respecvely. The sochasc N N symmerc marx H s almos surely posve defne for all. Le ε = y M, and obeys condonal mulvarae normal dsrbuon N(0, H ). The perssence s well characerzed by he nfluence of he nal condons on he fuure condonal varances as he forecas horzon ncreases. For beer llusrang he noon of perssence of a process Engle and Bollerslev gave he followng noaon: * H ( s) Es ( vech( H )) E0 ( vech( H )), > s > 0 (.3) Where vech ( ) denoes he vecor half operaor ha sacks he lower rangular elemens of an N N marx as an N( N + ) / vecor. Defnon : The sochasc { y } s defned o be perssen n varance f * lmsup { H ( s)} 0, a. s. for some s > 0 and some =,,, N( N + ) /. In order o research co-perssence among several me seres, consder he vecor GARCH (p, q) model nroduced by Bollerslev, Engle, and Wooldrdge (Bollerslev e al, 988). p q T ε ε j j = j = (.4) Vech( H ) = W + AVech( ) + B Vech( H ) =,... Condons on A ( =,..., p) and B j ( j =,..., q) for H o be posve defne a.s. Theorem : he vecor GARCH (p, q) process { ε } defned n (.4) s covarance saonary f and only f, all he roos of he characersc polynomal, de[ I A( λ ) B( λ )]=0 (.5) * Les nsde he un crcle, n whch case lmsup { H ( s)} = 0, a. s. for all s > 0. Many of emprcal resuls showed ha he sum of coeffcens of he unvarae GARCH (p, q) model p q - j j = j= σ = w + a ε + b σ =,... (.6) are ofen o be very close o one, so he IGARCH (p, q) emerged n response o he needs of mes. The shocks o he condonal varance wll have a permanen effec as * lmsup { H ( s)} 0, a. s. for some s > 0. Ths heorem exended he analyss of

4 perssence wh he unvarae IGARCH (p, q) model o he mulvarae GARCH process wh un characersc roo ( λ = ). Defnon :The mulvarae sochasc process {y } s defned o be co-perssen N n varance f here exs a nonzero vecor γ R such ha { vec( γ )} 0 and * lmsup { H ( s)} 0, a. s. for some s > 0 and some =,,, N( N + ) /, whle T T T * lmsup Es[ γ H γ ] E0[ γ H γ ] = lmsup Vec( γ ) H ( s) 0, a. s. for all s > 0. Where vec( γ ) = vech( γγ T ) - dag( γ ) dag( γ ). Theorem : Le λ λ L λr > λr + L λn denoe he ordered roos from he characersc polynomal for he vecor GARCH (p, q) process n (.4), and v, v, Lvn he correspondng N( N + ) / rgh egenvecors, A( λ ) v + B( λ ) v = v (.7) The process s hen co-perssence, f and only f vec( γ )' v = 0 ( =,...,r) (.8) N for some nonzero vecor γ R. Lemma : lnear combnaons, { γ ' ε } of he vecor GARCH (p, q) process n (.4) wll follow a unvarae GARCH (p, q) process f and only f for some scalar consans α, β j, =,...,q, j =,..., p; Vec( γ )' A = α Vec ( γ )' =,,...,q Vec( γ )' B = β Vec( γ )' j =,,..., p j j (.9) I clearly shows ha f he vecor GARCH (p, q) process s co-perssen n varance, and { γ ' ε } follows a unvarae GARCH (p, q) model, where γ s a co-perssen vecor, he sum of he scalar parameers α, β j, =,..., q, j =,..., p; mus be less han one. 3 THE FLAWS OF VECTOR GARCH METHOD FOR ANALYZING CO-PERSISTENCE The vecor GARCH mehod for obanng co-perssen vecor revewed n secon has four flaws: Frsly, he curse of dmensonaly problem may arse when esmang vecor GARCH model, he unresrced vecor GARCH n (.4) nvolves a oal of [ N( N + ) + N ( N + ) ( p + q)] / 4 unque parameers, esmaor may fal o converge or converge locally. Secondly, he proof of heorem resrcs he egenvalue of process (.4) o be real number, however, he egenvalue s ofen found o be magnary number. So he vecor GARCH model would fal o analyze he co-perssence propery when he egenvalues conan magnary number. Thrdly, Lemma mposes over-denfyng resrcons on vecor GARCH model (.4). Le p = q = n (.9) for ease of exposon hen reduces o vec( γ )' A = αvec( γ )' vec( γ )' B = βvec( γ )' (3.) Where nonzero vecor N γ R and scalar α, β R. I easly seen ha vec( γ )' s a lef egenvecor of coeffcen marx A and B whose correspondng egenvalue

5 s α and β. In fac, equaon (3.) makes vecor GARCH model (.4) o be over-resrced wh n resrcons whch could brng wh huge esmaor error of co-perssence vecor γ. Besdes, consderng condons on A and B for H o be posve defne, he esmaon mus be based. Fourhly, hs flaw s mos concealed and s he key o he problem. Even f here were no he prevous hree flaws hs defec s enough o make he heory hardly be applcable. We llusrae he problem wh one wo-dmensonal sochasc process and he case of mul-dmenson could be deduced smlarly. Gven ha boh { y } and { y } follow a non-saonary GARCH model and hey are co-perssen, so here exss a leas one co-perssence vecor γ = ( γ, γ ) make he lnear combnaon { γ * y + γ * y} follow a covarance-saonary unvarae GARCH model accordng o lemma. And now we ry o oban ( γ, γ ) by he co-perssence heory, for smplcy we sandardze he frs fracon o hen becomes no γ = (, γ ). I s evden ha he number of co-perssence vecors ofen more han one. I should be a se f exss. Bu he vecor GARCH model wh he resrcon mposed by lemma esmaes he co-perssence vecor γ by maxmum lkelhood esmaon, whch can oban one and only co-perssence vecor γ ˆ, whch could lead o ncorrec concluson. I s analyzed n deal n he followng. Le L( θ ) = L( α, β, γ ) denoe maxmum lkelhood funcon for esmang he parameer of he vecor GARCH model and θ denoe he parameer vecor ( α, β, γ ). The parameer s a hree-dmensonal vecor θ = ( α, β, γ ), we can analyze he co-perssence propery of { y, y} accordng o he value of ˆα + βˆ by lemma. For beer exposon of he problem we reduce hree-dmensonal parameer vecor ( αˆ, βˆ, ˆ γ ) o wo-dmensonal parameer vecor ( αˆ + βˆ, γˆ ) so ha he maxmum lkelhood esmaon could be llusraed n wo-dmensonal coordnaes graph. Fgure. Dervng co-perssence vecor γ by MLE αˆ + βˆ θˆ 3 θˆ θˆ γˆ γˆ 0 γˆ 3 In fgure, he abscssa axs denoes he co-perssence γˆ and he ordnae axs denoes αˆ + βˆ. There are more han one lkelhood value correspondng o one γˆ

6 coordnae pon snce each value of αˆ + βˆ could be composed of many combnaons of αˆ and ˆβ. For ease of exposon we suppose he closed regon s he defnon doman of parameer vecor θ n fgure (hs closed regon may dffer from he real one, bu would no essenally change he followng concluson). The vecor GARCH model selecs he MLE θˆ ( αˆ, βˆ, γˆ ) n he closed regon by maxmzng lkelhood funcon. Suppose he defnon doman of parameer conans only hree vecors θˆ ˆ, θ and θ ˆ 3, whch represened by hree pons n fgure. These hree pons le n he vercal lne of he closed regon wh γ ˆ, γˆ and γˆ 3 respecvely. The pons θ ˆ and θˆ are under he horzonal dashed lne whch means αˆ ˆ + β < ( =, ) and he co-perssence vecor are γ ˆ, γˆ and he pon θ ˆ 3 s jus he oppose. When dervng co-perssence vecor γˆ by vecor GARCH model, we may encouner he followng wo cases: Case : The lkelhood funcon value reaches maxmum a he pon θ ˆ 3. In hs case he co-perssence vecor (, γˆ ) and (, γˆ ) can no be obaned, and wha can be obaned s (, γˆ 3) whch makes he lnear combnaon of he combnaon { y + γ * y} follow an IGARCH model whch leads o msjudgemen of he co-perssence propery of sochasc process { y, y} as s acually co-perssen wh wo co-perssence vecors θ ˆ and θ ˆ. And wha abou he probably hs case occurs? Each lnear combnaon of he sochasc process { y, y} wh he vecor of porfolo wegh γ ( equals o co-perssence vecor, he same below) could be fed by a GARCH model. Tha s o say, each γˆ could make { y + γ * y} follow a GARCH model, even f { y, y} s co-perssen, here ceranly exss some γ make { y + γ * y} follow an IGARCH model( consder he exreme case of and he combnaon change no he non-saonary { y } ). So he probably ha lkelhood funcon reaches maxmum when γ = γˆ 3 s src posve. Ths case s wors and even f hs case dd no happen we may sll encouner he followng case. Case : The lkelhood funcon value reaches maxmum a he pon θ ˆ. In hs case we can only oban he parameer θˆ ( αˆ ˆ ˆ, β, γ ) and only ge one of he co-perssence vecor (, γ ˆ), he oher co-perssence vecor (, γˆ ) s mssed ou. Ths case s mperfec even s no worse han case. If we esmae all of he parameers wh every γ n he doman and hen choose he vecor of porfolo wegh wh αˆ + βˆ <. Tha means we ry all he porfolo wegh γ ˆ, γˆ and γˆ 3 and hen oban ( αˆ ˆ, β ), ( αˆ ˆ, β ) and ( αˆ ˆ 3, β 3), we can oban cach boh γˆ and γ ˆ, here s no oher way avalable. Ths s jus he dea of exhausve search algorhm whch esmae all of he parameers and hen oban he co-perssence vecor by he sandard αˆ + βˆ <. 4 EXHAUSTIVE SEARCH ALGORITHM

7 Le's vew he noon of decay coeffcen p q α + β j and half-lfe K, consder = j= GARCH(, ) model σ = w + αε + βσ =,... (4.) And he s seps expecaon of condonal varance s s ( ) s E ( ) α + β σ + s = ω + ( α + β ) σ + (4.) ( α + β ) I can be seen clearly ha he shock o he volaly of ε s subjec o a exponenal decay. Then he parameer α + β can be called decay coeffcen. A more nuve characersc of exponenal decay s he me requred for he decayng quany o fall o one half of s nal value. Ths me s called he half-lfe. Defnon gves he srong defnon of he perssence n varance whch requres ha he decay coeffcen mus be very close o one. Ths perssence s he permanen perssence, bu here s nearly no shock affecs economy permanenly. We can jus ake he perssence as long-erm perssence whch doesn mean permanen perssence. The formal noon of long memory could explan hs ssue (as he long memory GARCH nroduced by Zhuanxn Dng and Granger (996)). So we ake he half-lfe of he decay coeffcen as measure of perssence n varance. Ths way s very nuve and flexble because he nvesor could choose he half-lfe of decay coeffcen he can afford accordng o hs own rsk preference as crcal value. The porfolo volaly possesses long memory propery f he half-lfe of decay coeffcen s oo long. In ha suaon, he shocks of he curren condonal varance would las a long me. Defnon s so srong o applcable o emprcal analyss, so we pu forward he weak defnon of he perssence and he co-perssence n varance: Defnon 3: Gven ha he crcal value s K, he sochasc process { y } (one dmenson) s perssen n varance f he half-lfe of decay coeffcen of s volaly model s longer han K. Defnon 4: Gven ha he crcal value s K, he sochasc process { y } (mul-dmenson) s co-perssen n varance f all he componens of are perssen ' n varance and here exss a lnear combnaon of hem { γ y } whose half-lfe of decay coeffcen of s volaly model s shorer han K. Based on hese wo weak defnons we pu forward he seps for esng co-perssence:. Tes for he perssence n varance of each componen of he sochasc process { y } '. Tes for wheher exss a lnear combnaon { γ y } whch s no perssence n varance. The co-perssence es requres all he componens of sochasc process are perssen n varance, f here are some componens of he sochasc process are no perssen n varance we can only es co-perssence for he res componens, whch are enrely n analogy o co-negraon es dscussed by Engle and Granger (987). Now we nroduce exhausve search algorhm. In fac, we can bypassng vecor GARCH model whch can avod he four flaws we elaboraed above and analyze he perssence of all her lnear combnaons hen choose he ones are no perssen n varance drecly, ha s he advanage. As poned ou n he fourh flaw n secon, exhausve search algorhm mus ake all he porfolo weghs vecor γ n her defnon doman no consderaon,

8 ' ' hen analyze he perssence of { γ y } and choose he γ make { γ y } follow a covarance-saonary GARCH model. Tha s why s named exhausve search algorhm. We don' use vecor GARCH model n he exhausve search algorhm so he four flaws are avoded. Consderng N dmensonal sochasc process { y }, whose all componens { y } are perssence n varance, we need o judge wheher s co-perssen n varance. Accordng o he dea above, we fx porfolo weghs γ = ( γ,..., γ N )' frs and ' analyze he co-perssence of he lnear combnaon { γ y} = { γy + γ y + + γ N yn}, fnally esmae he unvarae GARCH model for lnear combnaon. Generally speakng, GARCH (, ) model could nearly f all fnancal me seres very well, so s reasonable o choose GARCH (, ) model o f he lnear combnaon as mos scholars do. Afer esmang he parameers αˆ, βˆ of GARCH model by maxmum lkelhood esmaon, he mappng can be obaned f : γˆ αˆ + βˆ. Then search all he co-perssence vecors whch correspondng decay coeffcen αˆ + βˆ < ( α + β) k whose half-lfe s k, ha s co-perssence vecor se we need. The case of wo-dmensons s shown n fgure and he dashed curve lne denoe he mappng. In he fourh flaw we reduce he hs dashed lne no jus hree pons θˆ ˆ, θ and θ ˆ 3 for easy exposon. The dashed lne below horzonal lne ( α + β ) k s he co-perssence vecor se we need. Fgure. Exhausve search algorhm ˆ α + βˆ ( α + β ) k 0 Besdes, exhausve search algorhm can also analyze he co-perssence of any * * * subse of he sochasc process { y }. Le γ = ( γ,..., γ N )' o be he co-perssence * * vecor se obaned, f he componen γ of γ s zero (he number of γ could be more han only one), hen all he subses of process bu { y } are co-perssen. Take a four-dmensonal sochasc process ( y, y, y3, y4 )' as example, he co-perssence vecors se conan zero componens are (0, γ, γ 3, γ 4 ), (0, 0, γ 3, γ 4 ), ( γ, γ, γ 3,0 ) hen can be known ha ( y, y3, y 4 )', ( y3, y 4 )', ( y, y, y3 )' all are co-perssen. For smplcy we normalze he frs componen γ of γ o uny, and hen s only need o consder he res componen γ ( N) (, + ). Exhausve search algorhm searches all he co-perssence vecor componens γ * ( N) γˆ

9 ha make f ( γ ) = α( γ ) + β ( γ ) o be smaller han he gven crcal value ( α + β ) k. When programmng n compuer we le he co-perssence vecor defnon doman o be γ ( N ) ( M, + M ), where M s a very bg posve number. The lnear combnaon of he sochasc process can be wren as {( / M)y + ( / M) γ y + + ( / M) γ N yn}, f γ don end o nfny γ / M mus end o zero, so he h wegh could be negleced. Afer massve emprcal analyses we fnd he value of M performs well from 5 o 5. Exhausve search algorhm s nally used o es he applcaon of co-perssence heory, unavodably has dsadvanages: Frsly, needs huge amoun of calculaon and he runnng me would las long f we analyze hree or more dmensonal sochasc process. Bu hen agan he vecor GARCH model also need much runnng me and can hardly converge when analyze a more dmensonal sochasc process. Secondly, he search sep may be no small enough o cach all he co-perssence vecor componen γ ( N), here may ex some sngular pons. Bu doesn' make sense n applcaon because nvesor can no maser hese sngular pons as well. I s deduced clearly ha here s no oher effecve way bu exhausve search algorhm o overcome he nheren dsadvanages of he mehod proposed by Bollerslev and Engle (993). Anyhow, exhausve search algorhm s he suable mehod o es he applcably of co-perssence heory and help us o reconsder. 5 DATA AND EMPIRICAL ANALYSIS 5. Daa In order o llusrae he lmaon of he vecor GARCH mehod for obanng he co-perssence vecor hs paper uses some counres sock ndex n Europe o es wheher hey are co-perssen and hen o know her degree of economc negraon. In vew of he roubles come from day-of-he-week effecs and non-synchronous radng n daly fnancal me seres, and consderng a very wde me span may falure o capure he nformaon conen of changes n levels and reurns. The daa are herefore sampled weekly. These dfferen sock ndces daa are colleced from se: hp://fnance.yahoo.com. Snce he vecor GARCH model nvolves a large number of parameers, he daa sample s chosen o be enough long n order o mnmze small sample ssues. The sample perod ranges from February, 00 o Augus 30, 00 and he daa nclude 499 observaons all ogeher. These daa nclude 0 Europe counres (French, Germany, Englsh, Swzerland, Holland, Ausra, Belgum, Dansh, Sweden and Norway). Some oher counres are no consdered n he esmaons because of he daa unavalably. Bu hese counres are represenave so her daa are enough o explan he ssue. The Europe counres correspondng sock ndces are CAC 40 (French), DAX (Germany), FTSE 00 (Englsh), SMI (Swzerland), AEX (Holland), ATX (Ausra), EURONEXT BEL-0 (Belgum), OMXC0.CO (Dansh), OMX Sockholm PI (Sweden), OSLO EXCH ALL SHARE (Norway). Then compose weekly oal sock reurns R, = log( P, / P, ), where R, denoes he connuously compounded reurn for ndex a me, and P, denoes he prce level of ndex a me. Fgure 3. Graphs for each counry s sock reurn seres

10 ENGLISH SWITZERLAND HOLLAND AUSTRIA DANISH SWEDEN NORWAY BELGIUM Table. Descrpve sascs for Europe counres sock ndces Mean Sd. Dev. Skewness Kuross French Germany Englsh Swzerland Holland Ausra Dansh Sweden Norway Belgum Noe: All he reurn seres bu Germany s negavely skewed and her Kuross all excess 3, ha ndcae all he counres sock reurn seres are lepokuross and fa-al.

11 Table. Tes for heeroscedascy LB(3.) LB(0) Tes resul French H Germany H Englsh H Swzerland H Holland H Ausra H Dansh H Sweden H Norway H Belgum H Noe: Daa summary sascs for weekly reurns daa. Ljung Box (LB) a 0 lag lenghs and 0 lag lenghs sascs are compued for reurns and squared reurns. And he P-value s gven n he parenhess under he LB sascs. Tes resuls all are H ndcae hs hypohess rejec he null hypohess of no heeroscedascy. Table 3. The esmaon of unvarae GARCH (, ) model for each sock reurn consan αˆ ˆβ ˆα + βˆ Ausra 3.70E-05 (0.0073) Belgum 5.09E-05 (0.00) Dansh 6.65E-05 Englsh.8E-05 (0.049) French.3E-05 (0.75) Germany.68E-05 (0.047) Holland 4.93E-05 (0.0090) Norway Sweden.0E-05 (0.039) Swzerland 5.60E-05 (0.0006)

12 Noe: The P-value s gven n he parenhess under he parameers. Only he p-values of he consan n he model of French and Germany sock reurn s more han 5% whch means all parameers are sgnfcan bu hese wo consans. Addonally, he decay coeffcen ˆα + βˆ s very close o whch mples all hese seres are perssen n varance. Fgure 3 shows each of hese counres sock reurn whch srongly ndcae ha he reurn seres are saonary. Table presens some descrpve sascs for he reurn seres. As s shown n Table, all he reurn seres bu Germany s negavely skewed and her Kuross all excess 3, ha ndcae all he counres sock reurn seres are lepokuross and fa-al. Furhermore, Table shows he Ljung Box ess whch clearly sugges he presence of GARCH effecs. 5. Emprcal analyss In order o compare vecor GARCH mehod and exhausve search algorhm, he wo mehods are mplemened o analyze co-perssence below. Accordng o he frs sep of co-perssence es poned ou n secon 4, he esmaed unvarae GARCH models of each sock reurn seres are lsed n able 3: Table 3. The esmaon of unvarae GARCH (, ) model for each sock reurn consan ˆα ˆβ ˆα + βˆ Ausra 3.70E (0.0073) Belgum 5.09E (0.00) Dansh 6.65E Englsh.8E (0.049) French.3E (0.75) Germany.68E (0.047) Holland 4.93E (0.0090) Norway Sweden.0E (0.039) Swzerland 5.60E-05 (0.0006) Noe: The P-value s gven n he parenhess under he parameers. Only he p-values of he consan n he model of French and Germany sock reurn s more han 5% whch means all parameers are sgnfcan bu hese wo consans. Addonally, he decay coeffcen ˆα + βˆ s very close o whch mples all hese seres are perssen n varance. I s legble n able 3 ha mos of he parameers of unvarae GARCH (, ) are sgnfcan and all hese sock reurn are perssen n varance. Two dfferen mehods are mplemened o analyze he co-perssence of hese sock reurns below. Frsly, we esmae he bvarae GARCH (, ) model of he sock reurn of

13 Germany and Swzerland wh he consrans mposed by lemma. The sock volaly of Germany and Swzerland exhb apparen perssence n able 3. Ther connuously compounded percenage daly rae of sock reurns, y = ( y, y )' s here parameerzed as bvarae GARCH(, ) model (0.035) y = + ε, (0.05) T Vech( H ) = + Vech( ε ε ) (-) (-) (-) (-) (-) Vech( H ) (-) (0.0 00) (-) (-) γˆ = ( γˆ ˆ ˆ )' = (.039)', α = , β = In he prelmnary esmaon he parameers { A } 3,{ A } 3,{ A } 3,{ B},{ B} 3, { B}, { B} 3, { B} 3 were all found o be small and nsgnfcan and hen were se o be zero for ease convergence of he nonlnear opmzaon algorhm. The resuls show ha he porfolo should follow a unvarae GARCH (, ) wh he parameers αˆ = , βˆ = , however, f nvesor allocae porfolo wh weghs vecor γˆ = ( γˆ )' = (.039)', he porfolo acually follows a unvarae GARCH (, ) wh he parameers αˆ = 0.67, βˆ = Ths s because of he prevous hree flaws. The esmaed resul dffers so much from he real suaon ha vecor GARCH can no analyze co-perssence accuraely. Even he esmaed resul s unbased we sll have o face he dffculy brough by he fourh flaw. Turnng o exhausve search algorhm. Dvdng hese en European counres no hree regons, ha s, he Germanc area (Germany, Swzerland, Ausra), he French area (Belgum, France, Holland), he Scandnavan area (Dansh, Sweden, Norway), plus England and akng any wo counres as one par (alogeher 45 pars). There are only 7 pars are found o be co-perssen. In addon, hs algorhm s exended o deal wh hree-dmensonal sochasc process o sudy he co-perssence of he hree nernal counres sock reurn n each regon respecvely. Ths paper ses he crcal value of decay coeffcen o be (s correspondng half-lfe s 5), accordng he weak defnon of perssence n varance (defnon 3), hey are perssen n varance,.e., each decay coeffcen ˆα + βˆ of he GARCH model s bgger han he gven crcal value Ths arcle ses γ =, γ (-M, M ), M = 0, dγ = 0. n he exhausve search algorhm when analyzng he co-perssence of 45 pars of socks reurn. The graphs

14 of he decay coeffcen funcon f ( γ ) = α( γ ) + β ( γ ) of he 7 co-perssen pars of sock reurn are showed n Fgure 4. Fgure 4. Graphs of decay coeffcen funcon f ( γ ) of all porfolos Decay coeffcen Crcal value Decay coeffcen Crcal value

15 0.95 Decay coeffcen Crcal value Decay coeffcen Crcal value

16 0.95 Decay coeffcen Crcal value Decay coeffcen Crcal value

17 0.95 Decay coeffcen Crcal value In Fgure 4, he ordnae axs and abscssa axs denoe decay coeffcen and he second componen of he vecor of porfolo γ respecvely, he doed lnes are he funcon curves of decay coeffcen f ( γ ) = α( γ ) + β ( γ ), and he sold lne s he crcal value of decay coeffcen (0.8706). The nersecon of he doed lne wh he sold lne ndcaes he wo sock reurns are co-perssen n varance. The abscssas nerval ha he curve under he sold lne s he co-perssence nerval, whch means nvesor can elmnae perssence by allocang porfolo wh he co-perssence vecor (, γ ). The resuls are summarzed no able 4. Table 4. The resuls of he co-perssence analyss * porfolo ( α + β ) γ co-perssence nerval Dansh and Germany (0.3,0.7) Dansh and Swzerland Dansh and Norway (0.5,3.3) Germany and Belgum (,.6) Germany and Swzerland (0.7,.7) Holland and Germany (0.5,.6) Norway and Ausra (-0.6,-0.) * Noe: The porfolo can reach he mnmum of perssence when γ = γ wh s correspondng coeffcen ( α + β). And he perssence can be elmnaed by mn ' allocang porfolo weghs vecor (, γ ), where γ s nsde of he co-perssence nerval. Ths algorhm s easly exended o analyze mul-dmenson sochasc process. Ths paper sudes he co-perssence of he hree nernal counres sock reurn n he each regon respecvely. Vecor GARCH model mus encouner curse of dmensonaly n hree dmensonal suaon as have o esmae 78 unque parameers. Se γ (- 5, 5), γ 3 (- 5, 5), and he sep dγ = dγ = 0., and search mn

18 he co-perssence vecor, we can see Fgure 5. Fgure 5. Graph of decay coeffcen funcon f ( γ ) for he Germanc area Fgure 5 presens he graph of decay coeffcen funcon f ( γ ) for he Germanc area. The axes of coordnaes are respecvely decay coeffcen, he second componen of he vecor of porfolo γ and he hrd componen of he vecor of porfolo γ 3. The surface s he funcon surface of decay coeffcen f ( γ ) = α( γ ) + β ( γ ) and he horzonal plane s he crcal value of decay coeffcen (s half-lfe s 5). The nersecon of he surface wh he horzonal plane ndcaes he hree sock reurns n Germanc area are volaly co-perssen. The area ha he surface under he horzonal lne s he co-perssence area, whch means ha nvesors can elmnae perssence by allocang porfolo wh he co-perssence ' vecor (, γ, γ 3). Fgure 6 s he conour graph of he surface whose alude s : Fgure 6. Conour for he surface of decay coeffcen funcon f ( γ ) for he Germanc area

19 In Fgure 6, he loop lne s conour lne wh alude of Obvously, he Swzerland, Germany and Ausra sock reurn are co-perssen. The area of co-perssence s locaed nsde he closed regon. And he porfolo reaches he mnmum of perssence n he coordnae pon ( γ, γ 3) = (.7, 0.7) wh s correspondng coeffcen αˆ + βˆ = And nvesor can elmnae perssence by ' allocang porfolo weghs vecor (, γ, γ 3), where ( γ, γ 3) s nsde of co-perssence area. As poned ou n secon 4, exhausve search algorhm can analyze he co-perssence of any subse of he sochasc process. The area of co-perssence conans he axs of coordnaes γ 3 = 0, whch mples he co-perssence vecor se has zero componen, so he frs componen( Swzerland) and he second componen( Germany) of hs sochasc process are co-perssen whch s n accordance wh he wo dmensonal resul ( see Fgure 4). Fgure 7. Graph of decay coeffcen funcon f ( γ ) for he French area

20 Fgure 7 presens he graph of decay coeffcen funcon f ( γ ) for he French area, and he porfolo can reach he mnmum of perssence n he coordnae pon ˆ + β = 0.947, bu he perssence sll can no be elmnaed as he decay coeffcen funcon surface s dsjon wh he horzonal plane. There s no any subse of hs sochasc process s co-perssen whch s n accordance wh n accordance wh he wo dmensonal ( γ, γ 3) = (., 0.7) wh s correspondng coeffcen α ˆ resul ( see Fgure 4). Fgure 8. Graph of decay coeffcen funcon f ( γ ) for he Scandnavan area Fgure 8 presens he graph of decay coeffcen funcon f ( γ ) for he Scandnavan area, he surface nersecs wh he horzonal plane, means he Dansh, Sweden and Norway sock reurn are co-perssen.

21 Fgure 9. Conour for he surface of decay coeffcen funcon f ( γ ) Scandnavan area for he Fgure 9 s he conour graph of he surface whose alude s The loop lne s conour lne wh alude of The area of co-perssence s locaed nsde he loop lne. And he porfolo can reach he mnmum of perssence n he coordnae pon ( γ, γ 3) = (0., 0.8) wh s correspondng coeffcen αˆ + βˆ = The ' perssence can be elmnaed by allocang porfolo weghs vecor (, γ, γ 3), where ( γ, γ 3) s nsde of co-perssence area. The area of co-perssence conans axs of coordnae γ = 0, whch mples he co-perssence vecor se has zero componen, so he frs componen( Dansh) and he second componen( Norway) of hs sochasc process are co-perssen whch s n accordance wh n accordance wh he wo dmensonal resul ( see Fgure 4). Fgure 0. The comprehensve resuls of co-perssence n he hree regons French area Belgum French Germanc area Holland Scandnavan area Swzerland Germany Ausra Dansh Sweden Norway Fgure 0 presens he above abundan resuls n a bref way. In he char, here are hree blocks sand for hree regons. The lne connecs wo counres means he sock

22 reurn of he wo counres are co-perssen, and he blocks flled wh grey colour means he hree sock reurn of hese hree counres are co-perssen. There s only one par of sock reurn are co-perssen n he Germanc area, and no any par of sock reurn are co-perssen n he French area and one par of sock reurn are co-perssen he Scandnavan. There are wo pars of sock reurn are co-perssen beween he Germanc area and he French area, hree pars of sock reurn are co-perssen beween he Germanc area and he Scandnavan area and no any par of sock reurn are co-perssen beween he French area and he Scandnavan area. The sock reurns of he counres n he Germanc area and he Scandnavan area are co-perssen bu he French area. Jus as s name mples, exhausve search algorhm may lack of echncal conen, bu wha we focus on s o es he acual applcaon of co-perssence by hs algorhm. Though hs mehod s smple bu he resuls derved by offer a huge amoun of nformaon. I can be easy realzed, easly exended o mul-dmensonal and comprehensve analyss of he co-perssence of sochasc process. Ths algorhm ncreases he praccal applcably of co-perssence heory. 6 CONCLUSION Afer analyzng he co-perssence heory hs paper pons ou he four flaws of he vecor GARCH mehod for sudyng co-perssence propery. In order o surmoun hese flaws we ake he half-lfe of decay coeffcen as he measure of he perssence and pu forward he weak defnon of he perssence and he co-perssence n varance. On hese bases we use exhausve search algorhm for obanng co-perssen vecor. In addon o overcome he four flaws can also analyze he co-perssence of any subse of he sochasc process. Ths mehod s llusraed by applyng o sudy he co-perssence of sock reurn volaly n 0 European counres, he concluson s drawn ha here are seven counres socks are co-perssen wh ohers, and he socks n he Germanc area and he Scandnavan area are volaly co-perssen, he socks n he French area are no co-perssen, whch also mples he hgh degree of negraon of he economes n Europe. Our sudy can be exended n hree ways: Frsly, he daa of hs paper are colleced weekly whle he hgh-frequency daa are easly o be obaned, so he co-perssence of he sochasc process may be dfferen by usng he hgh-frequency daa. Secondly, SV model (sochasc volaly model) has some good feaures for fng fnancal me seres, so researchers could use SV model nsead of GARCH model o analyze he co-perssence. Thrdly, some emprcal resuls obaned by Lamoureux e al (990) show ha srucural change n some exen accouns for he perssence n varance, so GARCH model or SV model wh srucural change dscussed by Xu and Zhang (005) can be aken no consderaon for co-perssence analyss. REFERENCES Bollerslev, T., R. F. Engle, and J. M. Wooldrge, (988). A asse prcng model wh me varyng covarances. Journal of polcal economy, 96, 6-3. Bollerslev T, Engle R F, (993). Common perssence n condonal varances. Economerca, 6, DU Z-png, ZHANG Sh-yng, (003). Sudy on co-perssence of vecor GARCH processes. Journal of sysems engneerng, 8(.5), Engle R F, Bollerslev T, (986). Modellng he perssence of condonal varances.

23 Economerc Revews, 5: -50; Engle R F. Granger C W J, (987). Co-negraon and error correcon: represenaon, esmaon, and esng. Economerca, 55:5 76. Engle R F, Kroner K F, (995). Mulvarae smulaneous generalzed ARCH. Economerc Theory,, -50. FAN Zh, ZHANG Sh-yng, (003). Mulvarae GARCH modelng and s applcaon n volaly analyss of Chnese sock markes. Journal of Managemen Scences n Chna, 6(.), Karnasos M, Psaradaks z, Sola M, (999). Cross seconal Aggregaon and Perssence n Condonal Varance. Helsngon, York, Unversy of York, Workng paper. Lamoureux, Chrsopher G, Wllan D. Lasrapes, (990). Perssence n varance, srucural change and he GARCH model. Journal of Busness and Economc Sascs 8, LI Han-dong, ZHANG Sh-yng, (00). Research on common perssence of BEKK model. Journal of sysems engneerng, 6(.3), 5 3. LI Han-dong, ZHANG Sh-yng, (00). Common perssence and error-correcon model n condonal varance. Journal of Sysem Scence and Sysems Engneerng, 0(.3), LI Han-dong, ZHANG Sh-yng, (00). Research on volaly perssence and co-perssence n sochasc volaly model. Journal of sysems engneerng, 7(.4), LIU Dan-hong, XU Zheng-guo, ZHANG Sh-yng, (004). The Nonlnear Common Perssence of Mulvarae GARCH Model. Sysems Engneerng, (.6), Xu Q-fa, ZHANG Sh-yng, (005). Perssence of Vecor FIGARCH Process. Sysems Engneerng, 3(.7), -6. Zhuanxn DING, Clve W.J Granger, (996). Modelng volaly perssence of speculave reurns: A new approach. Journal of economercs, 73, 85-5.