ֻ13जֻ3௹ܵ ॓ ࿐ ࿐ Б 2010୍3ᄅ ഈᆣᆷඔۚඔऌ֥؟ᇗٳྙհत 周炜星(ת۽ն࿐അ࿐ჽ,ഈݚ200237)ᅋေ:以上证指数5分钟取样的高频数据为例,采用配分函数法对每一交易日的数据进行多重分形分析,发现质量指数 (q)为线性函数.用统计自举生成随机时间序列以深入剖析多重分形谱f( ),发现约有51%的交易日,其多重分形特性无法通过显著性检验.进一步分析发现,所有真实时间序列的奇异性强度与随机序列的奇异性强度相差无几,因而完全可以用后者加以解释.因此,上证指数本身并不具多重分形特性.ܱՍ:金融物理学;上证指数;多重分形分析;统计检验ᇏٳোݼ:F830;;N93 ໓ངѓ്:A ໓ᅣщݼ:1007-9807(2010)03-0081-060 ႄ ྙหྟᄝఃࣁವൈࡗਙᇏ္ႵնਈБ֡,ೂߛࣁࡎ۬aഅࡎ۬aܢௐ۱ܢࡎ۬aܢ൧ᆷඔ֥ࣁವ࿐൞Ⴈ࠹aંaگᄖ༢൬ၭੱ֩[2].္Ⴕ၂ུ࿐ᆀᆰࢤ࣮ࡎ۬Чദ֥ંa٤ཌྟ॓࿐aႋႨඔ࿐֥֩ۀaٚمބ؟ᇗٳྙྟᇉ,ೂཐۖޚളᆷඔ[14-15]aഈᆣᆷඔં࣮ࣁವ൧ӆ๙ݖሱቆᆮطႤགྷ֥ޡܴܿੰބധᎪӮᆷ[16-20]aնഈ൧֥҆ٳ۱ܢ[21]֩,ࠣఃگᄖྟ֥၂ྍྖࢌҭ࿐॓[1-2].ࣁವૌҐႨۚඔऌ(ೂ5ٳᇒ),ؓૄ۱ࢌၞರࣉྛ࿐֥ᇶေ࣮ଽಸЇও4۱ٚ૫[2-3]:ֻ1,ࣁವ؟ᇗٳྙٳ༅,࠹ෘԛ؟ᇗٳྙఅၳ௶,ѩᆷԛᆃ൧ӆэਈ֥࠹ܿੰ,ఃᇏቋࠎЧ֥ྟᇉ൞ܱႿུ؟ᇗٳྙྟᇉაࡎّ۬ӈѯaڄགܵ֩ૡ൬ၭੱ֥ࡕڂແٳ҃[4-6].ֻ2,ᆣಊ֥ཌྷ్ܱཌྷܱ.൞,ࣉ၂҄ٳ༅ᆃུࢲݔ,ॖၛؿགྷྟaࠞ؊൙ࡱaࣁವڄགܵބሧቆކ֩,ٳྙਆ۱໙ี:൧ӆࡌඪა؟ᇗٳྙ֥ંބٚمФܼٗႋႨႿֻ1,۴ऌ؟ᇗٳྙં,֒Ԅ؇l 0ൈ,ٳ༅ࣁವൈࡗਙ[7-10].ֻ3,ޡܴ൧ӆ֥ࡹଆބყҩ,ЇওႨෛࠏݖӱؓ൬ၭੱࡹଆaؓඔᇛ௹ྟթᄝӈඔ (x)֤ҩ؇ ᄝׄx֥ਣთB(x,l)ૢੰଆ֩ഈડቀૢੰܱ༢[11].ֻ4,ࣁವ൧ӆ֥ັܴଆ,ᇶေЇওࠎЧ૫ሧᆀބᄮലࢌၞᆀѰᒮaჇറଆa (x)(B(x,l))~l(1)၈ྌଆaഒඔᆀѰᒮଆ֩,ၛࠣႮՎطစളԛᄵӫҩ؇ ᄝׄxԩఅၳ,ఃఅၳྟ఼؇ູ (x).ট֥۲ᇕଆ[12-13]֒ ູܢௐࡎ۬ࠇᆀܢ൧ᆷඔൈ, (B(x,l))࣍.ቋᄪႨ؟ᇗٳྙંٳ༅ࣁವൈࡗਙ֥࿐රᆞбႿl,ࠧؓ෮Ⴕׄx,అၳྟᆷඔ (x) 1.ᆀॖି൞Ghashghaie֩[10],ૌࡼຓ߸൧ӆაߐᆭ,ھҩ؇ᄝંഈ҂ऎС؟ᇗٳྙห.ྟ൙ੀোбൌഈ,ൌᆣ࣮֤֥֞ࢲݔ၂২ຓဒᆣਔഈඍ,ؿགྷૅჭބ֣ݓઔक़ຓ߸ࡎ۬ѯ֥इݦඔऎႵ٤ཌྟ֥ѓ؇ੰ,طఃܱႿ߸൧؟ᇗંҩ,అၳྟ఼؇ٳ҃! max- min 0.ٳྙหྟ֥ൌᆣ္࣮֤֞ਔܼܱٗᇿ.؟ᇗٳֻ2,ᄝੀࠇݚਈۚࣁವඔऌ֥؟ᇗٳ ൬۠ರ௹:2007-05-10;ྩרರ௹:2007-10-08.ࠎࣁཛଢ:ݓࡅሱಖ॓࿐ࠎࣁሧᇹཛଢ(70501011);ࠉႇת࢝ტࠎࣁ߶ۚ֩ჽౝ୍࢝ഽࠎࣁሧᇹཛଢ(101086).ቔᆀࡥࢺ:ᇛྒ(1974∀),ଳ,ᆄࡾᇭ᳝ದ,Ѱൖ,࢝൱,Ѱൖള֝ഽ.Emai:lwxzhou@
∀82∀ܵ ॓ ࿐ ࿐ Б2010୍3ᄅྙٳ༅ᇏ[22-24],ࠧ൞Ϥຣਈ֥ࠩඔऌ,၂Ϯ္ (n;l) ([(n-1)l+1,nl])ေࢨඔqཬႿ8,ҌିЌᆣٳݦඔ֥൬৻ྟl,ၹՎٳ༅۷ۚࢨ֥ٳݦඔᄝ࠹ഈીႵၩၬ=( ((n-1)l+i)(2).i=1طᄝഈ૫ษં֥࣮ᇏ,ૄ၂ࢌၞರ֥ۚൈࡗఃᇏ,l֥౼ᆴ:ູ1,2,3,4,6,8,12,16,24,ਙ(1ٳᇒࠇ5ٳᇒඔऌ)֥Ӊ؇҂ӑݖ240,ಏ48.Ⴟ൞,qࢨٳݦඔູ௴ђҐႨਔޓն֥ࢨඔq,മᇀ֞ղNq=#210 q(n;l),ᆃࠞնֹࢆ֮ਔٳ༅ࢲݔ֥ॖྐ!q(l)=(N(3).ྟn=1( (m;l)൙ൌഈm=1,ն҆ٳ໓ངᆺಣᇓႿБ֡ൌᆣ࣮֥ࢲݔ,ಏޭਔ۷Чᇉ֥ଽಸ,ࠧӁള؟ᇗٳྙҩ؇ ऎႵሱཌྷර,ྟᄵթᄝѓ؇ᆷඔݦඔหྟ֥ჰၹ (q)֤ೂ༯ѓ؇ܱ༢Ӯ৫.࣮іૼ,ࠧ൞҂ऎ؟ᇗٳྙหྟ (q)֥ٳྙଆ!q(l)~l(4) ,္ॖିӁള෮໌֥؟ᇗٳྙྟᇉ[25],ၹطᆃུൌᆣ֤֥֞؟ᇗٳྙห,ྟ۷ಒҩ؇ ູֆٳ,ྙᄵ (q)ູཌྟݦඔ; ్ູ֥іඍႋھ൞∃ࣜဒ؟ᇗٳྙหྟ%؟ᇗٳྙ,ᄵ (q)ູ٤ཌྟݦඔ.అၳྟᆷඔ.ൈࡗਙᇏ֥ࣜဒ؟ᇗٳྙหྟႵਆ۱ॖି֥টჷ,၂൞ (q)ބ؟ᇗٳྙఅၳ௶f( )ॖؓ (q)ࣉྛীѯྟᇏթᄝ֥Ӊӱཌྷܱ,ྟؽ൞൬ၭੱ֥ແಞ֣эߐ,֤֞[32]ٳ҃[26].ؓޓ؟ࣁವൈࡗਙ,ࣜဒ؟ᇗٳྙห (q)=d (q)/dq(5)ྟჷႿແٳ֥҃ਬࡌഡمऋध[27],ᆃ၂ࢲંf( )=q - (q)ၘູࣜ၂ུ࣮෮ᆣൌ.ؓႿࡎ۬ൈࡗਙطॖၛႨඔᆴ࠹ෘ֤֞ࢲݔ.,֤֥֞؟ᇗٳྙ௶ऎႵޓᅎ֥అၳྟٳ,҃ဢູࢲݔऎႵॖб,ྟЧ໓q֥౼ᆴٓຶಯഡ֤ޓն19-20]ླေؓᆭࣉྛ࠹ဒ,౼[-120)q)120.֒.Ч໓ᇗྍ࣮ഈᆣᆷඔರଽۚඔऌ֥؟ᇗ (n;l) 1ٳྙห( (m;l),ྟѩҐႨ࠹ሱई(bootstrapping)֥ٚمဒ֤֥֞ࣜဒ؟ᇗٳྙหྟ൞ڎটሱෛࠏᅨq 1ൈ,ູਔх૧ඔᆴ࠹ෘ֥ࢲݔӑԛ||ઋ,ࢲݔіૼ,ᄝ࣍϶ඔࢌၞರᇏ,ෛࠏ߄ު֥ඔଷթԥି৯,ॖႨ༯ൔ࠹ෘٳݦඔ֥ؓඔऌିܔӁളбᆇൌඔऌ۷఼֥؟ᇗٳྙห,ྟطN q(n;l)ᄝਸ਼၂϶ࢌၞರᇏ,ෛࠏ߄ު֥ඔऌ֤֥֞అၳln!q(l)=ln(+n=1max{ (m;l)}mྟ఼؇ିܔᄝޓնӱ؇ഈࢳᆇൌඔऌ֥అၳྟmax{ (m;l)}఼؇m.ၹط,ܱႿࡎ֥۬෮໌؟ᇗٳྙหྟൌ࠽ഈ qlnN҂ݖ൞၂ᇕհत.( (n;l)n=1Ֆطᄹն࠹ෘٓຶ.1 ඔऌაٳݦඔم2 ؟ᇗٳྙٳ༅Ч໓ҐႨഈᆣᆷඔ֥5ٳᇒۚඔऌ,࠺Ֆ2001୍2ᄅ28ರᇀ2006୍8ᄅ10ರ,ัԢ҂ູಒಪഈᆣᆷඔ֥ѓ؇ྛ,ູၛ2001୍7ᄅປᆜ֥ඔऌު,܋Ⴥ༯1201۱ࢌၞರ.ؓႿૄ۱24ರູ২,Ⴈٳݦඔمؓ5ٳᇒࢌၞඔऌࣉྛࢌၞರ,܋ႵT=48۱ඔऌ,ׄ҂ٞ࠺ູ{ (t)&؟ᇗٳྙٳ༅.t=1,2,∋,48}.1۳ԛਔ҂qᆴ֥ٳݦඔ!q(l)აԄაཌྷܱ໓ང[14-21]၂ᇁ,Ч໓ҐႨٳݦඔ؇lᆭࡗᄝචؓඔቕѓ༢ᇏ֥ܱ༢,ॖၛؿགྷ,ؓ(partitionfunction)م[28-32]ࣉྛ؟ᇗٳྙٳ༅.ૄ۱qᆴ!q(l)აlӯޓݺ֥ૢੰܱ༢,ఃᇏ֥ᆰࡼھҩ؇ ֥ᆦӪࠢٳູԄ؇ູl֥އሰ,ॖ֤ཌູཌྟቋཬؽӰކ෮,֤ѓ؇ڭۂਔ෮N=T/l۱އሰ,ֻn۱އሰଽ֥ҩ؇ູॉҳ֥෮ႵԄ؇,ࠧl∗[1,48].
ֻ3௹ᇛྒ:ഈᆣᆷඔۚඔऌ֥؟ᇗٳྙհत∀83∀13 ഈᆣᆷඔ֥؟ᇗٳྙ௶f( ) ഈᆣᆷඔٳݦඔ֥ѓ؇ྛູ( )ofSSECႮൔ(4),1ᇏᆰཌོੱ֥ཌྷّඔ۳ԛਔཌྷࢌၞರ֥ൌ࠽ࢌၞඔऌ,ࡼఃՑյ֤ఃႋqᆴ֥ᇉਈᆷඔ (q),ࢲݔೂ2෮ൕ.҂ुᇏॖିթᄝ֥ཌྷܱྟཨാ,ᄵ֤֥֞ෛࠏ߄ਙԛ,ᇉਈᆷඔ (q)ཌྷؓႿࢨඔqӯགྷਅݺ֥ཌྟ҂ऎ؟ᇗٳྙหྟ.ؓႿഈࢫษં֥২ሰ,Ч໓࠹ܱ༢,ೂᇏᆰཌ෮ൕ,ఃོੱູ (q)ູ٤ཌྟݦඔᆃ၂ࢲ,ંطఃࢌၞರ֥ؓႋෛࠏ߄ਙ,Ҭᄵ൞ؓᇶቐ༯֥ٚ٢ն. (q)ݦඔဢऎႵਅݺ֥ཌྟܱ༢,ၹط,ഈᆣॖၛु֞,ഈᆣᆷඔ֥؟ᇗٳྙ௶f( )აෛࠏ߄ᆷඔЧദ҂ऎႵ؟ᇗٳྙห.ྟਙ֥؟ᇗٳྙ௶frnd( rnd)مٳ,ࠧൌ࠽ਙ֥෮໌؟ᇗٳྙหྟѩ҂ཁᇷ.2 ഈᆣᆷඔ֥ᇉਈᆷඔݦඔ (q) (q)ofSSEC3۳ԛਔ๙ݖؓ (q)ࣉྛীಞ֣эߐ֤4 ൌ࠽ഈᆣᆷඔაෛࠏ߄ਙ؟ᇗٳྙ௶֥бࢠ֥֞؟ᇗٳྙఅၳ௶౷ཌf( ).ֆՖྙሑط,andshuffledSSECھ౷ཌऎС൯ޚ؟ᇗٳྙҩ؇అၳ௶౷ཌ֥෮Ⴕ؟ᇗٳྙ఼֥؇ॖၛ๙ݖఅၳྟ఼؇ٳ҃ࠫޅྟᇉ[2].ಖط,ࠧؓႿ-120)q)120ᆃ∀ = max- minটख़߂,࠹ဒ؟ᇗٳྙหᆘહն֥౼ᆴٓ,ຶ∀ ֥ᆴ၇ಖޓཬ,ط minބ max֥թᄝྟ,֩ࡎႿဒఅၳྟᆷඔ ၳႿ1,ࠇ֥ᆴٳࢤ࣍1,ဢཁൕ෮໌֥؟ᇗٳྙหྟ∀ +0.ູՎ,ؓႿૄ၂ࢌၞರ֥ඔऌਙ,ॖള҂ݖ൞၂ᇕհत.Ӯn=1000۱ෛࠏ߄ਙ,ఃఅၳྟᆷඔބఅၳ௶ݦඔႨ༯ѓrndіൕ,ᄵ࠹ဒ֥ਬࡌഡູ3 ࠹ဒ∀ )∀ rndྴࡌࣞБ֥ۀੱູູࣉ၂҄קਈဒ෮֤֥֞؟ᇗٳྙหᆘ֥p1=Pr(∀ )∀ rnd)࠹ཁᇷ,ྟҐႨ࠹ሱई֥ٚم.ؓႿ၂۱۳קp1),ᄵ໌ᆭ֤֥֞؟ᇗٳྙหྟթᄝѩ
∀84∀ܵ ॓ ࿐ ࿐ Б2010୍3ᄅཁᇷ,ڎᄵ,ᄵӫᄝཁᇷྟඣ༯مऋൌ࠽ਙ֥؟ᇗٳྙఅၳ௶ݦඔf( )აෛࠏधھਬࡌഡ.োරֹ,קၬਙ֥؟ᇗٳྙఅၳ௶frnd( rnd)ٳࢤ࣍,ࣇෛࠏf( ѯьିࢳf( )౷ཌ֥అၳྟٳ.҃ၹՎ,Fmin)+f( max)=2ဢॖၛಪ,ູႨٳݦඔمؓഈᆣᆷඔЧദࣉྛᄵ၂۱࣍ර֩ࡎ֥ਬࡌഡॖཿቔೂ༯ྙൔٳ༅֤֥֞؟ᇗٳྙห,ྟᄝ࠹ഈѩ҂ཁᇷ,҂F,Frnd.ݖ൞၂ᇕհतطၘ.ྴࡌࣞБ֥ۀੱູp2=Pr(F,Frnd).5۳ԛਔؓႋႿଖ၂۳קࢌၞರ(2001୍7ᄅ24ರ)֥Frndა∀ rndᆭࡗ֥ׄ.Ⴎؿགྷ,Frndა∀ rndᆭࡗթᄝਅݺ֥ཌྟܱ༢Frnd=k∀ rnd+b(6)ఃᇏ,k=-29!7,b=1!0,ཌྟཌྷܱ༢ඔູ.ᇏ֥ჵಁᄵіൕൌ࠽ഈᆣᆷඔਙ෮۳ԛ֥F-4=ა∀ =−10ᆴ,ᆞݺઋᄝᆰཌ(6)ഈ.ॖၛ֤֞,p1=0!289,p2=0!293.ॖ6 ෛࠏ߄ඔऌაൌ࠽ඔऌ۳ԛ֥Fა∀ ֥бࢠ∀ extractedfromreal,مࡼൌ࠽ඔऌՖෛࠏ߄ඔऌᇏٳԛট.ؓႿఃࢌၞರ,္֤֞ਔোර֥ࢲݔandshuffledSSECseries,ѩ,k=-29!6#0!7,b=1!000#0!003.࠹ෘؿགྷ,෮Ⴕࢌၞರ֥p1 p2.ᄝ1201۱ࢌၞರᇏ,p1)0!054 ࢲ ં֥ࢌၞರᅝ49!1%,p2)0!05֥ࢌၞರᅝ49!1%,ߐᆭ,Ⴕӑݖ၂϶֥ࢌၞರ,ఃਬࡌഡЧ໓ၛഈᆣᆷඔ5ٳᇒ౼ဢ֥ۚඔऌູ(ࠧ҂թᄝ؟ᇗٳྙหྟ)مऋध২,Ⴈٳݦඔمؓૄ၂ࢌၞರ֥ඔऌࣉྛ؟ᇗ.ٳྙٳ༅,ѩႨ࠹ሱई֥ٚمളӮෛࠏൈࡗਙ֤֥ؓ֞؟ᇗٳྙ௶ࣉྛധೆ௩༅.ࢲݔіૼ,նჿႵ51%֥ࢌၞರ,ః؟ᇗٳྙหྟم๙ݖཁᇷྟဒ.ࣉ၂҄ٳ༅ؿགྷ,ࠧ൞ପུ๙ݖཁᇷྟဒ֥ൈࡗਙ,ఃf( )౷ཌ֥అၳྟ఼؇,္აෛࠏਙ֥అၳྟ఼؇ཌྷҵࠫ,ၹطປಆॖၛႨުᆀ֤ၛࢳ.Վຓ,෮Ⴕࢌၞರ֥ᇉਈᆷඔ (q)ӯགྷٳਅݺ֥ཌྟܱ༢,٤ཌྟم֤֞ಒಪ.ཁಖ,໓ངᇏ෮໌֥؟ᇗٳྙห,ྟᆺ5൞၂ᇕհत.҂ҩ,ರଽࡎ۬(ೂ۱ܢ)ࠇᆷ ෛࠏ߄ഈᆣᆷඔਙ֥Frndა∀ rndᆭࡗ֥ׄ∀ rndof1000shuffledඔ(ೂޚളᆷඔaധᎪӮᆷ֩)Чദѩ҂ऎ؟ᇗٳSSECseriesྙห.ྟପહ,൞ڎॖၛಪູ,ᆃ%ࠇ%֥ܢ൧ᆷඔ֥෮໌؟ᇗٳྙห,ྟၘФ࿐ᆀႨࢌၞರ,ః؟ᇗٳྙหྟᄝ࠹ၩၬഈ൞ཁᇷ֥Ⴟࣉྛܢ൧ყҩ[15]ބܒࡹ൧ӆڄགᆷඔ[19,20],?ູՎ,ॖၛᆰࢤбࢠ۳קࢌၞರଽൌ࠽ඔऌ۳Ч໓֥࣮ؓՎิԛਔᇉၐ.ླေᆷԛ֥൞,Ч໓ԛ֥∀ ބFᆴაෛࠏਙ֥नᆴ.∀ rnd/ބѩીႵڎקႨ؟ᇗٳྙંࣉྛ൧ӆყҩބڄག.Frnd/,ࢲݔೂ6෮ൕ,ఃᇏ֥ൌཌ۳ԛਔіൕܵᆃ၂නਫ਼,ҐႨఃࣁವэਈ(ೂ൬ၭੱ)∃ൌ࠽ඔऌ֩Ⴟෛࠏ߄ඔऌ%֥ؓ࢘ཌ.6і࠹ෘᆇൌ؟ᇗٳྙห,ྟࠇࡼ֤֞۷Ⴕࡎᆴ֥ૼ,෮Ⴕࢌၞರ֥∀ .∀ rnd/F .Frnd/,Ӯݔ.
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