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*:2004-11-25:黄诒蓉(1976-),男,江西万安人,经济学博士,中山大学管理学院财务与投资系讲师,研究方向为分形市场与金融计量分析20052()No.2200545JOURNALOFSUNYATSENUNIVERSITYVol.45(194)(SOCIALSCIENCEEDITION)GeneralNo.194*黄诒蓉(中山大学管理学院,广东广州510275):Hurst4,,4:;;:F83091:A:1000-9639(2005)02-0097-07,,(B.B.Mandelbrot),(FractalGeometry),(FractalTheory),E(EdgarE.Peters),(FractalMarketHypothesis,FMH),,[1,2],,(Huang)[3],(Matsushita)[4],,[5][6][7][8],RS,Hurst,,4,RS,Hurst(H)(D)()(),,,(JarqueBera)(KolmogorovSmirnov),(Ljung&Box)Q,BDS(BrockDechert&Sheinkman),97Hurst4(H.E.Hurst)1951Hurst(H),:(1)H=12,();(2)12H1,();(3)0H12,();(4)H1,RS()H.E.Hurst(1951)H,(Mandelbrot&Wallis)(Wallis&Matalas)Peters(1991,1994)RSPeters[2](P59~60),HHausdorffHausdorff,,,:,,,,,115,,,,15,,152,,(ZhaoXingqiu)4(RS)[9],Mandelbrot(fractaldistribution)(),,[10,11]:另有一种广泛研究的系统相空间的分形维,因为相空间中包括系统的所有变量,其维数依赖于所研究系统的复杂性,取值一般大于2,而本文研究的时间序列的分形维与之不同,只有一个变量,取值在1和2之间即为一个具有统计自相似性的概率密度函数,也就是说,在不同的时间标度下,时间序列的统计特征保持相同值得注意的是,有时也称为分形维,与前面的分形维不是同一概念此处是时间序列概率空间的分形维,度量概率密度函数尾部的肥胖程度,而前面是时间序列(轨迹)的分形维,度量时间序列的参差不齐程度Eexp(iuX)=exp(-|u|[1+i(tan2)(sign(u)(|u|1--1)]+iu),1exp(-|u|[1+i2(sign(u)ln(|u|)]+iu),=1(1)4:(1)(0,2],,=2,;(2)[-1,1],;(3)(-,+),;(4)(0,+),(Nolan)Stable304,,0120,01,[11]:{Xt}klimnnk=-n|k|,(FractionalBrownMotion,FBM),(FractionalDifferencingNoise,FDN),[12]:(1-L)Xt=t(2)98(-05,05)(),{t},L,{Xt}005,FDN,,(MiChael)RS[13](Geweke&PoterHudak),GPH[12]:ln{Ix()}=0-ln{4sin2(2)}+(3)Ix()=2T(=1,,g(T))g(T),OLS,(-)s()(s())26,g(T)T0.54,,,,HD=2-H;H=1HH=H-05,,,,()200398(),,sz0(sc0)szz(scz),3133643,19961213(),sz1sz2(sc1sc2),,,:Rt=log(Pt+1)-log(Pt)(4)log(),{Pt},{Rt}8119961213200398,0,3,JBKS5%,,LB,19961213200398,5%,50,BDS5%,()Matlab6584,3,,,RSH,:10,(RS)n,V;Vm;10m,H,H,21RS1(b)Vn306,,1(a)2n=99306,06581,52576,5%H05,,04596,05,3063,sz2szz,Hurst,056070,05,,表1股票指数的统计检验JBKSLBBDSm=5m=50M=2,r=0.25M=10,r=2sz05.8461134.662276900*0.4602*31.70*101.33*17.99*26.39*sz15.343993.239518810*0.4555*21.20*66.86*19.92*16.42*sz2-0.24059.65253498.7*0.4716*8.7598.94*7.33*14.89*Szz5.367561.48693970*0.4392*10.2097.19*8.93*9.87*sc00.5337818.08829392*0.4648*39.37*123.75*17.54*24.67*sc10.6474115.6029783.6*0.4632*25.90*110.93*15.16*14.83*sc2-0.19108.59682363.9*0.4705*15.46*82.48*8.01*17.01*Scz1.580815.8684537.3*0.4414*22.37*78.22*8.05*8.73*:*5%,LBBDS表2H指数的回归及检验结果:上证指数日收益10n306307n1566E(RS)E(RS)-0.4171(-44.71*)-0.1803(-34.30*)0.6050(29.51*)0.0724(190.60*)0.6581(345.50*)0.5641(524.9*)0.4596(151.71*)0.5178(9230.20*)R2=0.9975F=119370P=0.0000R2=0.9989F=275000P=0.0000R2=0.9481F=23020P=0.0000R2=1.0000F=8519700P=0.0000H0.65810.56410.45960.51785.2576*-3.2577*:t,H5%196,*5%图1H指数的估计:上证指数日收益图2上证指数收益序列的盒维测量100,,N2,N4,N8,,Nm,,N,mint[log2(N)],M(),1M(),;,OLS,81225013350,115,,,2,132307,,012,01,8,1163714967,2,,4^=14084,^=-00426,^=00096705,^=00006939231020,,,,,,,PP,3,PP,PP,,,,,图3上证指数日收益的分形分布拟合及检验GPH83,v=T0.5-000260158367,sz2,,,,H,,3,1股市日收益与周收益之间存在相似性,而且上证指数比深圳成指表现更为强烈,H0646006653,30662,,H00072,4(5),,101表3股票指数收益的分形结构参数估计结果HD2-H1HH-0.5sz00.6581*1.3231*1.34191.4084*1.5195259080.07711*0.1581sz10.7040*1.2276*1.2961.1637*1.4204545450.082093*0.204sz20.5581.3226*1.4421.4467*1.792114695-0.00260.058szz0.66531.2844*1.33471.3745*1.5030813170.052481*0.1653sc00.6460*1.3350*1.3541.3672*1.5479876160.073423*0.146sc10.6727*1.2250*1.32731.3755*1.4865467520.076603*0.1727sc20.6194*1.2364*1.38061.4967*1.6144656120.043860*0.1194scz0.7041*1.2714*1.29591.3496*1.4202528050.158367*0.2041:*5%2上证指数日收益比深圳成指日收益表现出更强的长记忆特性,而深圳成指周收益却比上证指数周收益更强,,,,3.1996年12月13日之前的日收益比其后的日收益的长记忆性更强,19961213,4.4种分析方法得到的结论是一致的,而且它们的分形参数估计值在数量变动上也存在一定的内在关系4,,34,H,图4分形参数的估计值与理论值之间的比较H4,,4,,,,,102,,,(MonofractalProcess),(,H),,Mandelbrot(1997),,,[14],(MultifractalTheory)(MultifractalStructure)1Peters,E.ChaosandOrderintheCapitalMarket[M].JohnWiley&Sons,Inc.,NewYork,1991.2Peters,E.FractalMarketAnalysis:ApplyingChaosTheorytoInvestmentandEconomics[M].JohnWiley&Sons,Inc.,NewYork,1994.3Huang,Bwo-NuangandChinW.Yang.TheFractureStructureinMulti-nationalStockReturns[J].AppliedEconomicsLetter,2,67~71,1995.4Matsushita,Ranl,IramGleria,AnnibalFigueiredo,andSilva.FractalStructureintheChineseyuanUSdollarrate[J].EconomicsBulletin,7,No.2,1~13,2003.5,.RS[J].,1999,(2).6.[J].,2000,(5).7,.:[J].,2001,(6).8,.[J].,2002,(1).9ZhaoXingqiu,RenFuyao,JiangFeng.EstimationmethodoffractaldimensionformodeloffractionalBrownianmotion[J].MathematicsinEconornics,3,60~67,1999.10Nolan,J.P.NumericalComputationofStableDensitiesandDistributionFunctions[J].Comm.inStat.StochasticModels13,759~774,1997.11Nolan,J.P.MaximumLikelihoodEstimationandDiagnosticforStableDistribution[J].DepartmentofMathematicsandStatistics,AmericanUniversity,Washington,1999.12Bollerslev.T.andH.Mikkelsen.Modellingandpricinglongmemoryinstockmarketvolatility[J].JournalofEconometrics,73,151~184,1996.13MichaelA.HauserErhardReschenhofer.EstimationofthefractionallydifferencingparameterwiththeRSmethod[J].Computationalstatistics&DateAnalysis,20,569~579,1995.14Mandelbrot,B.B.,Fisher,A.,andCalvet,L..AMultifractalModelofAssetReturns[J].YaleUniversity,WorkingPaper,1997.:103AStudyofCriterionofEnvironmentalTortHUDanying(DepartmentofLaw,Guangdong