基于主体建模的股市交易者模仿行为研究
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摘要
社会经济系统是一个复杂适应系统,它是由许多不同层次的相互影响、相互学习、相互竞争的主体组成的。社会经济系统的复杂性导致它产生了传统线性计量经济学无法解释的非线性现象。新兴的理论产生,为经济学的研究提供了新的研究工具,如分形、混沌、非线性动力学等。然而这些研究工具的引入但仍然没有摆脱通过对时间序列的分析来研究经济管理统的传统方式。
最近十年来,基于主体建模的系统仿真技术逐渐被应用到社会经济研究中。这种建模技术是在微观层次上构建个体行为者,进而由微观个体行为的推导出宏观效应,是一种自下而上的研究方法。而复杂适应系统理论正是这种建模技术的理论基础。复杂适应系统理论的最基本的概念是具有适应能力的、主动的个体,简称主体。这种主体在与环境的交互作用中遵循一般的刺激—反应模型,所谓适应能力表现在它能够根据行为的效果修改自己的行为规则,以便更好地在客观环境中生存。复杂适应系统理论的核心思想就是,适应性造就复杂性。因此,对复杂系统的研究,可以从构成该系统的适应性主体开始。
少数派博弈(Minority Game)模型在建立复杂适应系统的基于主体的仿真模型方面具有重要的应用。模型描述奇数个主体的动态博弈行为,主体在每个时刻都根据以往的公共历史信息各自独立的决定站在甲方或乙方,最终人数较少的一方获胜。虽然少数派模型非常简单,但它却抓住了社会经济系统中有限理性的个体对有限资源争夺的本质特征。社会经济系统是公认的复杂系统。应用复杂适应系统的思想对社会经济系统进行研究是基于这样一个大胆的假设:社会经济系统复杂行为建立在某些简单的规则之上,人们可以通过这些简单规则的长期相互作用来揭示它的复杂行为。
本文基于基本的少数派博弈模型构建了一个仿真的股票市场,并赋予了各主体相互模仿的投机行为。研究了在股票市场中模仿行为在交易者之间的流行程度对市场宏观层次上如效率和风险等因素的影响。其主要创新之处在于使用主体仿真建模的方法,在少数派博弈模型中引入了投资者的模仿行为,使用RePast平台予以实现并进行了分析研究。
The economic system is a complex adaptive system, it makes up with many agents in different hierarchies affecting, learning and competing each other. The complexity of the economic system leads to many nonlinear phenomena that cannot be explained by traditional econometrics. The emerging of new theories brings new tools for economics study, such as fractal, chaos and nonlinear dynamics, etc. But these tools do not break up the traditional pattern, which studies economics by analyzing time series.
In the last decades, the agent-base modeling was introduced into economic studies. Agent-base modeling builds agents in the micro level to conclude macro effects, which is a bottom-up method. And the Complex Adaptive System (CAS) theory is its theoretical foundation. The basic concept of CAS is adaptive agent. Agent follows IF-THEN method when interacting with environment. Adaptive agent changes its behavior according the effectiveness of acts to survive. The core idea of CAS theory is: adaptability makes complexity. Therefore, to study complex system can start with adaptive agents that consist it.
The Minority Game (MG) model has an important application in the agent-based modeling of complex adaptive system. MG model describes a continuous game of agents with odd numbers, at each time step an agent much choose to stand one of two sides and the minority group win. Since MG model is very easy to understand, it masters the key feature of agents with boundary rationality contesting for limit resources in economic society. It is recognized that economic society is a complex system. The application of CAS theory in the study of economic society is based on the courageous hypothesis: The complexity of economic society is emerged from some simple rules, so we can study the long-running interaction of these rules to reveal the complexity.
We build a simulation stock market based on MG model in this paper. And each agent has the capability to imitate others behaviors. Also we study the relationship of the popularity of imitation behavior and the macro factors of stock market such as effectiveness and risk. The creativity of this paper is to simulate the imitation behavior based on MG model using agent-based modeling method, and implement and analyze it with the RePast simulation toolkit.
最近十年来,基于主体建模的系统仿真技术逐渐被应用到社会经济研究中。这种建模技术是在微观层次上构建个体行为者,进而由微观个体行为的推导出宏观效应,是一种自下而上的研究方法。而复杂适应系统理论正是这种建模技术的理论基础。复杂适应系统理论的最基本的概念是具有适应能力的、主动的个体,简称主体。这种主体在与环境的交互作用中遵循一般的刺激—反应模型,所谓适应能力表现在它能够根据行为的效果修改自己的行为规则,以便更好地在客观环境中生存。复杂适应系统理论的核心思想就是,适应性造就复杂性。因此,对复杂系统的研究,可以从构成该系统的适应性主体开始。
少数派博弈(Minority Game)模型在建立复杂适应系统的基于主体的仿真模型方面具有重要的应用。模型描述奇数个主体的动态博弈行为,主体在每个时刻都根据以往的公共历史信息各自独立的决定站在甲方或乙方,最终人数较少的一方获胜。虽然少数派模型非常简单,但它却抓住了社会经济系统中有限理性的个体对有限资源争夺的本质特征。社会经济系统是公认的复杂系统。应用复杂适应系统的思想对社会经济系统进行研究是基于这样一个大胆的假设:社会经济系统复杂行为建立在某些简单的规则之上,人们可以通过这些简单规则的长期相互作用来揭示它的复杂行为。
本文基于基本的少数派博弈模型构建了一个仿真的股票市场,并赋予了各主体相互模仿的投机行为。研究了在股票市场中模仿行为在交易者之间的流行程度对市场宏观层次上如效率和风险等因素的影响。其主要创新之处在于使用主体仿真建模的方法,在少数派博弈模型中引入了投资者的模仿行为,使用RePast平台予以实现并进行了分析研究。
The economic system is a complex adaptive system, it makes up with many agents in different hierarchies affecting, learning and competing each other. The complexity of the economic system leads to many nonlinear phenomena that cannot be explained by traditional econometrics. The emerging of new theories brings new tools for economics study, such as fractal, chaos and nonlinear dynamics, etc. But these tools do not break up the traditional pattern, which studies economics by analyzing time series.
In the last decades, the agent-base modeling was introduced into economic studies. Agent-base modeling builds agents in the micro level to conclude macro effects, which is a bottom-up method. And the Complex Adaptive System (CAS) theory is its theoretical foundation. The basic concept of CAS is adaptive agent. Agent follows IF-THEN method when interacting with environment. Adaptive agent changes its behavior according the effectiveness of acts to survive. The core idea of CAS theory is: adaptability makes complexity. Therefore, to study complex system can start with adaptive agents that consist it.
The Minority Game (MG) model has an important application in the agent-based modeling of complex adaptive system. MG model describes a continuous game of agents with odd numbers, at each time step an agent much choose to stand one of two sides and the minority group win. Since MG model is very easy to understand, it masters the key feature of agents with boundary rationality contesting for limit resources in economic society. It is recognized that economic society is a complex system. The application of CAS theory in the study of economic society is based on the courageous hypothesis: The complexity of economic society is emerged from some simple rules, so we can study the long-running interaction of these rules to reveal the complexity.
We build a simulation stock market based on MG model in this paper. And each agent has the capability to imitate others behaviors. Also we study the relationship of the popularity of imitation behavior and the macro factors of stock market such as effectiveness and risk. The creativity of this paper is to simulate the imitation behavior based on MG model using agent-based modeling method, and implement and analyze it with the RePast simulation toolkit.
引文
[1] 约翰?H?霍兰著.周小牧等译.隐秩序,适应性造就复杂性.上海.上海上海科技教育出版社.2000年8月第一版
[2] 许国志等.系统科学.第1版.上海:上海科技教育出版社. 2000
[3] http://www.econ.iastate.edu/tesfatsi/ace.htm
[4] (英)斯密(Smith.A.)著.杨敬年译.国富论.陕西:陕西人民出版社.2002-12
[5] F. A. Hayek. The use of knowledge in society. The American Economic Review. 1945-09. XXXV(4):519-530
[6] N.J.Vriend.Was Hayek an ace? Southern Economic Journal.2002.68(4):811-840
[7] Tesfatsion,L.Agent-Based Computation Economics.ISU Economics Working Paper No.1.Revised Auguest 24,2003
[8] Axerlrod R.Advancing the Art of Simulation in the Social Sciences.In R.Conte, R.Hegselmann and P.Terna(eds.), Simulating Social Phenomena.1997.Springer,Berlin
[9] Schelling,T.C.Micromotives and macrobehavior.New York,NY: W.W.Norton and Co.1978.
[10] Poundstone W.The revursive universe.Contemporary Books. Chicago, IL.1985
[11] Sargent T.J.Bounded Ration in Macroeconomics.Oxford:Oxford Clarendon Press.1993
[12] Freeman R.War of the models: Which labour market institutions for the 21st century?.Labour Economics.1998.No.5
[13] Forester J.System Dynamics and the Lessons of 35 Years.The Systemic Basis of Policy Making in the 1990s.In K.De Greene(ed.).1991.MIT Press,Cambridge,Ma
[14] Ostrom T.Computer simulation: the third symbol system. Journal of Experimental Social Psychology.1988.No.4
[15] Gilbert N.,P.Terna.How to build and use agent-based models in social science. Mind& Society.2000.No.1
[16] Kirman A.Market Organization and Individual Behaviour: Evidence from Fish markets. In J.Rauch and A. Casella(eds.).2001.Networks and Markets.Russel Sage Foundation. New York
[17] Axtell R.Why Agents? On the Varied Motivations for Agent Computing in the Social Sciences. N Proceedings of the Workshop on Agent Simulation: Applications, Models and Tools.Argonne National Laboratory.2000.IL
[18] Leijonhufvud A.Towards a not-too-rational macroeconomics.Southern Economic Journal.1993.Vol.50.No.1
[19] Conlisk J.Why Bounded Rationality.Journal of Economic Literature.1996.Vol.34.No.2
[20] Tesfatision L.ed.Special Issue on Agent-Based Computational Economics.Computational Economics.2001.Vol.18.No.1
[21] Tesfatision L.ed..Special Issue on Agent-Based Computational Economics. Journal of Economic Dynamics and Control.2001.Vol.25.No.3-4
[22] Tesfatsion L.ed.Special Issue on Agent-Based Modeling of Evolutionary Economic Systems. IEEE Transactions on Evolutionary Computation.2001.Vol.5.No.5
[23] Tesfatsion L.Preferential partner selection in evolutionary labor markets: A study in agent-based computational economics.In V.W.Porto, N. Saravannan, D. Waagan and A.E. Eiben(eds.). Evolutionary programming VII. Proceedings of the Seventh Annual Conference on Evolutionary Programming.1998.Springer-Verlag,Berlin
[24] Tesfation L.Structure,Behavior,and market power in an evolutionary labor market with adaptive search.Journal of Economic Dynamics and Control.2001.Vol.25
[25] Tassier T.,F.menczer.Emerging small-world referral networks in evolutionary labor markets.IEEE Transactions on Evolutionary Computation.2001.Vol.5
[26] Dawid H.,M. Reimann,B. Bullnheimer.To innovate or not to innovate?.IEEE Transactions on Evolutionary Computation.2001.Vol.5
[27] Axtell R.Non-Cooperative Dynamics of Multi-Agent Teams.in Proceedings of the Second International Joint Conference on Autonomous Agents and Multi Agent System.Volume AAMAS032003.2003.
[28] Axtell R.,R.Florida.The Evolution of Cities: A Microeconomic Explanation of Zipf's Law.The Brookings Institution and Carnegie Mellon University Working Paper Studies. No.52.Princeton University Press, Princeton, N.J.2000.
[29] Gallegati M.,D.Delli Gati,G.Giulioni,A.Palestrini.Financial Fragility, Patterns of Firms' entry and Exit,and Aggregate Dynamics.Journal of Economic Behaviour and Organizations. 2003. Vol.51.No.1
[30] Gallegati M.,D. Delli Gatti,G. Giulioni,C. Di Guilmi.Finanicial Fragility,Industrial Dynamics and Business Fluctuations in an Agent Based Model.The conference Wild@Ace 2003. Turin, Italy.October 3-4,2003
[31] Zhang J.Growing Silicon Valley on a landscape:an agent-based approach to high-tech industrial clusters.Journal of Evolutinary Economics.2003.Vol.13
[32] Pyka A.,P.Windrum.The Self-Orgnaisation of Strategic Alliances.Economics of Innovation and New Technology. 2003.Vol.12.No.3
[33] Lux T.Herd Behavior,Bubbles and Crashes.Economics Journal 1995.vol.105:p881
[34] M.Levy,H.Levy,S.Solomon.A microsopic model of the stock market.Economics Letters.1994.45:103-111
[35] E.C.Zeeman.On the unstable behaviour of stock exchanges.Journal of Mathematical Economics.1974.1:39-49
[36] G.Genotte,H.Leland.Market liquidity,hedging,and crashes.The American Economic Review.1990.80(5):999-1021
[37] 曹文斌,何建敏.供应链管理的多Agent系统模型及其决策方法.系统工程理论方法应用.2001.10(1):32-35.
[38] 李英,刘豹.预测支持系统中的人机界面Agent及其机器学习.系统工程理论与实践.2000.12:73-76
[39] 胡代平,刘豹.多Agent股票预测支持系统的设计.系统工程.2001.19(2):54-57
[40] 胡代平,王浣尘.预测模型Agent的实现方法.系统工程学报.2001.16(5):330-334,381.
[41] 李宏亮.基于Agent的复杂系统分布仿真.长沙:国防科技大学.2000
[42] 刘大海,王治宝,孙洪军,王秀峰.基于多agent的虚拟股市仿真研究. 计算机工程与应用.2003.25
[43] 刘晓峰,刘晓光,田存志.涨、跌停板制度安排对于股市稳定性影响的研究.国际金融研究 2001.9:16-22
[44] W.B.Arthur.,Am.Econ.Assoc. Papers and Proc.84,406(1994).
[45] 邓宏钟,迟妍,谭跃进.复杂经济系统的微观仿真分析.小型微型计算机系统. 2002.12:1506-1509
[46] 郑晓彬,王浣尘.股票市场价格行为的复杂性.上海交通大学学报. 2002.4:581-583
[47] 陈禹. 复杂性研究的新动向——基于主体的建模方法及其启迪.系统 辩证学学报.2003.1:43-50
[48] 李群.贝叶斯金融市场的多代理人仿真研究[学位论文].西安.西安交通大学.2-4
[49] 张维,刘文财,王启文,刘豹. 面向资本市场复杂性建模:基于Agent计算实验金融学. 现代财经. 2003.1:3-8
[50] http://www.swarm.org
[51] http://repast.sourceforge.net
[52] http://www.econ.iastate.edu/tesfatsi/tnghome.htm
[53] http://www.brook.edu/dybdocroot/Collab/CSED/install.htm
[54] http://www.unifr.ch/econophysics/minority/
[55] 全宏俊,许伯铭,汪秉宏.金融市场中经纪人相互竞争和适应性行为的物理模型探索.物理.2001.10
[56] Challet D., Zhang Y C., 1998. Physica A, 276:264; Phisica A,276: 265-283
[57] Cavagna A. 1999 Phys.Rev.E, 59:R3783-R3786
[58] Johnson N F, Hui P M,Zheng D, et al. 1999, J.Phys.A: Math.Gen., 32: L 427-L431
[59] (美)滋维?博迪 亚历克斯?凯恩,艾伦 J?马库斯著,朱宝宪,吴洪 赵冬青等译.投资学.北京:机械工业出版社,2000
[60] 张昀.生物进化.北京:北京大学出版社.2003.5
[61] (英)特伦斯.C.米尔斯著,余卓菁译.金融时间序列的经济计量学模型.北京:经济科学出版社.2002.7
[62] (美)埃德加.E.彼得斯著,储海林,殷勤译.分形市场分析──将混沌理论应用到投资于经济理论上. 北京:经济科学出版社.2002.7
[63] 李水根,吴纪桃.分形与小波.北京:科学出版社.2002.10
[64] 徐龙炳.中国股票市场股票收益稳态特性.金融研究.2001.6
[65] Shiller.Market volatility.Cambridge,MA:MIT Press.1989
[66] Günter Bamberg,Gregor Dorfleitner.Fat Tails and Traditional Capital Market Theory.Working paper,Uinversity Augsburg.2001.8
[67] Devajyoti Ghose,Kenneth F. Kroner.The relationship between GARCH and symmetric stable processes: Finding the source of fat tails in financial data.Journal of Empirical Finance.1995.2:225-251
[68] Rama Cont,Jean-Philippe Bouchaud.Herd behavior and aggregate fluctuations in financial markets.[Dissertation].Université de Paris
[69] Granovetter, M. & Soong, R.Threshold models of diffusion and collective behavior.Journal of Mathematical Society,9:165-179
[2] 许国志等.系统科学.第1版.上海:上海科技教育出版社. 2000
[3] http://www.econ.iastate.edu/tesfatsi/ace.htm
[4] (英)斯密(Smith.A.)著.杨敬年译.国富论.陕西:陕西人民出版社.2002-12
[5] F. A. Hayek. The use of knowledge in society. The American Economic Review. 1945-09. XXXV(4):519-530
[6] N.J.Vriend.Was Hayek an ace? Southern Economic Journal.2002.68(4):811-840
[7] Tesfatsion,L.Agent-Based Computation Economics.ISU Economics Working Paper No.1.Revised Auguest 24,2003
[8] Axerlrod R.Advancing the Art of Simulation in the Social Sciences.In R.Conte, R.Hegselmann and P.Terna(eds.), Simulating Social Phenomena.1997.Springer,Berlin
[9] Schelling,T.C.Micromotives and macrobehavior.New York,NY: W.W.Norton and Co.1978.
[10] Poundstone W.The revursive universe.Contemporary Books. Chicago, IL.1985
[11] Sargent T.J.Bounded Ration in Macroeconomics.Oxford:Oxford Clarendon Press.1993
[12] Freeman R.War of the models: Which labour market institutions for the 21st century?.Labour Economics.1998.No.5
[13] Forester J.System Dynamics and the Lessons of 35 Years.The Systemic Basis of Policy Making in the 1990s.In K.De Greene(ed.).1991.MIT Press,Cambridge,Ma
[14] Ostrom T.Computer simulation: the third symbol system. Journal of Experimental Social Psychology.1988.No.4
[15] Gilbert N.,P.Terna.How to build and use agent-based models in social science. Mind& Society.2000.No.1
[16] Kirman A.Market Organization and Individual Behaviour: Evidence from Fish markets. In J.Rauch and A. Casella(eds.).2001.Networks and Markets.Russel Sage Foundation. New York
[17] Axtell R.Why Agents? On the Varied Motivations for Agent Computing in the Social Sciences. N Proceedings of the Workshop on Agent Simulation: Applications, Models and Tools.Argonne National Laboratory.2000.IL
[18] Leijonhufvud A.Towards a not-too-rational macroeconomics.Southern Economic Journal.1993.Vol.50.No.1
[19] Conlisk J.Why Bounded Rationality.Journal of Economic Literature.1996.Vol.34.No.2
[20] Tesfatision L.ed.Special Issue on Agent-Based Computational Economics.Computational Economics.2001.Vol.18.No.1
[21] Tesfatision L.ed..Special Issue on Agent-Based Computational Economics. Journal of Economic Dynamics and Control.2001.Vol.25.No.3-4
[22] Tesfatsion L.ed.Special Issue on Agent-Based Modeling of Evolutionary Economic Systems. IEEE Transactions on Evolutionary Computation.2001.Vol.5.No.5
[23] Tesfatsion L.Preferential partner selection in evolutionary labor markets: A study in agent-based computational economics.In V.W.Porto, N. Saravannan, D. Waagan and A.E. Eiben(eds.). Evolutionary programming VII. Proceedings of the Seventh Annual Conference on Evolutionary Programming.1998.Springer-Verlag,Berlin
[24] Tesfation L.Structure,Behavior,and market power in an evolutionary labor market with adaptive search.Journal of Economic Dynamics and Control.2001.Vol.25
[25] Tassier T.,F.menczer.Emerging small-world referral networks in evolutionary labor markets.IEEE Transactions on Evolutionary Computation.2001.Vol.5
[26] Dawid H.,M. Reimann,B. Bullnheimer.To innovate or not to innovate?.IEEE Transactions on Evolutionary Computation.2001.Vol.5
[27] Axtell R.Non-Cooperative Dynamics of Multi-Agent Teams.in Proceedings of the Second International Joint Conference on Autonomous Agents and Multi Agent System.Volume AAMAS032003.2003.
[28] Axtell R.,R.Florida.The Evolution of Cities: A Microeconomic Explanation of Zipf's Law.The Brookings Institution and Carnegie Mellon University Working Paper Studies. No.52.Princeton University Press, Princeton, N.J.2000.
[29] Gallegati M.,D.Delli Gati,G.Giulioni,A.Palestrini.Financial Fragility, Patterns of Firms' entry and Exit,and Aggregate Dynamics.Journal of Economic Behaviour and Organizations. 2003. Vol.51.No.1
[30] Gallegati M.,D. Delli Gatti,G. Giulioni,C. Di Guilmi.Finanicial Fragility,Industrial Dynamics and Business Fluctuations in an Agent Based Model.The conference Wild@Ace 2003. Turin, Italy.October 3-4,2003
[31] Zhang J.Growing Silicon Valley on a landscape:an agent-based approach to high-tech industrial clusters.Journal of Evolutinary Economics.2003.Vol.13
[32] Pyka A.,P.Windrum.The Self-Orgnaisation of Strategic Alliances.Economics of Innovation and New Technology. 2003.Vol.12.No.3
[33] Lux T.Herd Behavior,Bubbles and Crashes.Economics Journal 1995.vol.105:p881
[34] M.Levy,H.Levy,S.Solomon.A microsopic model of the stock market.Economics Letters.1994.45:103-111
[35] E.C.Zeeman.On the unstable behaviour of stock exchanges.Journal of Mathematical Economics.1974.1:39-49
[36] G.Genotte,H.Leland.Market liquidity,hedging,and crashes.The American Economic Review.1990.80(5):999-1021
[37] 曹文斌,何建敏.供应链管理的多Agent系统模型及其决策方法.系统工程理论方法应用.2001.10(1):32-35.
[38] 李英,刘豹.预测支持系统中的人机界面Agent及其机器学习.系统工程理论与实践.2000.12:73-76
[39] 胡代平,刘豹.多Agent股票预测支持系统的设计.系统工程.2001.19(2):54-57
[40] 胡代平,王浣尘.预测模型Agent的实现方法.系统工程学报.2001.16(5):330-334,381.
[41] 李宏亮.基于Agent的复杂系统分布仿真.长沙:国防科技大学.2000
[42] 刘大海,王治宝,孙洪军,王秀峰.基于多agent的虚拟股市仿真研究. 计算机工程与应用.2003.25
[43] 刘晓峰,刘晓光,田存志.涨、跌停板制度安排对于股市稳定性影响的研究.国际金融研究 2001.9:16-22
[44] W.B.Arthur.,Am.Econ.Assoc. Papers and Proc.84,406(1994).
[45] 邓宏钟,迟妍,谭跃进.复杂经济系统的微观仿真分析.小型微型计算机系统. 2002.12:1506-1509
[46] 郑晓彬,王浣尘.股票市场价格行为的复杂性.上海交通大学学报. 2002.4:581-583
[47] 陈禹. 复杂性研究的新动向——基于主体的建模方法及其启迪.系统 辩证学学报.2003.1:43-50
[48] 李群.贝叶斯金融市场的多代理人仿真研究[学位论文].西安.西安交通大学.2-4
[49] 张维,刘文财,王启文,刘豹. 面向资本市场复杂性建模:基于Agent计算实验金融学. 现代财经. 2003.1:3-8
[50] http://www.swarm.org
[51] http://repast.sourceforge.net
[52] http://www.econ.iastate.edu/tesfatsi/tnghome.htm
[53] http://www.brook.edu/dybdocroot/Collab/CSED/install.htm
[54] http://www.unifr.ch/econophysics/minority/
[55] 全宏俊,许伯铭,汪秉宏.金融市场中经纪人相互竞争和适应性行为的物理模型探索.物理.2001.10
[56] Challet D., Zhang Y C., 1998. Physica A, 276:264; Phisica A,276: 265-283
[57] Cavagna A. 1999 Phys.Rev.E, 59:R3783-R3786
[58] Johnson N F, Hui P M,Zheng D, et al. 1999, J.Phys.A: Math.Gen., 32: L 427-L431
[59] (美)滋维?博迪 亚历克斯?凯恩,艾伦 J?马库斯著,朱宝宪,吴洪 赵冬青等译.投资学.北京:机械工业出版社,2000
[60] 张昀.生物进化.北京:北京大学出版社.2003.5
[61] (英)特伦斯.C.米尔斯著,余卓菁译.金融时间序列的经济计量学模型.北京:经济科学出版社.2002.7
[62] (美)埃德加.E.彼得斯著,储海林,殷勤译.分形市场分析──将混沌理论应用到投资于经济理论上. 北京:经济科学出版社.2002.7
[63] 李水根,吴纪桃.分形与小波.北京:科学出版社.2002.10
[64] 徐龙炳.中国股票市场股票收益稳态特性.金融研究.2001.6
[65] Shiller.Market volatility.Cambridge,MA:MIT Press.1989
[66] Günter Bamberg,Gregor Dorfleitner.Fat Tails and Traditional Capital Market Theory.Working paper,Uinversity Augsburg.2001.8
[67] Devajyoti Ghose,Kenneth F. Kroner.The relationship between GARCH and symmetric stable processes: Finding the source of fat tails in financial data.Journal of Empirical Finance.1995.2:225-251
[68] Rama Cont,Jean-Philippe Bouchaud.Herd behavior and aggregate fluctuations in financial markets.[Dissertation].Université de Paris
[69] Granovetter, M. & Soong, R.Threshold models of diffusion and collective behavior.Journal of Mathematical Society,9:165-179