金融市场高维交叉关联矩阵结构演化分析
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摘要
金融时间序列的交叉关联矩阵(协方差矩阵)作为现代投资理论的基石,其研究在经济金融学中具有相当长时间的历史。传统的计量经济学研究主要关注于如何构造更有效的投资组合模型,而对交叉相关矩阵本身的性质却关注不足。计量经济模型构建时通常只需要计算少数几个时间序列的交叉关联矩阵。一个由某个市场所有股票价格时间序列计算得到的高维交叉关联矩阵用处甚微,但这个矩阵却可能包含了与这个市场结构特征紧密相关的关键信息。传统的计量经济学研究方法无法处理这个高维矩阵,因此我们从复杂性科学出发,用金融物理学和复杂网络的理论和方法对高维交叉关联矩阵的结构演化特征进行深入数据挖掘。
本文在介绍金融物理学、复杂网络这两个重要的复杂性科学研究方向之后,分别采取最小生成树、随机矩阵以及复杂网络社团划分这三种研究方法,同时对CSM市场249支股票构成的交叉关联矩阵以及NYSE市场259支股票构成的交叉关联矩阵展开分析。我们首先用1997年至2007(2008)年将近10年的股票价格数据计算静态交叉关联矩阵分析其长期稳定结构特征,然后将这10年的数据分成若干段,每段数据包含500个交易日,两段相邻数据有20个交易日的滚动,根据这若干段滚动的数据计算动态交叉关联矩阵并分析其动态演化特征。
三种不同的研究方法的实证结果得到一致的结论,即发现CSM市场的股票算法分类与标准市场板块分类不对应,而NYSE市场的股票算法分类与标准板块分类较为吻合。就其差异性的原因,本文给出两个经济含义解释:首先是两个市场投资者投资决策的关注点存在差异,中国市场的投资者侧重于关注上市公司业绩的好坏,而美国市场的投资者侧重于对上市公司所处行业前景的判断。其次是两个市场的基本面存在差异,中国市场是以非日常消费品、工业等行业为代表的第二产业为支柱产业,美国市场是以金融行业为代表的第三产业是NYSE市场的支柱产业。
本文的分析方法和研究结论将有助于加深人们对中美市场差异性的认识,为人们进行市场宏观分析、投资组合构建等提供有效的参考。
As the blockscone of modern investment theory, the research of the equal-time cross correlation matrix (CCM or covariance matrix) of financial time series has a long history. The traditional econometrics mainly focused on how to construct an effective investment portfolio but without any concerning about the basic properties of itself. While constructing the investment portfolio in such research, only a few (always less than 20) financial assets are considered. However, a high-dimension correlation matrix that is computed by a large number of the prices time series in the same market may hide some key information about the structure characteristics of this market. The methodology of the traditional econometrics cannot deal with such problem, which means we have to try some new. In this paper, I try to talk about this problem from the view of complexity science by using the mature theory and methodology in econophysics and complex network to analyze the structure and evolution properties of CCM.
After introducing the research progress in econophysics and complex network, I use three methods: the minimal spanning tree, the random matrix theory and community detection to simultaneously survey the CCM of Chinese stock market (CSM) and New York stock market (NYSE), which respectively contains 249 stocks and 259 stocks. The financial time series I collected are from the year 1997 to 2007(8). Firstly, I use the total data to construct the Static Cross-Correlation Matrix (S-CCM) to investigate the long term stable structure properties. Secondly, I split the ten years data into several small parts, each contains 500 prices, and two contiguous parts just have the difference of only 20 prices. Then we use these divided data to construct a series of Dynamics Cross-Correlation Matrix (D-CCM) to investigate the dynamics evolution properties.
The empirical result of the three different research methods shows highly consistency: we find the classification by algorithm does not correspond to the standard industry classification in CSM market, but is almost the same in the NYSE market. We give two financial meaning for the differences between these two markets: Firstly, Investors concern different information in the two markets. Chinese investors pay more attention on the performance of listed companies over the last few years while U.S. investors focus on the prospects of the industry of the listed companies. Secondly, the fundamental in the two markets is not the same. The secondary industry represented by Consumer goods and Industry is the core business sector of CSM, while in NYSE is the tertiary industry represented by Finance. In CSM, the correlation reflects the Industry chain relationship between manufacturing company, and the classification by community detection doesn’t correspond to the standard industry classification. By contrary in NYSE, the correlation relation mainly reflects the capital chain between financial enterprise, and the classification by community detection corresponds quite well to the standard industry classification.
This paper’s research methodology and conclusion will help people to deepen the understanding of differences between the Chinese and America stock market, providing an effective reference to the market macroscopic problem and investment portfolio.
本文在介绍金融物理学、复杂网络这两个重要的复杂性科学研究方向之后,分别采取最小生成树、随机矩阵以及复杂网络社团划分这三种研究方法,同时对CSM市场249支股票构成的交叉关联矩阵以及NYSE市场259支股票构成的交叉关联矩阵展开分析。我们首先用1997年至2007(2008)年将近10年的股票价格数据计算静态交叉关联矩阵分析其长期稳定结构特征,然后将这10年的数据分成若干段,每段数据包含500个交易日,两段相邻数据有20个交易日的滚动,根据这若干段滚动的数据计算动态交叉关联矩阵并分析其动态演化特征。
三种不同的研究方法的实证结果得到一致的结论,即发现CSM市场的股票算法分类与标准市场板块分类不对应,而NYSE市场的股票算法分类与标准板块分类较为吻合。就其差异性的原因,本文给出两个经济含义解释:首先是两个市场投资者投资决策的关注点存在差异,中国市场的投资者侧重于关注上市公司业绩的好坏,而美国市场的投资者侧重于对上市公司所处行业前景的判断。其次是两个市场的基本面存在差异,中国市场是以非日常消费品、工业等行业为代表的第二产业为支柱产业,美国市场是以金融行业为代表的第三产业是NYSE市场的支柱产业。
本文的分析方法和研究结论将有助于加深人们对中美市场差异性的认识,为人们进行市场宏观分析、投资组合构建等提供有效的参考。
As the blockscone of modern investment theory, the research of the equal-time cross correlation matrix (CCM or covariance matrix) of financial time series has a long history. The traditional econometrics mainly focused on how to construct an effective investment portfolio but without any concerning about the basic properties of itself. While constructing the investment portfolio in such research, only a few (always less than 20) financial assets are considered. However, a high-dimension correlation matrix that is computed by a large number of the prices time series in the same market may hide some key information about the structure characteristics of this market. The methodology of the traditional econometrics cannot deal with such problem, which means we have to try some new. In this paper, I try to talk about this problem from the view of complexity science by using the mature theory and methodology in econophysics and complex network to analyze the structure and evolution properties of CCM.
After introducing the research progress in econophysics and complex network, I use three methods: the minimal spanning tree, the random matrix theory and community detection to simultaneously survey the CCM of Chinese stock market (CSM) and New York stock market (NYSE), which respectively contains 249 stocks and 259 stocks. The financial time series I collected are from the year 1997 to 2007(8). Firstly, I use the total data to construct the Static Cross-Correlation Matrix (S-CCM) to investigate the long term stable structure properties. Secondly, I split the ten years data into several small parts, each contains 500 prices, and two contiguous parts just have the difference of only 20 prices. Then we use these divided data to construct a series of Dynamics Cross-Correlation Matrix (D-CCM) to investigate the dynamics evolution properties.
The empirical result of the three different research methods shows highly consistency: we find the classification by algorithm does not correspond to the standard industry classification in CSM market, but is almost the same in the NYSE market. We give two financial meaning for the differences between these two markets: Firstly, Investors concern different information in the two markets. Chinese investors pay more attention on the performance of listed companies over the last few years while U.S. investors focus on the prospects of the industry of the listed companies. Secondly, the fundamental in the two markets is not the same. The secondary industry represented by Consumer goods and Industry is the core business sector of CSM, while in NYSE is the tertiary industry represented by Finance. In CSM, the correlation reflects the Industry chain relationship between manufacturing company, and the classification by community detection doesn’t correspond to the standard industry classification. By contrary in NYSE, the correlation relation mainly reflects the capital chain between financial enterprise, and the classification by community detection corresponds quite well to the standard industry classification.
This paper’s research methodology and conclusion will help people to deepen the understanding of differences between the Chinese and America stock market, providing an effective reference to the market macroscopic problem and investment portfolio.
引文
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[30] J.Shen, B. Zheng. On return-volatility correlation in financial dynamics. Euro. Phys. Let., 2009, 88:28003
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[41] F.Schweitzer, G.Fagiolo, D.Sornette, F.V.Redondo, A. Vespignani, D.R.White. Economic Networks: The new Challenges. Science., 2009, 325(24):422-425
[42] T.Y. Choi, K.J. Dooley, M. Rungtusanatham. Supply networks and complex adaptive systems: control versus emergence. J. Opera. Mana., 2001, 19:351-366
[43] W.W.Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research., 1977, 33:452-473
[44] B.W. Kernighan, S. Lin. An efficient heuristic procedure for partitioning graphs. Bell System Technical Journal., 1970,49:291-307
[45] M. Girvan, M.E.J. Newman. Community structure in social and biological networks. Proc Natl Acad Sci., 2002, 99:7821-7826
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[2] R.F.Engle. Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation. Econometrica, 1982, 6:987-1007
[3] T.Bollerslev, R.Y.Chou, K.F.Kroner. ARCH Modeling in Finance. The Journal of Econometrics, 1992, 52:5-59
[4] I.T.Jolliffe. Principal Component Analysis(Springer Series in Statistics). Berlin:Springer, 1986
[5] L. Bauwens, S. Laurent. Multivariate GARCH models: a survey Journal of Applied Econometrics. 2006, 21:79-109
[6] C.H. Papadimitrou, K.Steiglitz. Combinatorial Optimization: algorithms and complexity. Prentice-Hall, 1982
[7] A.M.Sengupta, P.P.Mitra. Distributions of singular values for some random matrices. Phys. Rev. E, 1999, 60:3389-3392
[8] A.L.Barabasi, R.Albert. Emergence of Scaling in Random Networks. Nature,1999, 286(5439):509-512
[9] R.N.Mantegna, Hierarchical structure in financial markets. Eur.Phys. J. B., 1999, 11: 193-197
[10] R.N.Mantegna, H.E.Stanley. An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge: Cambridge University Press, 1999
[11] N.Vandewalle, F.Brisbois, X.Tordoir. Self-organized critical topology of stock markets. Physics A., 2000, 271:3-4
[12] G. Bonanno, F. Lillo, R.N. Mantegna. High-frequency Cross-correlation in a Set of Stocks. Quantitative Finance, 2001, 11:96-104
[13] G. Bonanno, G.Galdarelli, F.Lillo, S. Micciche, N.Vandewalle, R.N.Mantegna. Networks of equities in financial market. Eur. Phys. J. B., 2004, 38(2): 363-371
[14] J.P.Onnela, A.Chakraborti, K.Kaski, J.Kertész, A.Kanto. Dynamics of market correlations: Taxonomy and portfolio analysis. Phys. Rev. E, 2003, 68(5):056110
[15] J.P.Onnela, A.Chakraborti, K.Kaski and J.Kertész. Dynamics asset trees and Black Monday. Physica A, 2003, 324:247-252
[16] V.A. Mar?enko, L.A.Pastur. Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sbornik, 1967, 1(4):457-483
[17] L.Laloux, P.Cizeau, J.P.Bouchaud, M.Potters. Noise Dressing of Financial Correlation Matrices. Phys.Rev.Lett. 1999, 83:1467
[18] L. Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters. Random matrix theory and financial correlations. Int. J. Theor. Appl. Finance.,2000, 3:391
[19] V.Plerou, P.Gopikrishnan, B.Rosenow, L.A.N.Amaral, H.E.Stanley. Universal and Nonuniversial Properties of Cross Correlations in Financial Time Series. Phys. Rev. L. 1999,83(7)
[20] V.Plerou, P.Gopikrishnan, B.Rosenow, L.A.N. Amaral, T. Guhr, H.E.Stanley. Random matrix approach to cross correlations in financial data. Phys. Rev. E., 2002, 65:066126
[21] P.Gopikrishnan, B. Rosenow, V.Plerou, H.E.Stanley. Quantifying and interpreting collective behavior in financial markets. Phys. Rev. E. 2001, 64:035106
[22] J. Shen, B. Zheng. Cross-correlation in financial dynamics. Europhys. Letts., 2009, 86:28003
[23]郑波.金融动力学的时空关联与大波动特性—兼谈中西方金融市场的对比研究.物理., 2010, 39(2):95-100
[24] J.P.Bouchaud, M. Potters. Financial Applications of Random Matrix Theory: a short review. 2009. arXiv:0910.1205v1
[25] J. P. Onnela, A. Chakraborti, K. Kaski, J Kertész, A. Kanto. Asset Trees and Asset Graphs in Financial Markets. Phys. Scr. T. 2003, 106(48)
[26] J. P. Onnela, K. Kaski, J. Kertész. Clustering and information in correlation based financial networks. Eur. Phys. J. B., 2004, 38:353–362
[27] J. P. Onnela, J. Saram?ki, J. Kertész, K. Kaski. Intensity and coherence of motifs in weighted complex networks. Phys. Rev. E., 2005, 71:065103(R)
[28] T. Qiu, B. Zheng, G. Chen. Financial networks with static and dynamics thresholds. Eur. Phy. J. B., 2010, 12:043057
[29] R.N.Mantegna, H.E.Stanley. Scaling behavior in the dynamics of an economic index. Nature., 1995, 376(6):46-49
[30] J.Shen, B. Zheng. On return-volatility correlation in financial dynamics. Euro. Phys. Let., 2009, 88:28003
[31] T.Qiu, B.Zheng, F.Ren, S.Trimper. Return-volatility correlation in financial dynamics. Phy. Rev. E., 2006, 73:065103(R)
[32] A.L. Barabási, R. Albert. Emergence of scaling in random networks. Science., 286:509-512
[33] R. Albert, A. L. Barabási. Statistical mechanics of complex networks. Rev. Mod. Phys., 2002, 74:47-97
[34] G. Caldarelli, R. Marchetti, L. Pietronero. The Fractal Properties of Internet. Euro. Phys. Lett., 2000, 52:386-391
[35] S. Wasserman, K. Faust. Social Network Analysis. Cambridge: Cambridge Univ. Press, U.K., 1994
[36] J. Scott, Social Network Analysis: A Handbook. London, 2nd Ed., 2002
[37] R.J. Williams, N.D. Martinez. Simple rules yield complex food webs. Nature. 2000, 404:180-183
[38] M. Faloutsos, P. Faloutsos, C. Floutsos. On the power-law relationships of the Internet topology. Comput. Commun. Rev., 1999, 29:251-262
[39] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomins, J. Wiener. Graph structure in the Web. Comput. Networks., 33:309-320
[40] M.E.J. Newman. The Structure of scientific collaboration networks. Proc. Natl. Acad. Sci., 2001,98(2):404-409
[41] F.Schweitzer, G.Fagiolo, D.Sornette, F.V.Redondo, A. Vespignani, D.R.White. Economic Networks: The new Challenges. Science., 2009, 325(24):422-425
[42] T.Y. Choi, K.J. Dooley, M. Rungtusanatham. Supply networks and complex adaptive systems: control versus emergence. J. Opera. Mana., 2001, 19:351-366
[43] W.W.Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research., 1977, 33:452-473
[44] B.W. Kernighan, S. Lin. An efficient heuristic procedure for partitioning graphs. Bell System Technical Journal., 1970,49:291-307
[45] M. Girvan, M.E.J. Newman. Community structure in social and biological networks. Proc Natl Acad Sci., 2002, 99:7821-7826
[46] L. Donetti, M. A. Mu?oz. Detecting network communities: a new systematic and efficient algorithm. J.Stat. Mech., 2004, p10012
[47] A. Pothen, H. Simon, K.P. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal, Appl., 1990, 11:430
[48] J. Reichardt, S. Bornholdt. Phys. Rev. Lett. 2004, 93: 218701
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