Plerou Vasiliki, Gopikrishnan Parameswaran, Rosenow Bernd, Amaral Luís A Nunes, Guhr Thomas, Stanley H Eugene
Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Jun;65(6 Pt 2):066126. doi: 10.1103/PhysRevE.65.066126. Epub 2002 Jun 27.
We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis"--a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.
我们使用随机矩阵理论（RMT）的方法分析不同股票价格波动之间的交叉相关性。利用两个大型数据库，我们计算了由以下数据构建的收益交叉相关矩阵C：（i）1994 - 1995年两年期间1000只美国股票的30分钟收益；（ii）1996 - 1997年两年期间881只美国股票的30分钟收益；（iii）1962 - 1996年35年期间422只美国股票的日收益。我们针对一个“零假设”——由相互不相关的时间序列构建的随机相关矩阵，检验C的特征值λ(i)的统计量。我们发现，C的大多数特征值落在随机相关矩阵特征值的RMT界限[λ(-),λ(+)]内。我们在RMT界限内检验C的特征值以探究随机矩阵的普遍性质，并发现与随机矩阵的高斯正交系综的结果高度吻合——这意味着所测量的交叉相关系数具有很大程度的随机性。此外，我们发现，对应于RMT界限之外特征值的特征向量的特征向量分量分布与RMT预测存在系统性偏差。另外，我们发现这些“偏离特征向量”在时间上是稳定的。我们分析了偏离特征向量的分量，发现最大特征值对应于所有股票共有的一种影响。我们对其余偏离特征向量的分析显示出不同的组，其特征与传统上确定的商业部门相对应。最后，我们讨论了在构建风险与回报比率稳定的股票投资组合方面的应用。
