Serial Correlation, Periodicity and Scaling of Eigenmodes in an Emerging Market

We investigate serial correlation, periodic, aperiodic and scaling behavior of eigenmodes, i.e., daily price fluctuation time-series derived from eigenvectors, of correlation matrices of shares listed on the Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. Periodic, or calendar, components are detected by spectral analysis. We find that calendar effects are limited to eigenmodes which correspond to eigenvalues outside the Wishart range. Using a variance ratio test, we uncover serial correlation in the first eigenmodes and find slight negative serial correlation for eigenmodes within the Wishart range. Our spectral analysis and variance ratio investigations suggest that interpolating missing data or illiquid trading days with zero-order hold introduces high frequency noise and spurious serial correlation. Aperiodic and scaling behavior of the eigenmodes are investigated by using rescaled-range (R/S) methods and detrended fluctuation analysis (DFA). We find that DFA and classic and modified R/S exponents suggest the presence of long-term memory effects in the first five eigenmodes.

[1]  P. Cizeau,et al.  Statistical properties of the volatility of price fluctuations. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Bernd Rosenow,et al.  Portfolio optimization and the random magnet problem , 2002 .

[3]  A. Lo.  Long-Term Memory in Stock Market Prices , 1989 .

[4]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[5]  Jonathan H. Wright.  Alternative Variance-Ratio Tests Using Ranks and Signs , 2000 .

[6]  L. Bachelier,et al.  Théorie de la spéculation , 1900 .

[7]  Rosario N. Mantegna,et al.  Book Review: An Introduction to Econophysics, Correlations, and Complexity in Finance, N. Rosario, H. Mantegna, and H. E. Stanley, Cambridge University Press, Cambridge, 2000. , 2000 .

[8]  Gilbert Colletaz.  A Simple Multiple Variance-Ratio Test Based on Ranks , 2006 .

[9]  Paul H. Cootner.  The random character of stock market prices , 1968 .

[10]  F. Lillo,et al.  The Long Memory of the Efficient Market , 2003, cond-mat/0311053.

[11]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[12]  Sergei Maslov.  Measures of globalization based on cross-correlations of world financial indices , 2001 .

[13]  Jaroslaw Kwapien,et al.  Quantifying the dynamics of financial correlations , 2001 .

[14]  R. Mantegna.  Hierarchical structure in financial markets , 1998, cond-mat/9802256.

[15]  Anirvan M. Sengupta,et al.  Distributions of singular values for some random matrices. , 1997, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  S. Ross.  The arbitrage theory of capital asset pricing , 1976 .

[17]  H. Eugene Stanley,et al.  Dynamics of cross-correlations in the stock market , 2003 .

[18]  S. Ross,et al.  An Empirical Investigation of the Arbitrage Pricing Theory , 1980 .

[19]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[20]  E. Fama.  EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK* , 1970 .

[21]  Z. Burda,et al.  Levy Matrices and Financial Covariances , 2001 .

[22]  W. Willinger,et al.  ESTIMATORS FOR LONG-RANGE DEPENDENCE: AN EMPIRICAL STUDY , 1995 .

[23]  S. Havlin,et al.  Detecting long-range correlations with detrended fluctuation analysis , 2001, cond-mat/0102214.

[24]  M Marsili,et al.  Data clustering and noise undressing of correlation matrices. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[26]  J. Bouchaud,et al.  Theory of financial risks : from statistical physics to risk management , 2000 .

[27]  D. Wilcox,et al.  On the analysis of cross-correlations in South African market data , 2004 .

[28]  P. A. Mello,et al.  Random matrix physics: Spectrum and strength fluctuations , 1981 .

[29]  A. Lo,et al.  Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test , 1987 .

[30]  J. Bouchaud,et al.  Rational Decisions, Random Matrices and Spin Glasses , 1998, cond-mat/9801209.

[31]  CAPITALISATION AND WEAK-FORM EFFICIENCY IN THE JSE SECURITIES EXCHANGE , 2005 .

[32]  Walter Willinger,et al.  Stock market prices and long-range dependence , 1999, Finance Stochastics.

[33]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[34]  Maurice G. Kendall,et al.  The Analysis of Economic Time‐Series—Part I: Prices , 1953 .

[35]  K. V. Chow,et al.  A simple multiple variance ratio test , 1993 .

[36]  S. Alfarano.  An agent-based stochastic volatility model , 2006 .

[37]  A. Bowley.  The Analysis of Economic Time Series , 1942, Nature.

[38]  M. Marsili.  Dissecting financial markets: sectors and states , 2002, cond-mat/0207156.

[39]  J. McCauley,et al.  Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance , 2006, cond-mat/0602316.

[40]  Clive W. J. Granger,et al.  Spectral Analysis for Economic Time Series , 1964 .

[41]  A. Iacobucci.  Spectral Analysis for Economic Time Series , 2005 .

[42]  K. Staliūnas.  Spatial and temporal spectra of noise driven stripe patterns. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  W. Willinger,et al.  A critical look at Lo's modified R/S statistic , 1999 .

[44]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[45]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[46]  I. Kondor,et al.  Noisy Covariance Matrices and Portfolio Optimization II , 2002, cond-mat/0205119.

[47]  VARIANCE RATIO TESTS OF THE RANDOM WALK HYPOTHESIS FOR SOUTH AFRICAN STOCK FUTURES , 2006 .

[48]  T. Guhr,et al.  RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS , 1997, cond-mat/9707301.

[49]  M. Gaudin.  Sur la loi limite de l'espacement des valeurs propres d'une matrice ale´atoire , 1961 .

[50]  William A. Brock,et al.  Nonlinear Time Series, Complexity Theory, and Finance , 1995 .

[51]  A. Craig MacKinlay,et al.  Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test , 1988 .

[52]  F. Lillo,et al.  High-frequency cross-correlation in a set of stocks , 2000 .

[53]  The Analysis of Economic Time Series. , 1942 .

[54]  J. R. Wallis,et al.  Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence , 1969 .

[55]  V. Plerou,et al.  Identifying Business Sectors from Stock Price Fluctuations , 2000, cond-mat/0011145.

[56]  V. Plerou,et al.  Random matrix approach to cross correlations in financial data. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  J. Kwapień,et al.  Identifying Complexity by Means of Matrices , 2001, cond-mat/0112271.

[58]  M. L. Mehta,et al.  ON THE STATISTICAL PROPERTIES OF THE LEVEL-SPACINGS IN NUCLEAR SPECTRA , 1960 .

[59]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[60]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[61]  M. Osborne.  Brownian Motion in the Stock Market , 1959 .