Recurrence Quantification Analysis of Wavelet Pre-Filtered Index Returns

In this paper we investigate for the presence of non-stochastic, possibly nonlinear deterministic dynamical cycles in financial time series. Evidence of nonlinear dynamics is revealed in denoised daily stock market index returns for six countries by combining Recurrence Quantification Analysis (RQA: see Zbilut and Webber (J. Appl. Phys. 76(2) (1994) 965)) and wavelet filtering. Quantitative and qualitative results indicate that through wavelet pre-filtering we can obtain a clearer view of the underlying dynamical structure of returns generating processes. Our results also suggest the existence of high dimensional deterministic dynamics, unstable periodic orbits and chaos.

[1]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[2]  A. Oppenheim,et al.  Signal processing with fractals: a wavelet-based approach , 1996 .

[3]  Claire G. Gilmore,et al.  An examination of nonlinear dependence in exchange rates, using recent methods from chaos theory , 2001 .

[4]  William A. Barnett,et al.  Martingales, Nonlinearity, and Chaos , 2000 .

[5]  D. Peel,et al.  Some empirical evidence on the time series properties of four UK Asset Prices , 1996 .

[6]  William A. Brock,et al.  Non-linear dynamical systems: Instability and chaos in economics , 1991 .

[7]  R. Gencay,et al.  An Introduction to Wavelets and Other Filtering Methods in Finance and Economics , 2001 .

[8]  J. A. Stewart,et al.  Nonlinear Time Series Analysis , 2015 .

[9]  Douglas M. Patterson,et al.  Linear versus Nonlinear Macroeconomies: A Statistical Test , 1989 .

[10]  Apostolos Serletis,et al.  Evidence of a random multifractal turbulent structure in the Dow Jones Industrial Average , 2002 .

[11]  David M. Raup,et al.  How Nature Works: The Science of Self-Organized Criticality , 1997 .

[12]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[13]  Anastasios Malliaris,et al.  Methodological issues in asset pricing: Random walk or chaotic dynamics , 1999 .

[14]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[15]  Thanasis Stengos,et al.  Some evidence concerning macroeconomic chaos , 1988 .

[16]  D. Turcotte,et al.  Self-organized criticality , 1999 .

[17]  Holger Kantz,et al.  Practical implementation of nonlinear time series methods: The TISEAN package. , 1998, Chaos.

[18]  Richard Urbach.  Footprints of chaos in the markets : analysing non-linear time series in financial markets and other natural systems , 2000 .

[19]  A. Giuliani,et al.  Detecting deterministic signals in exceptionally noisy environments using cross-recurrence quantification , 1998 .

[20]  David Jerison.  The World According to Wavelets : The Story of a Mathematical Technique in the Making Reviewed by David Jerison , 1999 .

[21]  E. Fama,et al.  Efficient Capital Markets : II , 2007 .

[22]  B. LeBaron,et al.  Nonlinear Dynamics and Stock Returns , 2021, Cycles and Chaos in Economic Equilibrium.

[23]  Seth Greenblatt,et al.  Wavelets in Econometrics: An Application to Outlier Testing , 1994 .

[24]  J. A. Hołyst,et al.  Observations of deterministic chaos in financial time series by recurrence plots, can one control chaotic economy? , 2001 .

[25]  E. Capobianco.  Multiresolution approximation for volatility processes , 2002 .

[26]  Jorge Belaire-Franch,et al.  Assessing nonlinear structures in real exchange rates using recurrence plot strategies , 2002 .

[27]  Enrico Capobianco,et al.  WAVELET TRANSFORMS FOR THE STATISTICAL ANALYSIS OF RETURNS GENERATING STOCHASTIC PROCESSES , 2001 .

[28]  J. Theiler,et al.  Don't bleach chaotic data. , 1993, Chaos.

[29]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[30]  Claire G. Gilmore,et al.  A new test for chaos , 1993 .

[31]  Philip Rothman,et al.  The Statistical Properties of Dimension Calculations Using Small Data Sets: Some Economic Applications , 1990 .

[32]  Edgar E. Peters.  Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility , 1996 .

[33]  Chen Ping,et al.  A Random Walk or Color Chaos on the Stock Market? Time-Frequency Analysis of S&P Indexes , 1996 .

[34]  Hong-Ye Gao,et al.  Applied wavelet analysis with S-plus , 1996 .

[35]  W. Brock,et al.  Is the business cycle characterized by deterministic chaos , 1988 .

[36]  William A. Barnett,et al.  The Aggregation-Theoretic Monetary Aggregates Are Chaotic and Have Strange Attractors: An Econometric Application of Mathematical Chaos , 2004 .

[37]  E. Bradley,et al.  Recurrence plots of experimental data: To embed or not to embed? , 1998, Chaos.

[38]  Mark J. Jensen.  An Alternative Maximum Likelihood Estimator of Long-Memeory Processes Using Compactly Supported Wavelets , 1997 .

[39]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[40]  J. Zbilut,et al.  Embeddings and delays as derived from quantification of recurrence plots , 1992 .

[41]  D. Kugiumtzis.  State space reconstruction parameters in the analysis of chaotic time series—the role of the time window length , 1996, comp-gas/9602002.

[42]  G. Wornell.  Wavelet-based representations for the 1/f family of fractal processes , 1993, Proc. IEEE.

[43]  Dimitris Kugiumtzis,et al.  On the Reliability of the surrogate Data Test for nonlinearity in the Analysis of noisy Time Series , 2001, Int. J. Bifurc. Chaos.

[44]  T. Schreiber.  Interdisciplinary application of nonlinear time series methods , 1998, chao-dyn/9807001.

[45]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[46]  J. Zbilut,et al.  Recurrence quantification analysis as a tool for nonlinear exploration of nonstationary cardiac signals. , 2002, Medical engineering & physics.

[47]  Michael R. Chernick,et al.  Wavelet Methods for Time Series Analysis , 2001, Technometrics.

[48]  Sherrill Shaffer,et al.  Structural shifts and the volatility of chaotic markets , 1991 .

[49]  Apostolos Serletis,et al.  No evidence of chaos but some evidence of dependence in the US stock market , 2003 .

[50]  Mark J. Jensen.  An Approximate Wavelet MLE of Short- and Long-Memory Parameters , 1999 .

[51]  D. Ruelle,et al.  Recurrence Plots of Dynamical Systems , 1987 .

[52]  Enrico Capobianco,et al.  Statistical Analysis of Financial Volatility by Wavelet Shrinkage , 1999 .

[53]  Bruce Mizrach,et al.  On Determining the Dimension of Real-Time Stock-Price Data , 1992 .

[54]  Kari Takala,et al.  Chaos and nonlinear dynamics in financial and nonfinancial time series: Evidence from Finland , 1996 .

[55]  C L Webber,et al.  Dynamical assessment of physiological systems and states using recurrence plot strategies. , 1994, Journal of applied physiology.

[56]  Mehmet Emre Çek,et al.  Analysis of observed chaotic data , 2004 .

[57]  Douglas M. Patterson,et al.  Evidence of Nonlinearity in Daily Stock Returns , 1985 .

[58]  B. LeBaron,et al.  Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence , 1991 .

[59]  TheilerJames,et al.  Testing for nonlinearity in time series , 1992 .

[60]  James B. Ramsey,et al.  Wavelets in Economics and Finance: Past and Future , 2002 .

[61]  C. Granger,et al.  USING THE MUTUAL INFORMATION COEFFICIENT TO IDENTIFY LAGS IN NONLINEAR MODELS , 1994 .

[62]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[63]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

[64]  Robert Savit,et al.  When random is not random: An introduction to chaos in market prices , 1988 .

[65]  David Hsieh.  Chaos and Nonlinear Dynamics: Application to Financial Markets , 1991 .

[66]  T. Schreiber,et al.  Surrogate time series , 1999, chao-dyn/9909037.

[67]  A. Bruce,et al.  Understanding WaveShrink: Variance and bias estimation , 1996 .

[68]  G. Nason.  Choice of the Threshold Parameter in Wavelet Function Estimation , 1995 .

[69]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[70]  G. Hansen.  Indivisible Labor and the Business Cycle , 1985 .

[71]  B. Wade Brorsen,et al.  Nonlinear dynamics of daily futures prices: Conditional heteroskedasticity or chaos? , 1993 .

[72]  Christoph Schleicher,et al.  An Introduction to Wavelets for Economists , 2002 .