Dominique, C-Rene and Rivera-Solis, Luis Eduardo (2012): Short-term Dependence in Time Series as an Index of Complexity: Example from the S&P-500 Index. Published in: International Business Research , Vol. Volume, No. No. 9 (8 August 2012): pp. 38-48.
Download (369kB) | Preview
The capital market is a reflexive dynamical input/output construct whose output (time series) is usually assessed by an index of roughness known as Hurst’s exponent (H). Oddly enough, H has no theoretical foundation, but recently it has been found experimentally to vary from persistence (H > 1/2) or long-term dependence to anti-persistence (H < 1/2) or short-term dependence. This paper uses the thrown-offs of quadratic maps (modeled asymptotically) and singularity spectra of fractal sets to characterize H, the alternateness of dependence, and market crashes while proposing a simpler method of computing the correlation dimension than the Grassberger-Procaccia procedure.
|Item Type:||MPRA Paper|
|Original Title:||Short-term Dependence in Time Series as an Index of Complexity: Example from the S&P-500 Index|
|Keywords:||Hurst Exponent, anti-persistence, fractal attractors, SDIC, chaos, inherent noise, market crashes, Renyi’s generalized fractal dimensions|
|Subjects:||G - Financial Economics > G1 - General Financial Markets
C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling
A - General Economics and Teaching > A1 - General Economics
G - Financial Economics > G0 - General > G01 - Financial Crises
|Depositing User:||Dr. Luis Rivera|
|Date Deposited:||19 Sep 2012 11:44|
|Last Modified:||17 Jun 2016 17:51|
Alvarez-Ramirez, J., Alvarez, J., Rodriguez, E. & Fernandez A. (2008). Time-varying Hurst exponent for US stock markets. Physica A, 1959- 1969.
Arneodo, A., De Coster, N. & Roux, S. G. (2000). A wavelet-based method for multifractal image analysis. European Physical Journal B, 15, 567-600.
Arneodo, A. et al. (1995). The thermodynamics of fractals revisited with wavelets. Physica A, 213, 232-275.
Calvert, Laurent. E. & Fisher, Adlai, J. (2002). Multifractal in asset returns: theory and evidence. Review of Econom-ics and Statistics, 84, 381-406.
Cutland, N. J. et al. (1995). Stock price returns and the Joseph effect. A fractal version of the Black-Scholes model. Progress in Probability, 36, 327-351.
De Coster, G. P.. & Mitchell, D. (1991). The efficiency of the correlation dimension technique in detecting determin-ism in small samples. Journal of Statistical Computation and Simulation. 39, 221-229.
Dominique, C-R. & Rivera, S L. (2011). Mixed fractional Brownian motion, short and long-term dependence and economic conditions: The case of the S&P-500 Index. International Business and Management, 3, 1-6.
Frish, U. (1995). Turbulence. Cambridge University Press: Cambridge.
Eckmann, J. P. & Ruelle, D. (1985). Ergotic theory of chaos and strange attractors. Review of Modern Physics, 57, 617-656.
Grassberger, Peter (1981). On the Hausdorff dimension of fractal attractors. Journal of Statistical Physics, 26, 173-179.
Grassberger, Peter & Procaccia, I. (1983). Characterization of strange attractors. Physical Review Letters, 50, 346-349.
Greene, M. T. & Fielitz, B. D., (1977). Long-term dependence in common stock returns. Journal of Financial Eco-nomics, 4, 339-349
Hurst, E. et al. (1951). Long-term storage: An Engineering study. Transactions of the American Society of Civil Engi-neers, 116, 770-790.
Invernizzi, S. & Medio, A. (1991). On lags and chaos in economic dynamic models. Journal of Mathematical Eco-nomics. 20, 521-550.
Kaplan, L. M. & Jay Kuo, C. C., (1993). Fractal estimation from noisy data via discrete fractional Gaussian noise and the Haar Basis. IEEE Transactions, 41, 3554-3562.
Kesterner, P. & Arneodo, A. (2003). Three-dimensional wavelet-based multifractal method: The need for revisiting the multifractal description of turbulence dissipation data. Physical Review Letters, 91, 194501.
Li, Y. L & Yorke, A. (1975). Period three implies chaos. American Mathematical Monthly, 82, 985-992.
Lobato, I. N.,& Savin, N. E. (1998). Real and spurious long-memory properties of stock market data. Journal of Business and Economic Statistics, 16, 261-268.
Los, Cornelis, A. (2000). Visualization of chaos for finance majors. Working Paper 00-7, School of Economics, Adelaide Univer-sity.
Lux, T. (1996). Long-term stochastic dependence in financial prices: evidence from the German stock market. Applied Econom-ics Letters, 3, 701-706.
Maio, Y. , Ren, W. & Ren, Z. (2008). On the fractional mixed fractional Brownian motion. Applied Mathematical. Science, 35, 1729-1738.
Mandelbrot, Bernard (1974). Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. Journal of Fluid Mechanics, 62, 331-358.
Mandelbrot, B. & van Ness, J., W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review, 10, 422-437.
Medio, Alfredo (1992). Chaotic dynamics: Theory and applications to economics. Cambridge University Press: Cambridge, uk.
Peters, Edgar (1991). A chaotic attractor for the S&P-500. Financial Analyst Journal, March/April.
Renyi, A. (1970). Probability Theory. North-Holland: Amsterdam.
Schroeder, Manfred (2009). Fractals, Chaos, Power Laws, Dover Pub., Inc.: New York.
Sottinen, T. (2003) Fractional Brownian motion in finance and queuing. (Doctoral dissertation, University of Hel-sinki, Finland, 2003).
Thale, C. (2009). Further remarks on mixed fractional Brownian motion. Applied Mathematical Sciences, 3, 1-17. Warwick. ac.uk (2012). Lectures on fractals and dimension theory. Homepages.warwick.ac.uk/mas dbl/dimension-total.pdf.
Zili, M. (2006). On the mixed fractional Brownian motion. Journal of Applied Mathematics and Stochastic Analy-ses, 2006, 1-9.