Dominique, CRene and RiveraSolis, Luis Eduardo (2012): Could Investors’ Expectations Explain Temporal Variations in Hurst’s Exponent, Loci of Multifractal Spectra, and Statistical Prediction Errors? The Case of the S&P 500 Index. Published in: International Business Research , Vol. Volume, No. No. 5 (1. May 2012): pp. 815.

PDF
MPRA_paper_41407.pdf Download (150kB)  Preview 
Abstract
Over the periods 19982002 and 20092011, the S&P500 Index went from persistence to antipersistence mode, as measured by the Hurst index H. To uncover the reasons that characterize such a change, this paper uses a simple method that consists in treating quasi selfsimilar segments of the Index as initiators, and then finding appropriate generators with two intervals each to asymptotically model the strange attractor. The multifractal formalism shows that the change in persistence implies a corresponding change in the multifractal spectrum, and an enlargement of the invariant equilibrium set, making a market crash more likely, most probably due to a collapse of investors’ expectations. This also means that all statistical predictions made in one mode would have been off by an amount proportional to change in any element of the generalized set of dimensions in the other.
Item Type:  MPRA Paper 

Original Title:  Could Investors’ Expectations Explain Temporal Variations in Hurst’s Exponent, Loci of Multifractal Spectra, and Statistical Prediction Errors? The Case of the S&P 500 Index 
Language:  English 
Keywords:  Persistence, Strange Equilibrium sets, Scaling Exponents, Multifractal Spectra, Generalized Dimensions of order q, Statisticalpredictionerror 
Subjects:  G  Financial Economics > G1  General Financial Markets > G14  Information and Market Efficiency ; Event Studies ; Insider Trading D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D84  Expectations ; Speculations C  Mathematical and Quantitative Methods > C0  General G  Financial Economics > G0  General > G00  General A  General Economics and Teaching > A1  General Economics > A10  General G  Financial Economics > G0  General > G01  Financial Crises 
Item ID:  41407 
Depositing User:  Dr. Luis Rivera 
Date Deposited:  19. Sep 2012 11:34 
Last Modified:  21. Oct 2015 22:01 
References:  AlvarezRamirez, J., Alvarez, J., Rodriguez, E., & Fernandez, A. (2008). Timevarying Hurst exponent for US stock markets. Physica A, 1959 1969. Baraktur, E., Poor, V. H., & Sircar, R. K. (2003). Estimating the fractal dimension of the S&P500 index using wavelet analysis. EQuad Paper, Department of Electrical Engineering, Princeton Univ., Princeton, N.J. 08544. Cutland, N. J., Kopp, P. E. and Willinger, W. (1993). Stock price returns and the Joseph effect: A fractal version of the Black Scholes model. Progress in Probability, 36, 327351. Dominique, CR., & Rivera, L. S. (2011). Mixed fractional brownian motion, short and longterm dependence and economic conditions: The case of the S&P500 index. International Business and Management, 3, 16. Eckmann, J. P., & Ruelle, D. (1985). Ergotic theory of chaos and strange attractors. Review of Modern Physics, 57, 617656. http://dx.doi.org/10.1103/RevModPhys.57.617 Falconer, K. (2003). fractal geometry. New York: John Wiley & Sons. Flandrin, P. (1992). Wavelet analysis and synthesis of fractional Brownian motion. IEEE Transactions, 28, 910917. Fisher, A. J., Calvet, L. E., & Mandelbrot, B. B. (1997). Multifractality of deutschmark / US dollar exchange rates. Cowles Foundation Discussion Paper no 1165. Genyuk, J. (1999). Topics in multifractal formalism: local dimension, gobal scaling. Dept of Mathematics Ohio Unoiversity. Retrieved from www.math.osu.edu/node/19421. Gopikrishan, P. (1999). Scaling of the distribution of fluctuations in financial market indices. Physica E, 60, 53055310. Grassberger, P. (1981). On the Hausdorff dimension of fractal attractors. Journal of Statistical Physics, 26, 173179. http://dx.doi.org/10.1007/BF01106792 Grassberger, P., & Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D, 9(12), 189208. http://dx.doi.org/10.1016/01672789(83)902981 Greene, M. T., & Fielitz, B. D. (1977). Longterm dependence in common stock returns. Journal of Financial Economics, 4, 339349. http://dx.doi.org/10.1016/0304405X(77)90006X Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., & Shaiman, B. I. (1986). Fractal measures and their singularities: The characterization of strange sets. Physical Review A, 33, 11411151. http://dx.doi.org/10.1103/PhysRevA.33.1141 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, 35543562. http://dx.doi.org/10.1109/78.258096 Miao, Y., Ren, W., & Ren, Z. (2008). On the fractional mixed fractional brownian motion. Applied Mathematical. Science, 35, 17291738. Mandelbrot, B. (1974). Intermittent turbulence in selfsimilar cascades: Divergence of high moments and dimension of the carrier. Journal of Fluid Mechanics, 62, 331358. http://dx.doi.org/10.1017/S0022112074000711 Mandelbrot, B., & van Ness, J. W. (1968). Fractional brownian motions, fractional noises and applications. SIAM Review, 10, 422437. http://dx.doi.org/10.1137/1010093 Medio, A. (1992). Chaotic dynamics: Theory and applications to economics. United Kingdom: Cambridge University Press. Muller, U., Dacorna, M. M., Dave, R. D., Olson, R. B., Pictet, O. V., & von Weizsacker, J. E. (1997). Volatilities of different time resolution: Analyzing the dynamics of market components. Journal of Empirical Finance, 4, 213239. http://dx.doi.org/10.1016/S09275398(97)000078 Preciado, J., & Morris, H. (2008, Oct.). The varying behavior of U. S. market persistence. Paper presented at the World Congress on Engineering and Computer Science 2008, San Francisco, USA. Renyi, A. (1955). On a new axiomatic theory of probability. Acta Mathematica Hungaria, 6, 285335. http://dx.doi.org/10.1007/BF02024393 Renyi, A. (1970). Probability theory. Amsterdam: NorthHolland. Schroeder, M. (2009). Fractals, chaos, power laws. New York: Dover Pub., Inc. Sottinen, T. (2003). Fractional brownian motion in finance and queuing. (Doctoral dissertation). University of Helsinki, Finland, 2003. Takens, F. (1981). Detecting strange attractors in turbulence. In D. A. Rand and L. S. Young (Eds.), Dynamical systems and turbulence (pp.366381). New York: Springer Verlag. http://dx.doi.org/10.1007/BFb0091924 Thale, C. (2009). Further remarks on mixed fractional Brownian motion. Applied Mathematical Sciences, 3, 117. Zili, M. (2006). On the mixed fractional Brownian motion. Journal of Applied Mathematics and Stochastic Analyses, 2006, 19. http://dx.doi.org/10.1155/JAMSA/2006/32435 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/41407 