Statistical Inference for Stochastic Processes

, Volume 3, Issue 3, pp 263–276 | Cite as

Semiparametric Bootstrap Approach to Hypothesis Tests and Confidence Intervals for the Hurst Coefficient

  • Peter Hall
  • Wolfgang Härdle
  • Torsten Kleinow
  • Peter Schmidt


A major application of rescaled adjusted range analysis (R–S analysis) is to the study of price fluctuations in financial markets. There, the value of the Hurst constant, H, in a time series may be interpreted as an indicator of the irregularity of the price of a commodity, currency or similar quantity. Interval estimation and hypothesis testing for H are central to comparative quantitative analysis. In this paper we propose a new bootstrap, or Monte Carlo, approach to such problems. Traditional bootstrap methods in this context are based on fitting a process chosen from a wide but relatively conventional range of discrete time series models, including autoregressions, moving averages, autoregressive moving averages and many more. By way of contrast we suggest simulation using a single type of continuous-time process, with its fractal dimension. We provide theoretical justification for this method, and explore its numerical properties and statistical performance by application to real data on commodity prices and exchange rates.

Box-counting method commodity price financial market fractal dimension fractional Brownian motion Gaussian process long-range dependence Monte Carlo R–S analysis self affineness self similarity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adler, R. J.: 1981, The Geometry of Random Fields, Wiley, New York.Google Scholar
  2. Barnsley, M.: 1988, Fractals Everywhere, Academic Press, New York.Google Scholar
  3. Beran, J.: 1994, Statistics for Long-Memory Processes, Chapman and Hall, London.Google Scholar
  4. Berry, M. V. and Hannay, J. H.: 1978, Topography of random surfaces, Nature 273, 573.CrossRefGoogle Scholar
  5. Calvet, L., Fisher, A. and Mandelbrot, B. B.: 1997, A Multifractal Model of Asset Returns, Cowles Foundation Discussion Paper #1164Google Scholar
  6. Carter, P. H., Cawley, R. and Mauldin, R. D.: 1988, Mathematics of dimension measurements of graphs of functions. in Proc. Symp. Fractal Aspects of Materials, Disordered Systems, D. A. Weitz, L. M. Sander and B. B. Mandelbrot (eds) Materials Research Society, Pittsburgh, PA, pp. 183–186.Google Scholar
  7. Constantine, A. G. and Hall, P.: 1994, Characterizing surface smoothness via estimation of effective fractal dimension, J. Roy. Statist. Soc. Ser. B 56, 97–113.zbMATHMathSciNetGoogle Scholar
  8. Davies, S. and Hall, P.: 1998, Fractal analysis of surface roughness using spatial data, J. Roy. Statist. Soc. Ser. B, to appear.Google Scholar
  9. Davies, R. B. and Harte, D. S.: 1987, Tests for Hurst effect, Biometrika 74, 95–101.zbMATHMathSciNetCrossRefGoogle Scholar
  10. Davison, A. C. and Hinkley, D. V.: 1997, Bootstrap Methods and their Application, Cambridge: Cambridge University Press.Google Scholar
  11. Efron, B. and Tibshirani, R.: 1993, An Introduction to the Bootstrap, Chapman and Hall, London.Google Scholar
  12. Hall, P.: 1992, The Bootstrap and Edgeworth Expansion, Springer, New York.Google Scholar
  13. Hall, P., Matthews, D. and Platen, E.: 1996, Algorithms for analyzing nonstationary time series with fractal noise, J. Computat. Graph. Statist. 5, 351–364.MathSciNetCrossRefGoogle Scholar
  14. Hall, P. and Roy, R.: 1994, On the relationship between fractal dimension and fractal index for stationary stochastic processes, Ann. Appl. Probab. 4, 241–253.zbMATHMathSciNetGoogle Scholar
  15. Hall, P. and Wood, A. T. A.: 1993, On the performance of box-counting estimators of fractal dimension, Biometrika 80, 246–252.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Hunt, F.: 1990, Error analysis and convergence of capacity dimension algorithms, SIAM J. Appl. Math. 50, 307–321.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Hurst, H. E. 1951, Long-term storage capacity of reservoirs, Trans. Amer. Soc. Civil Engineers 116, 770–799.Google Scholar
  18. Kent, J. T. and Wood, A. T. A.: 1993, Estimating the fractal dimension of a locally self-similar Gaussian process by using increments, J. Roy. Statist. Soc. Ser. B 59, 679–699.MathSciNetGoogle Scholar
  19. Mandelbrot, B. B., Passoja, D. E. and Paullay, A. J.: 1984, Fractal character of surfaces of metals, Nature 308, 721–722.CrossRefGoogle Scholar
  20. Peters, E. E.: 1994, Fractal Market Analysis: Applying Chaos Theory to Investment and Economics, Wiley, New York.Google Scholar
  21. Rosenblatt, M.: 1961, Independence and dependence, in Proc. 4th Berkeley Symp. Math. Statist. Probab., J. Neyman (ed) University of California Press, Berkeley, pp. 411–433.Google Scholar
  22. Sayles, R. S. and Thomas, T. R.: 1978, Surface topography as a nonstationary random process. Nature 271, 431–434.CrossRefGoogle Scholar
  23. Shao, J. and Tu, D.: 1995, The Jackknife and Bootstrap, Springer, New York.Google Scholar
  24. Sullivan, F. and Hunt, F.: 1988, How to estimate capacity dimension, Nuclear Phys. B. (Proc. Suppl.) 5A, 125–128.Google Scholar
  25. Taqqu, M. S.: 1975, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. Verw. Gebiete 31, 287–302.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Peter Hall
    • 1
  • Wolfgang Härdle
    • 2
  • Torsten Kleinow
    • 2
  • Peter Schmidt
    • 3
  1. 1.Centre for Mathematics and its ApplicationsAustralian National UniversityCanberraAustralia
  2. 2.Institut für Statistik und ÖkonometrieHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Quantitative ResearchBankgesellschaft Berlin AGGermany

Personalised recommendations