Characterizing Heteroscedasticity

  • Gilles Zumbach
Part of the Springer Finance book series (FINANCE)


The largest deviation from a simple random walk founds in the empirical time series is the volatility clustering, or heteroskedasticity. It is important to characterize at best the decay of the volatility lagged correlation because of its practical implications for the construction of processes. First, the volatility and correlation estimators most suited for this task are studied. Then, they are applied to a broad panel of financial time series. Finally, simple analytical shapes are estimated on the empirical lagged correlation. The best overall characterization is given by a logarithmic decay for increasing lags, whereas an exponential decay and power law decay can be eliminated. This shows that a multiscale structure should be used in the processes, capturing the observed long memory of the volatility clusters.


Stock Index Hurst Exponent Robust Estimator Asset Class Correlation Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 9.
    Arneodo, A., Muzy, J.-F., Sornette, D.: Causal cascade in the stock market from the “infrared” to the “ultraviolet”. Eur. Phys. J. B 2, 277–282 (1998) CrossRefGoogle Scholar
  2. 13.
    Bacry, E., Delour, J., Muzy, J.-F.: Multifractal random walk. Phys. Rev. E 64, 26103 (2001) CrossRefGoogle Scholar
  3. 14.
    Baillie, R.T., Bollerslev, T., Mikkelsen, H.-O.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econom. 74(1), 3–30 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 25.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 49.
    Dacorogna, M.M., Müller, U.A., Nagler, R.J., Olsen, R.B., Pictet, O.V.: A geographical model for the daily and weekly seasonal volatility in the FX market. J. Int. Money Finance 12 (1993) Google Scholar
  6. 53.
    Ding, Z., Granger, C.W.J., Engle, R.F.: A long memory property of stock market returns and a new model. J. Empir. Finance 1, 83–106 (1993) CrossRefGoogle Scholar
  7. 59.
    Engle, R.F.: Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 61.
    Engle, R.F., Bollerslev, T.: Modelling the persistence of conditional variances. Econom. Rev. 5, 1–50 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 64.
    Fama, E.F.: Mandelbrot and the stable Paretian distribution. J. Bus. 36, 420–429 (1965) Google Scholar
  10. 76.
    Gnanadeskian, R., Kettenring, J.R.: Robust estimates, residuals, and outlier detection with multiresponse data. Biometrika 28, 81–124 (1972) Google Scholar
  11. 79.
    Hauksson, H., Dacorogna, M.M., Domenig, T., Müller, U.A.: Multivariate extremes, aggregation and risk estimation. Quant. Finance 1(1), 79–95 (2001) CrossRefGoogle Scholar
  12. 89.
    Lindskog, F.: Linear correlation estimation. Technical report RiskLab, ETH, Zürich, Switzerland (2000) Google Scholar
  13. 90.
    Liu, Y., Cizeau, P., Meyer, M., Peng, C.-K., Stanley, H.E.: Correlation in economic time series. Physica A 245, 437 (1997) MathSciNetCrossRefGoogle Scholar
  14. 92.
    Lynch, P., Zumbach, G.: Market heterogeneities and the causal structure of the volatility. Quant. Finance 3, 320–331 (2003) CrossRefGoogle Scholar
  15. 94.
    Mandelbrot, B.B.: Fractals and Scaling in Finance. Springer, Berlin (1997) zbMATHGoogle Scholar
  16. 95.
    Mandelbrot, B.B.: The variation of certain speculative prices. J. Bus. 36, 394 (1963) Google Scholar
  17. 96.
    Mandelbrot, B.B., Fisher, A.J., Calvet, L.E.: A multifractal model of asset returns. Cowles Found. Discuss. Pap. 1164, 1–44 (1997) Google Scholar
  18. 103.
    Di Matteo, T., Aste, T., Dacorogna, M.M.: Long-term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development. J. Bank. Finance 29, 827–851 (2005) CrossRefGoogle Scholar
  19. 112.
    Muzy, J.-F., Delour, J., Bacry, E.: Modelling fluctuations of financial time series: from cascade process to stochastic volatility model. Eur. Phys. J. B 17, 537–548 (2000) CrossRefGoogle Scholar
  20. 114.
    Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 125.
    Poon, S.-H., Granger, C.W.J.: Forecasting volatility in financial markets. J. Econ. Lit. XLI, 478–539 (2003) CrossRefGoogle Scholar
  22. 126.
    Potters, M., Cont, R., Bouchaud, J.-P.: Financial markets as adaptative ecosystems. Europhys. Lett. 41, 239 (1998) CrossRefGoogle Scholar
  23. 129.
    Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in C++. Cambridge University Press, Cambridge (2002) Google Scholar
  24. 154.
    Zumbach, G.: The riskmetrics 2006 methodology. Technical report, RiskMetrics Group (2006). Available at: and
  25. 159.
    Zumbach, G.: Characterizing heteroskedasticity. Quantitative Finance (2011) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gilles Zumbach
    • 1
  1. 1.Consulting in Financial ResearchSaconnex d’ArveSwitzerland

Personalised recommendations