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The Markov-switching multi-fractal model of asset returns: GMM estimation and linear forecasting of volatility

  • Lux, Thomas

Multi-fractal processes have recently been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns - a feature that has been found in virtually all financial data. Initial difficulties stemming from non-stationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markov-switching multi-fractal model in Calvet and Fisher (2001) which allows for estimation of its parameters via maximum likelihood and Bayesian forecasting of volatility. However, applicability of MLE is restricted to cases with a discrete distribution of volatility components. From a practical point of view, ML also becomes computationally unfeasible for large numbers of components even if they are drawn from a discrete distribution. Here we propose an alternative GMM estimator together with linear forecasts which in principle is applicable for any continuous distribution with any number of volatility components. Monte Carlo studies show that GMM performs reasonably well for the popular Binomial and Lognormal models and that the loss incured with linear compared to optimal forecasts is small. Extending the number of volatility components beyond what is feasible with MLE leads to gains in forecasting accuracy for some time series.

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File URL: http://econstor.eu/bitstream/10419/3407/1/EWP-2004-11.pdf
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Paper provided by Christian-Albrechts-University of Kiel, Department of Economics in its series Economics Working Papers with number 2004,11.

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Date of creation: 2004
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Handle: RePEc:zbw:cauewp:2442
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  1. Laurent Calvet & Adlai Fisher, 2003. "Regime-Switching and the Estimation of Multifractal Processes," Harvard Institute of Economic Research Working Papers 1999, Harvard - Institute of Economic Research.
  2. Hansen, Lars Peter & Heaton, John & Yaron, Amir, 1996. "Finite-Sample Properties of Some Alternative GMM Estimators," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(3), pages 262-80, July.
  3. Lobato, Ignacio N & Savin, N E, 1998. "Real and Spurious Long-Memory Properties of Stock-Market Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 261-68, July.
  4. Vilasuso, Jon, 2002. "Forecasting exchange rate volatility," Economics Letters, Elsevier, vol. 76(1), pages 59-64, June.
  5. Diebold, Francis X & Mariano, Roberto S, 2002. "Comparing Predictive Accuracy," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 134-44, January.
  6. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
  7. Torben G. Andersen & Hyung-Jin Chung & Bent E. Sorensen, . "EMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Computing in Economics and Finance 1997 6, Society for Computational Economics.
  8. Ser-Huang Poon & Clive W.J. Granger, 2003. "Forecasting Volatility in Financial Markets: A Review," Journal of Economic Literature, American Economic Association, vol. 41(2), pages 478-539, June.
  9. Calvet, Laurent E. & Fisher, Adlai J. & Thompson, Samuel B., 2006. "Volatility comovement: a multifrequency approach," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 179-215.
  10. Calvet, Laurent & Fisher, Adlai, 2001. "Forecasting multifractal volatility," Journal of Econometrics, Elsevier, vol. 105(1), pages 27-58, November.
  11. Benoit Mandelbrot & Adlai Fisher & Laurent Calvet, 1997. "A Multifractal Model of Asset Returns," Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University.
  12. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
  13. Torben G. Andersen & Tim Bollerslev, 1996. "Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long-Run in High Frequency Returns," NBER Working Papers 5752, National Bureau of Economic Research, Inc.
  14. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
  15. Laurent Calvet & Adlai Fisher, 2002. "Multifractality In Asset Returns: Theory And Evidence," The Review of Economics and Statistics, MIT Press, vol. 84(3), pages 381-406, August.
  16. Brockwell, P. J. & Dahlhaus, R., 2004. "Generalized Levinson-Durbin and Burg algorithms," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 129-149.
  17. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
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