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

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  • Lux, Thomas

Abstract

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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Bibliographic Info

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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Keywords: Markov-switching; Multifractal; Forecasting; Volatility; GMM estimation;

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  3. Laurent E. Calvet & Adlai J. Fisher & Samuel B. Thompson, 2004. "Volatility Comovement: A Multifrequency Approach," NBER Technical Working Papers 0300, National Bureau of Economic Research, Inc.
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  9. Laurent Calvet & Adlai Fisher & Benoit Mandelbrot, 1999. "A Multifractal Model of Assets Returns," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-072, New York University, Leonard N. Stern School of Business-.
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  17. 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.
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Cited by:
  1. Nasr, Adnen Ben & Lux, Thomas & Ajm, Ahdi Noomen & Gupta, Rangan, 2014. "Forecasting the volatility of the dow jones islamic stock market index: Long memory vs. regime switching," Economics Working Papers 2014-07, Christian-Albrechts-University of Kiel, Department of Economics.
  2. Batten, Jonathan A. & Kinateder, Harald & Wagner, Niklas, 2014. "Multifractality and value-at-risk forecasting of exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 71-81.
  3. Ruipeng Liu & Thomas Lux, 2010. "Flexible and Robust Modelling of Volatility Comovements: A Comparison of Two Multifractal Models," Kiel Working Papers 1594, Kiel Institute for the World Economy.
  4. G.-F. Gu & W.-X. Zhou, 2009. "On the probability distribution of stock returns in the Mike-Farmer model," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 67(4), pages 585-592, February.
  5. Thomas Lux & Leonardo Morales-Arias & Cristina Sattarhoff, 2011. "A Markov-switching Multifractal Approach to Forecasting Realized Volatility," Kiel Working Papers 1737, Kiel Institute for the World Economy.
  6. Calvet, Laurent E. & Fisher, Adlai J., 2008. "Multifrequency jump-diffusions: An equilibrium approach," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 207-226, January.
  7. Adnen Ben Nasr & Ahdi N. Ajmi & Rangan Gupta, 2013. "Modeling the Volatility of the Dow Jones Islamic Market World Index Using a Fractionally Integrated Time Varying GARCH (FITVGARCH) Model," Working Papers 201357, University of Pretoria, Department of Economics.
  8. Filip Zikes & Jozef Barunik & Nikhil Shenai, 2012. "Modeling and Forecasting Persistent Financial Durations," Papers 1208.3087, arXiv.org, revised Apr 2013.
  9. Idier, J., 2008. "Long term vs. short term comovements in stock markets: the use of Markov-switching multifractal models," Working papers 218, Banque de France.
  10. Ola L{\o}vsletten & Martin Rypdal, 2012. "A multifractal approach towards inference in finance," Papers 1202.5376, arXiv.org.
  11. M. Rypdal & O. L{\o}vsletten, 2011. "Multifractal modeling of short-term interest rates," Papers 1111.5265, arXiv.org.

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