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Journal of Financial Econometrics Vol. 2, No. 1, pp. 49-83
© 2004 Oxford University Press; all rights reserved.

How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes

Laurent E. Calvet
     Harvard University and NBER

Adlai J. Fisher
     University of British Columbia

Address correspondence to Department of Economics, Cambridge, MA 02138, or e-mail: lcalvet{at}aya.yale.edu.

Address correspondence to Finance Division, Sauder School of Business, 2053 Main Mall, Vancouver, BC, Canada V6T 1Z2, or e-mail: adlai.fisher{at}sauder.ubc.ca.

We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low-frequency variations. Second, they specify intermediate-frequency dynamics usually assigned to smooth autoregressive transitions. Finally, high-frequency switches generate substantial outliers. Thus a single mechanism captures three features that are typically viewed as distinct in the literature. Maximum-likelihood estimation is developed and performs well in finite samples. Using exchange rates, we estimate a version of the process with four parameters and more than a thousand states. The multifractal outperforms GARCH, MS-GARCH, and FIGARCH in- and out-of-sample. Considerable gains in forecasting accuracy are obtained at horizons of 10 to 50 days.

KEYWORDS: forecasting, long memory, Markov-switching multifractal (MSM), closed-form likelihood, scaling, stochastic volatility, volatility component, Vuong test

Received October 25, 2002; revised March 4, 2003; accepted October 29, 2003


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