The Markov-Switching Multifractal Model of Asset Returns: Estimation via GMM and Linear Forecasting of Volatility

Multifractal processes have recently been proposed as a new formalism for modeling 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 nonstationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markov-switching multifractal model which allows for estimation of its parameters via maximum likelihood (ML) 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 generalized method of moments (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 incurred 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.