The Multi-Fractal Model of Asset Returns: Simple Moment and GMM Estimation

Multi-fractal processes have been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their capability of generating various degrees of long-memory in different powers of returns - a feature that has been found to characterise virtually all financial prices. Furthermore, elementary variants of multi-fractal models are very parsimonious formalisations as they are essentially one-parameter families of stochastic processes. The aim of this paper is to provide a first assessment of the goodness-of-fit of this new class of models by applying them to four long time series from different financial markets (one exchange rate, two stock market indices and the price of gold). To this end, we complement the heuristic estimation methods from statistical physics by developing a GMM (Generalised Method of Moment) estimator for the Binomial and Lognormal multifractal models. Our results are very encouraging in that the estimated models provide an astonishingly good fit to the unconditional distribution of the data. Applying Hansen's test, we are, in fact, unable in three out of four cases to reject the multi-fractal model.