Inference for nonlinear state space models: A comparison of different methods applied to Markov-switching multifractal models

Nonlinear, non-Gaussian state space models have found wide applications in many areas. Since such models usually do not allow for an analytical representation of their likelihood function, sequential Monte Carlo or particle filter methods are mostly applied to estimate their parameters. Since such stochastic approximations lead to non-smooth likelihood functions, finding the best-fitting parameters of a model is a non-trivial task. In this paper, we compare recently proposed iterative filtering algorithms developed for this purpose with simpler online filters and more traditional methods of inference. We use a highly nonlinear class of Markov-switching models, the so called Markov-switching multifractal model (MSM), as our workhorse in the comparison of different optimisation routines. Besides the well-established univariate discrete-time MSM, we introduce univariate and multivariate continuous-time versions of MSM. Monte Carlo simulation experiments indicate that across a variety of MSM specifications, the classical Nelder-Mead or simplex algorithm appears still as more efficient and robust compared to a number of online and iterated filters. A very close competitor is the iterated filter recently proposed by Ionides et al. (2006) while other alternatives are mostly dominated by these two algorithms. An empirical application of both discrete and continuous-time MSM to seven financial time series shows that both models dominate GARCH and FIGARCH models in terms of in-sample goodness-of-fit. Out-of-sample forecast comparisons show in the majority of cases a clear dominance of the continuous-time MSM under a mean absolute error criterion, and less conclusive results under a mean squared error criterion.