HMM analysis of general state-space models

These notes explain how to use a hidden Markov model (HMM) approach for analysing possibly highly nonlinear time series data in a state-space formulation. The text introduces the general state-space model and gives an overview of other methods for filtering and smoothing ranging from the simple linear and Gaussian case to the fully general case. A discretization of the state-space is instrumental to the HMM approach and the choice of discretization is therefore discussed. The filter and smoothing recursions for the hidden Markov model are presented and applied to the standard benchmark model known from the literature on nonlinear time series. Finally, as an example, the parameters of a stochastic volatility model are estimated with maximum likelihood and the results are compared with an Monte Carlo based estimation procedure.