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Application of Hidden Markov Models and Hidden Semi-Markov Models to Financial Time Series

Bulla, Jan (2006): Application of Hidden Markov Models and Hidden Semi-Markov Models to Financial Time Series. Published in:

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Abstract

Hidden Markov Models (HMMs) and Hidden Semi-Markov Models (HSMMs) provide flexible, general-purpose models for univariate and multivariate time series. Although interest in HMMs and HSMMs has continuously increased during the past years, and numerous articles on theoretical and practical aspects have been published, several gaps remain. This thesis addresses some of them, divided into three main topics.

1. Computational issues in parameter estimation of stationary HMMs. The parameters of a HMM can be estimated by direct numerical maximization (DNM) of the log-likelihood function or, more popularly, using the Expectation-Maximization (EM) algorithm. We show how the EM algorithm could be modified to fit stationary HMMs. We propose a hybrid algorithm that is designed to combine the advantageous features of the EM and DNM algorithms, and compare the performance of the three algorithms (EM, DNM and the hybrid). We then describe the results of an experiment to assess the true coverage probability of bootstrap-based confidence intervals for the parameters.

2. A Markov switching approach to model time-varying Beta risk of pan-European Industry portfolios. The motive to take up this topic was the development of a joint model for many financial time series. We study two Markov switching models in a Capital Asset Pricing Model framework, and compare their forecast performances to three models, namely a bivariate t-GARCH(1,1) model, two Kalman filter based approaches and a bivariate stochastic volatility model.

3. Stylized facts of financial time series and HSMMs. The ability of a HMM to reproduce several stylized facts of daily return series was illustrated by Ryden et al. (1998). However, they point out that one stylized fact cannot be reproduced by a HMM, namely the slowly decaying autocorrelation function of squared returns. We present two HSMM-based approaches to model eighteen series of daily sector returns with about 5.000 observations. The key result is that, compared to a HMM, the slowly decaying autocorrelation function is significantly better described by a HSMM with negative binomial sojourn time and Normal conditional distributions.

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