Parameter Estimation of a Class of Hidden Markov Models using Sequential Monte Carlo Expectation-Maximization Algorithm
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Date
2012-07
Authors
Ahani, E.B
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Abstract
Much research has been advanced in the development of Monte Carlo methods for stochastic processes. A particular focus is on sequential Monte Carlo methods (particle filters and particle smoothers) and the Expectation-Maximization (EM) algorithm which allows the estimation of a class of Hidden Markov Models (HMMs) with nonlinear, non-Gaussian state-space models. The Stochastic Volatility (SV) model can be regarded as a nonlinear state space model. SV model has become increasingly popular for explaining the behaviour of financial variables (e.g. stock prices and exchange rates). This has resulted in several different proposed approaches to estimating the parameters of the model. This thesis proposes a Sequential Monte Carlo Expectation Maximization (SMCEM) approximation method for the nonlinear state space representation and applies it for estimating the SV model. The basic idea of our approach is to combine the Expectation-Maximization (EM) algorithm with particle filters and smoothers in order to estimate parameters of the model. In addition to mixture-of-normal distributions of Kim & Stoffer (2008), the scope of application of SV models is expanded by adopting a student-t and the Generalized Error Distribution (GED), for the observational error term. To establish the viability of the extended volatility models, simulation studies as well as real life data analysis results are presented. Furthermore, the research establishes the convergence properties of the proposed technique. The results obtained from the models indicate that the student-t and the GED are comparable to the normal mixture SV model but empirically more successful. The proposed model allows for a more robust fit, giving us a new tool to explore the tail fit. In the same vein, there are theoretical as well as empirical reasons to study multivariate volatility models. The application of the SMCEM approach to a multivariate factor model with stochastic volatility using the student-t distribution indicates that it performs quite well in explaining the joint dynamics in the volatility of a number of asset returns.
In the same vein, this work applies the proposed procedure to nonlinear problems in signal processing such as bearings-only tracking; again the procedure is successful in accommodating nonlinear model for a target tracking scenario.
Description
A Thesis Submitted to the School of Postgraduate Studies, University of Lagos
Keywords
Hidden Markov model (HMM) , Stochastic Volatility model , State-space model , EM algorithm , Research Subject Categories::MATHEMATICS::Algebra, geometry and mathematical analysis
Citation
Ahani, E.B (2012). Parameter Estimation of a Class of Hidden Markov Models using Sequential Monte Carlo Expectation-Maximization Algorithm. A Thesis Submitted to University of Lagos School of Postgraduate Studies Phd Thesis and Dissertation, 181pp.