Applications of state space models in finance

44 4 Markov regime switching
who employ a hidden Markov model to derive stylized facts of daily S&P 500 return
series.
The standard regime switching model can be extended to provide an even higher
degree of flexibility. Important specifications include Markov switching models with
time-varying switching probabilities (Diebold et al. 1994; Filardo 1994) and hidden
semi-Markov models, which allow for nonparametric state occupancy distributions, as
proposed by Ferguson (1980). As these extensions are beyond the scope of this thesis,
they will not find any consideration hereafter; for further reading, see, for example, Bulla
(2006, ¢ 5) and the references given therein.
HMMs are employed in a wide spectrum of applications due to their overall versatility
and mathematical tractability. Their attractiveness is grounded on the fact that
the likelihood can be evaluated in a straightforward fashion, either by numerical maximization
or by employing the EM algorithm. Moment properties can be easily derived,
missing observations can be easily dealt with, and HMMs are often interpretable in a
natural way. Besides, they are moderately parsimonious with a simple two-state model
providing a good fit in many cases. Comprehensive treatments of HMMs are provided
by Elliott et al. (1995), MacDonald and Zucchini (1997), Bhar and Hamori (2004) and
Hamilton (1993), with the latter two focusing on applications in economics and finance.
The organization of this chapter is as follows. Section 4.1 briefly reviews independent
mixture models and basic properties of Markov chains. Section 4.2 outlines the basic
hidden Markov model. Section 4.3 briefly discusses parameter estimation based on the
ML method. Section 4.4 looks at forecasting and decoding procedures, before Section 4.5
provides an overview of model selection and validation. To assure notational conformability,
the outline in this chapter is, unless stated otherwise, based on Bulla (2006, ¢ 2–3)
who also provided the code for conducting the computations related to HMMs in this
thesis using the statistical software package R 2.1.1 (R Development Core Team 2005).
4.1 Basic concepts
Before introducing the theory of HMMs, it is instructive to look at two fundamental
concepts as a basis to understand the basic structure of a hidden Markov model. Subsection
4.1.1 reviews the concept of mixture distributions. Subsection 4.1.2 provides a
brief introduction to the basic properties of Markov chains, which are needed to construct
HMMs.
4.1.1 Independent mixture distributions
Consider the weekly log-return series of the Technology sector. As discussed in ¢ 2.2.2,
the series of all sectors show signs of positive autocorrelation in the squared returns.
This characteristic is commonly referred to as volatility clustering, a common feature
of financial returns data that usually induces excess kurtosis. Panel (a) of Figure 4.1
shows the weekly percentage log-returns on the DJ Technology sector. It can be seen
that the sector’s volatility is not constant over time: the level of volatility is clearly
lower at the beginning and higher in the second half of the sample. As a consequence, a
normal distribution is not capable of describing the return series adequately. Panel (b)