Applications of state space models in finance

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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)
4.1 Basic concepts 45 R t (%) 20 10 0 −10 −20 (a) Technology log−returns 1990 1995 2000 2005 probability density 0.15 0.10 0.05 (b) Histogram and fitted normal 0.00 −20 −10 0 10 20 Figure 4.1: (a) Weekly percentage log-return series of the Technology sector and (b) histogram with a fitted normal distribution. displays a histogram of the weekly returns together with a fitted normal distribution. It becomes obvious that the chosen normal underestimates the probability of both low returns around zero and extremely high absolute returns. One possibility to overcome the shortcomings of the normal distribution is to employ a mixture of two normal distributions. Mixture distributions are useful in the context of overdispersed or multimodal data that may be caused by unobserved heterogeneity in the sample. An independent mixture distribution can either be modeled as a discrete mixture, which is characterized by a finite number of component distributions, or as a continuous mixture, which can be thought of as a discrete mixture consisting of infinitely many component distributions. Note that in both cases discrete or continuous distributions can be chosen as components. As only discrete mixtures are relevant for the HMMs considered in the context of this chapter and also for the applications in the empirical part of this thesis, the continuous case will not find any further consideration. 10 In the general m-component case, the characteristics of the mixture distribution are determined by m random variables X1, . . . , Xm, and their probability or probability density functions, denoted as pi(x) or fi(x), respectively, for i = 1, . . . , m. The mixture is performed using a discrete random variable S that determines from which random variable an observation is drawn. It can take values between 1 and m, each with a probability πi: ⎧ ⎪⎨ S := ⎪⎩ 1 with probability π1 2 with probability π2 . m with probability πm, (4.1) where � m i=1 πi = 1 and πi ≥ 0 for i = 1, . . . , m. With π1, . . . , πm representing the weights of the various components, the probability (density) function of the mixture distribution 10 A survey of mixture distributions, is given, for example, by Titterington et al. (1985). For more details on continuous mixture distributions, see the references provided by Zucchini et al. (2006, 2.1).
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Sascha Mergner Applications of Stat
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erschienen im Universitätsverlag G
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Bibliographische Information der De
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Contents List of figures . . . . .
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Contents ix 4.5 Model selection and
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Contents xi 7.5 Concluding remarks
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xiv List of figures 6.14 Histograms
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Notation and conventions The outlin
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xx Used abbreviations and symbols L
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xxii Used abbreviations and symbols
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Page 29: Preface The work on this book began
Page 33 and 34: Chapter 1 Introduction “Economist
Page 35 and 36: 1.2 Research objectives 3 1.2 Resea
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Page 40 and 41: 8 2 Some stylized facts of weekly s
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Page 50 and 51: 18 3 Linear Gaussian state space mo
Page 75: Chapter 4 Markov regime switching M
Page 79 and 80: 4.1 Basic concepts 47 probability d
Page 81 and 82: 4.3 Parameter estimation 49 X 1 X 2
Page 83 and 84: 4.3 Parameter estimation 51 4.3.2.1
Page 85 and 86: 4.4 Forecasting and decoding 53 4.4
Page 87 and 88: 4.4 Forecasting and decoding 55 mod
Page 89 and 90: Chapter 5 Conditional heteroskedast
Page 91 and 92: 5.1 Autoregressive conditional hete
Page 97 and 98: 5.2 Stochastic volatility 65 or the
Page 99 and 100: 5.2 Stochastic volatility 67 5.2.1.
Page 101 and 102: 5.2 Stochastic volatility 69 likeli
Page 103 and 104: 5.2 Stochastic volatility 71 linear
Page 105 and 106: 5.2 Stochastic volatility 73 3. An
Page 107 and 108: 5.3 Multivariate conditional hetero
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94 6 Time-varying market beta risk
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Chapter 7 A Kalman filter based con
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7.1 Factor modeling 125 uncondition
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7.1 Factor modeling 127 predictabil
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7.1 Factor modeling 129 usefulness,
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7.2 Specification of a conditional
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7.3 The risk factors 133 Table 7.1:
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7.3 The risk factors 135 introduced
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7.3 The risk factors 137 auxiliary
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7.4 Empirical results 139 Table 7.4
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7.4 Empirical results 141 are consi
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7.4 Empirical results 145 β TS 0.4
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7.4 Empirical results 153 cumulativ
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Chapter 8 Conclusion and outlook Th
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Conclusion and outlook 157 cannot o
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Appendix A Brief review of asset pr
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A.3 Alternative asset pricing model
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Appendix B Figures
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Figures 165 β t β t β t 2.0 1.5
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Appendix C Tables
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Tables 179 Table C.1 — continued
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Tables 181 Table C.3: In-sample mea
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Tables 183 Table C.5: Out-of-sample
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Tables 185 Table C.7: Parameter est
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Tables 187 Table C.7 — continued
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Tables 189 Table C.8: Out-of-sample
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References Abell, J. D. and T. M. K
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References 193 Bulla, J. (2006). Ap
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References 195 Fabozzi, F. J. and J
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References 197 Harvey, A. C., E. Ru
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References 199 Lie, F., R. Brooks,
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References 201 Shephard, N. (1993).
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State space models play a key role
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