Applications of state space models in finance

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20 3 Linear Gaussian state space models and the Kalman filter where a1 and P 1 are assumed to be known for now. The initialization of the state vector will be discussed in ¢ 3.3.6. The matrices T t, ct, Rt, Zt, dt, Q t and Ht are called system matrices. Unless stated otherwise, they are assumed to be non-stochastic, i.e. they change over time in a predetermined way. This leads to a system in which y t can be expressed as a linear combination of ξ 1 and past and present state and observation disturbances. A state space model with constant system matrices over time is referred to as being time-homogeneous. Usually, at least some of the elements of the system matrices T t and Rt in the state equation and Zt, Q t and Ht in the observation equation depend on a vector ψ of unknown parameters, which are referred to as hyperparameters. While the hyperparameters control for the stochastic characteristics of the model, the parameters related to the vectors ct and dt have a deterministic impact on the observations and the expected state value. The values of the system matrices are typically unknown. Their estimation on the basis of the available observations will be the subject of ¢ 3.4. For now, the system matrices are assumed to be known. Throughout this thesis, the software package SsfPack 2.3 by Koopman et al. (1999) is employed to apply the Kalman filter to estimate models in state space form. SsfPack is a comprehensive collection of C routines for estimating models in state space form with general routines for smoothing, filtering, likelihood evaluation and forecasting linked to the computing environment Ox 3.30 by Doornik (2001). With regard to the use of SsfPack, it will prove useful to represent the state space model introduced by (3.1)–(3.7) in compact form as � ξt+1 yt � = Φtξt + δt + υt, υt ∼ N(0, Ωt), t = 1, . . . , T, (3.8) with � T t Φt = Zt � � ct , δt = dt � � Rtηt , υt = ɛt � � Qt 0 , Ωt = 0 Ht � , (3.9) where Φt is of dimension (m + N) × m, δt is (m + N) × 1, υt is (m + N) × 1 and Ωt is (m + N) × (m + N). The initial values are collected in the (m + 1) × m matrix Σ which is defined as � P 1 Σ = a ′ 1 � . (3.10) For a survey on using SsfPack for state space modeling in macroeconomics and finance, see Zivot et al. (2002). 3.3 The Kalman filter and smoother Once a model is put into state space form, the Kalman filter can be employed to compute optimal forecasts of the mean and covariance matrix of the normally distributed state vector ξ t+1, based on the available information through time t. This section outlines how the Kalman filter can be used for estimation by filtering and smoothing: filtering bases an inference about the state vector only on the information up to time t; smoothing
3.3 The Kalman filter and smoother 21 incorporates the full set of information in the sample, where one distinguishes between state smoothing and disturbance smoothing. Subsection 3.3.4 shows how the Kalman filter deals with missing observations, which serves as a basis for the subsequent outline of using the Kalman filter for forecasting purposes. The initialization of the Kalman filter will be discussed in ¢ 3.3.6. The section closes with an interpretation of the Kalman filter when used in combination with non-Gaussian disturbances. The results in this section are given without proof and are primarily taken from Durbin and Koopman (2001, ¢ 4) where derivations can be found. 3.3.1 Filtering The purpose of filtering is to update our knowledge of the state vector as soon as a new observation y t becomes available. Reconsidering the state space model given by (3.1) through (3.7), the Kalman filter can be derived under the assumption that a1 and P 1 are known. The objective is to get the conditional distribution of ξ t+1 for t = 1, . . . , T , based on Y t, the set of observations up to time t, i.e. Y t = {y 1, . . . , y t}. 3.3.1.1 The general form of the Kalman filter Given the assumption of normality in all distributions of the system, all conditional distributions related to subsets of variables based on other subsets also have to be normally distributed. The required conditional distribution of ξ t+1 can be characterized by its mean at+1 and covariance P t+1: at+1 = E(ξ t+1|Y t), (3.11) P t+1 = V ar(ξ t+1 |Y t), (3.12) where the mean of the conditional distribution of ξ t+1 represents an optimal estimator of the state vector at time t + 1: it minimizes the mean squared error (MSE) matrix, E((ξ t+1 − at+1)(ξ t+1 − at+1) ′ |Y t), for all ξ t+1. Under the assumption that ξ t given Y t−1 is normally distributed with mean at and covariance P t, it can be shown that at+1 and P t+1 can be calculated recursively from at and P t as at+1 = T tat + Ktvt, (3.13) and with P t+1 = T tP tL ′ t + RtQtR ′ t , (3.14) vt = y t − E(y t|Y t−1) = y t − Ztat, (3.15) F t = V ar(vt) = ZtP tZ ′ t + Ht, (3.16) Kt = T tP tZ ′ tF −1 t , (3.17) Lt = T t − KtZt, (3.18) for t = 1, . . . , T . The m × N matrix Kt is referred to as the Kalman gain matrix. The N × 1 vector vt is the one-step ahead prediction error of y t given Y t−1. It represents
Page 1: Sascha Mergner Applications of Stat
Page 4 and 5: erschienen im Universitätsverlag G
Page 6 and 7: Bibliographische Information der De
Page 9 and 10: Contents List of figures . . . . .
Page 11 and 12: Contents ix 4.5 Model selection and
Page 13: Contents xi 7.5 Concluding remarks
Page 16 and 17: xiv List of figures 6.14 Histograms
Page 19: Notation and conventions The outlin
Page 22 and 23: xx Used abbreviations and symbols L
Page 24 and 25: xxii Used abbreviations and symbols
Page 26 and 27: xxiv Used abbreviations and symbols
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
Page 37: 1.3 Organization of the thesis 5 to
Page 40 and 41: 8 2 Some stylized facts of weekly s
Page 42 and 43: 10 2 Some stylized facts of weekly
Page 50 and 51: 18 3 Linear Gaussian state space mo
Page 75 and 76: Chapter 4 Markov regime switching M
Page 77 and 78: 4.1 Basic concepts 45 R t (%) 20 10
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
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5.2 Stochastic volatility 71 linear
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5.2 Stochastic volatility 73 3. An
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5.3 Multivariate conditional hetero
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84 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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Applications of state space models in finance

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