Applications of state space models in finance

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26 3 Linear Gaussian state space models and the Kalman filter The first is to employ a diffuse prior that fixes a1 at an arbitrary value and allows the diagonal elements of P 1 to go to infinity. Following Durbin and Koopman (2001, ¢ 5.1) a general specification for the initial state vector is given by ξ 1 = a + Aδ + R0η 0, δ ∼ N(0, κIq), η 0 ∼ N(0, Q 0), (3.49) where the m × 1 vector a can be treated as a zero vector whenever none of the elements of of ξ 1 are known constants. The m × q matrix A and the m × (m − q) matrix R0 are fixed and known selection matrices with A ′ R0 = 0. The initial covariance matrix Q 0 is assumed to be known and positive definite. The q × 1 vector δ is treated as a random variable with infinite variance and is called the diffuse vector as κ → ∞. This leads to P 1 = P ∗ + κP ∞, (3.50) with P ∗ = R0Q 0 R ′ 0 and P ∞ = AA ′ . With some elements of ξ 1 being diffuse, the initialization of the Kalman filter is referred to as diffuse initialization. However, in cases where P ∞ is a nonzero matrix, the standard Kalman filter cannot be employed as no real value can represent κ as κ → ∞. It is necessary to find an approximation or to modify the Kalman filter in an appropriate way. The technique that will be used throughout this thesis is based on Harvey and Phillips (1979). The two authors propose to replace κ by a large but finite numerical value, which enables the use of the standard Kalman filter. They showed that this will yield a good approximation, where the remaining rounding errors are small. As an alternative to the large κ approximation, exact initialization techniques have been developed for κ → ∞. However, exact treatments turn out to be difficult to implement and cannot deal with the general multivariate linear Gaussian state space model directly. They will not find any consideration hereafter. For details on alternative exact treatments of the initial Kalman filter, see, for example, Ansley and Kohn (1985), de Jong (1988b, 1991) and Koopman (1997). The assumption of an infinite variance might be regarded unnatural as all observed time series have a finite variance. This leads to the third way of initialization: Rosenberg (1973) considers ξ 1 to be an unknown constant that can be estimated from the first observation y 1 by maximum likelihood. It can be shown that by this procedure, the same initialization of the filter is obtained as by assuming that ξ 1 is a random variable with infinite variance (cf. Durbin and Koopman 2001, ¢ 2.9). In the following, unless genuine prior information on a1 and P 1 is available, a diffuse prior with a = 0, P ∗ = 0 and P ∞ = I will be used such that ξ 1 ∼ N(0, κI). Following the recommendation of Koopman et al. (1999), κ is first set to 10 6 and then multiplied by the maximum diagonal value of the residual covariances to adjust for scale: κ = 10 6 � � Qt 0 × max 1, 0 Ht �� . (3.51) It should be noted that the initialization is not trivial for the multivariate case. It is possible that the part of F −1 t that is linked to P ∞ is sometimes singular with different cannot simply be expanded in powers of κ−1 for the first few ranks. In these cases F −1 t
3.4 Maximum likelihood estimation 27 elements of a multivariate series. As it is straightforward to deal with this problem for univariate series, Durbin and Koopman (2001, ¢ 5.1) propose to bring in the components of a multivariate series into the analysis one at a time. Thus, a multivariate series is converted into a univariate series. This leads to computational gains and significantly simplifies the process of diffuse initialization for multivariate series. For a detailed discussion of the univariate treatment of multivariate series, the reader is referred to Durbin and Koopman (2001, ¢ 6.4). 3.3.7 The Kalman filter with non-Gaussian errors The derivation of the Kalman filter presented above and the related estimation procedures are based on the assumption of normally distributed disturbances. It was shown how the conditional distribution of the state vector ξ t can be calculated recursively given the information set at time t, for all t = 1, . . . , T . As these conditional distributions are also normal, they are fully specified by their first two moments. These can be computed by the Kalman filter. The conditional mean of the state vector represents an optimal estimator in the sense that is has minimum MSE matrix. In case of non-Gaussian disturbances, the Kalman filter is no longer guaranteed to yield the conditional mean of the state vector. However, it nevertheless represents an optimal estimator in the sense that no other linear estimator has a smaller MSE. Therefore, the Kalman filter can still be employed when the normality assumption is dropped. The estimators obtained by maximizing the Gaussian likelihood function with observations that are not normally distributed are referred to as quasi-maximum likelihood (QML) estimators; cf. Hamilton (1994a). 6 3.4 Maximum likelihood estimation So far, the system matrices have been assumed to be known. In the more general case, they depend at least partially on ψ, the vector of unknown parameters. This section explains how the vector of unknowns can be estimated by means of maximum likelihood (ML). After briefly summarizing the general idea behind the concept of ML, Subsection 3.4.1 introduces the loglikelihood function for the general Gaussian state space model. It will be demonstrated how the general model can be reparameterized in order to reduce the dimension of the vector of parameters by one. Subsections 3.4.2 and 3.4.3 briefly overview the two alternative concepts of maximizing the loglikelihood: direct numerical maximization and the EM algorithm. Finally, Subsection 3.4.4 shows how parameter restrictions can be implemented. 3.4.1 The loglikelihood function In order to estimate a model by ML, it has to be parametric and fully specified through the joint probability density function; the parameter values have to contain all the necessary information for a simulation of the dependent variables. For the T sets of 6 For an introduction to state space models that take non-Gaussianity explicitly into account, which is beyond the scope of this thesis, see, for example, Durbin and Koopman (2001, Part II).
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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
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
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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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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