Applications of state space models in finance

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70 5 Conditional heteroskedasticity models The enhancement of MCMC methods for the estimation of SV models and their extensions to cope with non-Gaussian errors and the leverage effect (cf. ¢ 5.1.2), is an active field with recent contributions, for example, by Jacquier et al. (2004), Omori et al. (2004) and Yu (2005). Overall, Bayesian estimators have been shown to outperform method of moments and QML approaches with respect to both, computing filtered volatility estimates and the estimation of parameters. On the other hand, they demand a large amount of computationally intensive simulations. Nontrivial modifications are required for certain extensions, such as the incorporation of explanatory variables. 5.2.2.3 Monte Carlo likelihood These undesirable features of MCMC procedures have led to the second branch of inference that attempts to evaluate the full likelihood: simulated maximum likelihood, also referred to as Monte Carlo likelihood (MCL). According to Durbin and Koopman (2001, ¢ 8.3) MCL techniques are more transparent and computationally more convenient for the estimation of SV models than MCMC methods. As these two features are particularly valuable to practitioners who do not dispose of expert knowledge of simulation techniques, MCL will be preferred over MCMC in this thesis. Danielsson (1994) was the first to apply MCL to the estimation of SV models. General contributions to the MCL literature were made by Shephard and Pitt (1997) and Durbin and Koopman (1997) who improved computational efficiency by employing importance sampling techniques. Sandmann and Koopman (1998) proposed an efficient MCL estimator, which was demonstrated to be a veritable alternative to MCMC procedures. They showed that efficient MCL, while offering comparable finite sample properties, is less computationally demanding than MCMC. As it is possible to approximate the likelihood arbitrarily close, inference can be performed by making use of Likelihood Ratio test statistics. Another comparative advantage of the proposed MCL technique is that only trivial modifications have to be imposed to extend the basic SV model to allow for heavy-tailed errors, leverage effects and explanatory variables; see, for example, Sandmann and Koopman (1998). Koopman and Hol-Uspensky (2001) proposed the SVin-Mean model, in which the mean may also be influenced by changes in the conditional volatility. Liesenfeld and Richard (2003) generalized the importance sampling method employed by Danielsson (1994) by making use of the efficient importance sampling procedure proposed by Richard and Zhang (1996). Lee and Koopman (2004) compared the two different importance sampling techniques by considering a generalized SV model with Student-t distributed observation errors. For a general introduction to SV models and a more detailed discussion of the various estimation techniques, see, for example, Shephard (1996), Ghysels et al. (1996) or Broto and Ruiz (2004). A collection of some of the most important papers on the topic is presented by Shephard (2005). 5.2.3 Efficient Monte Carlo likelihood estimation The method of MCL will be used throughout this thesis for the estimation of SV models. In contrast to Sandmann and Koopman (1998), who employed MCL combined with the
5.2 Stochastic volatility 71 linearized representation — thus dealing with a non-Gaussian state space model — the basic SV model without transformations will be considered in the following. As a comprehensive treatment of sampling methods would be beyond the scope of this thesis, only the main ideas behind the concepts of MCL and importance sampling will be outlined. For a discussion of importance sampling techniques, the reader is referred to Ripley (1987, ¢ 5). A comprehensive treatment of the handling of non-Gaussian and nonlinear state space models using simulation techniques, both from a classical and a Bayesian perspective, is given by Durbin and Koopman (2000) and Durbin and Koopman (2001, Part II). All inferential aspects will be considered from a classical standpoint. 5.2.3.1 The likelihood function Following Durbin and Koopman (1997), the likelihood of the basic SV model in (5.25) and (5.26) is given by � � L(ψ) = p(y|ψ) = p(y, θ|ψ)dθ = p(y|θ, ψ)p(θ|ψ)dθ, (5.35) where p(y|ψ), p(θ|ψ), p(y|θ, ψ) and p(y, θ|ψ) denote the marginal densities of y and θ, the conditional density of y given θ for given ψ and the joint density of y and θ given ψ, respectively. For the SV model, no analytical solution for this integral is available; direct numerical maximization is not feasible. The idea of employing MCL is to estimate the loglikelihood by simulation and find an estimate for ψ by numerical maximization. In principle, simulation could be used to estimate L(ψ) by generating M independent draws θ (i) from the density p(θ|ψ), for i = 1, . . . , M. In a second step, the likelihood could be estimated by M −1 �M i=1 p(y|θ(i) ). However, as most replications θ (i) would not resemble the process of θ under which y was observed, M would have to become large to gain some accuracy. As a consequence, this procedure turns out to be highly inefficient in practice (Lee and Koopman 2004). 5.2.3.2 Importance sampling Computational efficiency can be improved by applying importance sampling techniques. A conditional Gaussian approximating density, denoted as g(θ|y, ψ), for which random draws can be obtained is chosen as importance density. The approximating density should be as close as possible to p(θ|y, ψ). Sampling from this density and estimating the original likelihood by appropriately adjusting the Gaussian likelihood of the approximating model using a correction factor, is referred to as importance sampling (cf. Durbin and Koopman 2001, ¢ 11.1). To simplify the notation, in the following a density’s dependency on ψ will be suppressed as all densities depend on ψ. With the Gaussian importance density g(θ|y) the likelihood for the approximating Gaussian SV model can be rewritten as LG(ψ) = g(y) = g(y, θ) g(θ|y) g(y|θ)p(θ) = . (5.36) g(θ|y)
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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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xxiv Used abbreviations and symbols
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Preface The work on this book began
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Chapter 1 Introduction “Economist
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1.2 Research objectives 3 1.2 Resea
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1.3 Organization of the thesis 5 to
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8 2 Some stylized facts of weekly s
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10 2 Some stylized facts of weekly
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18 3 Linear Gaussian state space mo
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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
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Page 101: 5.2 Stochastic volatility 69 likeli
Page 105 and 106: 5.2 Stochastic volatility 73 3. An
Page 107 and 108: 5.3 Multivariate conditional hetero
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Page 116 and 117: 84 6 Time-varying market beta risk
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120 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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