Applications of state space models in finance

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viii Contents 3.3.7 The Kalman filter with non-Gaussian errors . . . . . . . . . . . . 27 3.4 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . 27 3.4.1 The loglikelihood function . . . . . . . . . . . . . . . . . . . . . . 27 3.4.1.1 Prediction error decomposition . . . . . . . . . . . . . . 28 3.4.1.2 Concentrated loglikelihood . . . . . . . . . . . . . . . . 29 3.4.2 Numerical maximization . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.3 The EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.4 Parameter restrictions . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Introduction of explanatory variables . . . . . . . . . . . . . . . . . . . . 32 3.5.1 Incorporation of regression effects . . . . . . . . . . . . . . . . . . 32 3.5.2 Time-varying parameter models . . . . . . . . . . . . . . . . . . . 33 3.5.2.1 The random coefficient model . . . . . . . . . . . . . . 34 3.5.2.2 The random walk model . . . . . . . . . . . . . . . . . 34 3.5.2.3 The mean reverting model . . . . . . . . . . . . . . . . 35 3.5.2.4 The moving mean reverting model . . . . . . . . . . . . 36 3.5.3 Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6.1 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6.1.1 Generalized recursive residuals . . . . . . . . . . . . . . 37 3.6.1.2 Generalized least squares residuals . . . . . . . . . . . . 37 3.6.2 Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6.2.1 Prediction error variance . . . . . . . . . . . . . . . . . 38 3.6.2.2 Coefficient of determination . . . . . . . . . . . . . . . . 39 3.6.2.3 Information criteria . . . . . . . . . . . . . . . . . . . . 39 3.6.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 Illustration: How to specify the MMR model for estimation using SsfPack 40 4 Markov regime switching 43 4.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Independent mixture distributions . . . . . . . . . . . . . . . . . 44 4.1.2 Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 The basic hidden Markov model . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3.1 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . 50 4.3.2 Direct numerical maximization . . . . . . . . . . . . . . . . . . . 50 4.3.2.1 Forward-backward probabilities . . . . . . . . . . . . . 51 4.3.2.2 Recursive evaluation of the loglikelihood . . . . . . . . 51 4.3.3 Standard errors of ML estimates . . . . . . . . . . . . . . . . . . 52 4.4 Forecasting and decoding . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4.1 Forecast distributions . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.2 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.2.1 Local decoding . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.2.2 State predictions . . . . . . . . . . . . . . . . . . . . . . 54 4.4.2.3 Global decoding . . . . . . . . . . . . . . . . . . . . . . 54
Contents ix 4.5 Model selection and validation . . . . . . . . . . . . . . . . . . . . . . . 56 5 Conditional heteroskedasticity models 57 5.1 Autoregressive conditional heteroskedasticity . . . . . . . . . . . . . . . 58 5.1.1 The GARCH(p, q) model . . . . . . . . . . . . . . . . . . . . . . 59 5.1.1.1 Statistical properties . . . . . . . . . . . . . . . . . . . . 60 5.1.1.2 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.2 Nonlinear extensions . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.2.1 Exponential GARCH . . . . . . . . . . . . . . . . . . . 62 5.1.2.2 GJR-GARCH . . . . . . . . . . . . . . . . . . . . . . . 62 5.1.2.3 Testing for asymmetric effects . . . . . . . . . . . . . . 63 5.1.3 Non-Gaussian conditional densities . . . . . . . . . . . . . . . . . 64 5.1.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 The basic stochastic volatility model . . . . . . . . . . . . . . . . 66 5.2.1.1 Linearized representation . . . . . . . . . . . . . . . . . 67 5.2.1.2 Statistical properties . . . . . . . . . . . . . . . . . . . . 67 5.2.2 Alternative estimation procedures . . . . . . . . . . . . . . . . . 68 5.2.2.1 Methods of moments and quasi maximum likelihood . . 68 5.2.2.2 Markov chain Monte Carlo . . . . . . . . . . . . . . . . 69 5.2.2.3 Monte Carlo likelihood . . . . . . . . . . . . . . . . . . 70 5.2.3 Efficient Monte Carlo likelihood estimation . . . . . . . . . . . . 70 5.2.3.1 The likelihood function . . . . . . . . . . . . . . . . . . 71 5.2.3.2 Importance sampling . . . . . . . . . . . . . . . . . . . 71 5.2.3.3 Filtering, smoothing and forecasting . . . . . . . . . . . 73 5.2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.4.1 Heavy-tailed distributed errors . . . . . . . . . . . . . . 74 5.2.4.2 Asymmetric effects . . . . . . . . . . . . . . . . . . . . . 74 5.3 Multivariate conditional heteroskedasticity . . . . . . . . . . . . . . . . . 75 5.3.1 Multivariate GARCH . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3.1.1 The vech model . . . . . . . . . . . . . . . . . . . . . . 76 5.3.1.2 The diagonal vech model . . . . . . . . . . . . . . . . . 77 5.3.1.3 The BEKK model . . . . . . . . . . . . . . . . . . . . . 77 5.3.1.4 The constant conditional correlation model . . . . . . . 78 5.3.1.5 The dynamic conditional correlation model . . . . . . . 79 5.3.2 Multivariate stochastic volatility . . . . . . . . . . . . . . . . . . 80 6 Time-varying market beta risk of pan-European sectors 83 6.1 The unconditional beta in the CAPM . . . . . . . . . . . . . . . . . . . 84 6.2 Modeling conditional betas . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.1 GARCH conditional betas . . . . . . . . . . . . . . . . . . . . . . 88 6.2.2 Stochastic volatility conditional betas . . . . . . . . . . . . . . . 92 6.2.3 Kalman filter based approaches . . . . . . . . . . . . . . . . . . . 94 6.2.3.1 The random walk model . . . . . . . . . . . . . . . . . 95
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: Contents List of figures . . . . .
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
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28 3 Linear Gaussian state space mo
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Chapter 4 Markov regime switching M
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4.1 Basic concepts 45 R t (%) 20 10
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4.1 Basic concepts 47 probability d
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4.3 Parameter estimation 49 X 1 X 2
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4.3 Parameter estimation 51 4.3.2.1
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4.4 Forecasting and decoding 53 4.4
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4.4 Forecasting and decoding 55 mod
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Chapter 5 Conditional heteroskedast
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5.1 Autoregressive conditional hete
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5.2 Stochastic volatility 65 or the
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5.2 Stochastic volatility 67 5.2.1.
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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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