Applications of state space models in finance

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58 5 Conditional heteroskedasticity models intensive procedures. This is the major reason why practitioners prefer ARCH models to model time-varying volatilities in financial markets. However, the progress made with respect to the estimation of latent variable models over the past ten years increased the relative competitiveness of SV models. Yu (2002), for example, compares the ability of SV models to that of alternative ARCH-type models to predict stock price volatility and concludes that the SV model outperforms its competitors. Major advantages of the SV model, which have allowed the concept to grow up as a viable alternative to the modeling of conditional volatility, are: The ability to capture the stylized facts of excess kurtosis and leverage more naturally than a GARCH model. The provision of both filtered and smoothed estimates of conditional volatility. A treatment in continuous time, which is essential in mathematical finance and modern option pricing theory (not to be explored here). Explicit comparisons of the basic ideas, estimation and inference issues related to ARCH and SV models are provided, for example, by Danielsson (1994), Jacquier et al. (1994), Pagan (1996), Shephard (1996) and Andersen et al. (2005). This chapter is structured as follows. Section 5.1 introduces the basic GARCH framework. The concept of SV is the subject of Section 5.2. Alternative estimation methods with a focus on efficient Monte Carlo likelihood estimation are discussed. The chapter closes with a brief presentation of multivariate conditional heteroskedasticity models in Section 5.3. 5.1 Autoregressive conditional heteroskedasticity The aim of this section is to provide a summary of the basic theory of GARCH models as a prerequisite for subsequent analyses. For more details and extensions of the standard GARCH model, the reader is referred to one of the excellent surveys that have been made available over the last fifteen years. For example, Bollerslev et al. (1992) give an overview of the numerous empirical applications related to finance; Bera and Higgins (1993) comprehensively treat many of the GARCH extensions; Bollerslev et al. (1994) evaluate the most important theoretical aspects; Palm (1996) gives a survey on GARCH modeling related to finance including multivariate models. More recent articles include Andersen and Bollerslev (1998), Engle (2001b) and Diebold (2004). A collection of the most influential papers on ARCH/ GARCH is presented by Engle (1995). Subsection 5.1.1 introduces the basic univariate GARCH representation, summarizes the statistical properties of the model and shows how forecasts of conditional volatility can be produced. Subsection 5.1.2 looks at the two most influential nonlinear extensions of the GARCH model and demonstrates how to test for asymmetric effects. Subsection 5.1.3 discusses how to deal with non-Gaussianity. Subsection 5.1.4 closes with a summary of parameter estimation.
5.1 Autoregressive conditional heteroskedasticity 59 5.1.1 The GARCH(p, q) model For an observed univariate time series yt and a given information set Ωt−1, which contains the relevant information through period t − 1, the functional form of yt can be defined as yt = E(yt|Ωt−1) + ɛt, (5.1) with E(ɛt) = 0 and E(ɛtɛs) = 0 for all t �= s. The conditional mean E(yt|Ωt−1) represents the predictable part of yt; the disturbance term ɛt represents the unpredictable part of yt. In contrast to the standard regression model, in which ɛt is assumed to be unconditionally and conditionally homoskedastic, i.e., E � ɛ2 � 2 t = E(ɛt |Ωt−1) = σ2 for all t, the conditional variance of ɛt is allowed to vary over time: E(ɛ 2 t |Ωt−1) = ht, (5.2) with nonnegative function ht := ht(Ωt−1). A general expression for the conditional heteroskedasticity of ɛt is given by � ɛt = zt ht, (5.3) where zt is an IID process with E(zt) = 0 and V ar(zt) = 1. For now, zt is assumed to be normally distributed. Taking (5.3) and the assumed properties of zt together, it follows that ɛt is conditionally normal distributed with mean zero and variance ht. With the unconditional variance of ɛt being assumed to be constant, it can be shown that the unconditional expectation of ht is itself constant with σ2 := E � ɛ2 � 2 t = E(E(ɛt |Ωt−1)) = E(ht); cf. Franses and van Dijk (2000, ¢ 4) who, unless mentioned otherwise, serve as a basis for the outline on GARCH modeling in this section. The model is completed by finding a specification for the behavior of ht over time. In the basic ARCH(q) model by Engle (1982), the conditional variance of the error term at time t depends on the realized values of past squared disturbance terms: ht = ω + q� i=1 γiɛ 2 t−i , (5.4) Nonnegativeness and stationarity of ht are guaranteed for ω > 0, γi ≥ 0 for i = 1, . . . , q, and �q i=1 γi < 1. With ∀ γi = 0 the conditional variance reduces to a constant. For the case with q = 1, it becomes obvious that the ARCH model is able to capture volatility clustering: as ht is a function of the previous squared shock, large shocks of either sign tend to be precedented by large shocks and vice versa. It can be shown that the ARCH(1) model captures the excess kurtosis usually inherent to financial time series. While the ARCH(1) model captures the stylized facts of volatility clustering and excess kurtosis, it is unlikely that the model also accommodates for the characteristic features related to the autocorrelation function of squared disturbances ɛ2 t . As illustrated by Figure 2.3, the first-order autocorrelation function of squared weekly returns is not very high, which implies a low value for γ1. At the same time, the autocorrelation function is observed to be persistent, hence demanding a value of γ1 that is close to unity. One possibility to cope with the persistent autocorrelation, is to add additional
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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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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
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Page 101 and 102: 5.2 Stochastic volatility 69 likeli
Page 103 and 104: 5.2 Stochastic volatility 71 linear
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108 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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