Applications of state space models in finance

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78 5 Conditional heteroskedasticity models further reduced by imposing diagonality on Γ and δ, or by setting these matrices equal to scalars; both results in a loss of generality. As the number of parameters to be estimated simultaneously still becomes large with increasing N, the resulting computational problems soon become insurmountable. Various approximations to compute multivariate GARCH conditional covariance matrices based on univariate GARCH models have been proposed. Two of the most influential specifications will be summarized below. 18 5.3.1.4 The constant conditional correlation model The constant conditional correlation model (CCC) by Bollerslev (1990) reduces the computational complexity of the general multivariate GARCH(1,1) model by assuming the conditional correlations between ɛi,t and ɛj,t, denoted by ρij, to be constant. The diagonal vech model with constant correlations can be defined as hii,t = ωii + γiiɛ 2 i,t−1 + δiihii,t−1, (5.53) � � hij,t = ρij hii,t hjj,t, for all i �= j. (5.54) Equation (5.53) denotes the conditional variance of the returns of index i; Equation (5.54) is the conditional covariance between the returns of indices i and j. Alternatively, Bollerslev’s model can be written as Ht = DtRDt, (5.55) where Dt is an N × N matrix with the conditional standard deviations � hii,t on the diagonal; R denotes a symmetric positive definite N × N matrix that contains the unconditional correlations ρij. In the bivariate case, the CCC model becomes Ht = � � h11,t 0 0 � h22,t � � 1 ρ12 ρ12 1 � � � h11,t 0 0 � h22,t � , (5.56) with only seven (= N(N + 5)/2) parameters left to be estimated. The conditional covariance matrix is guaranteed to be positive definite, provided that R is positive definite and that the conditional variances hii,t are positive (cf. Franses and van Dijk 2000, ¢ 4.7). The consideration of different functional forms for the GARCH process, for example, allowing for asymmetric effects or non-Gaussian conditional densities, is straightforward: as the CCC model is exclusively based on univariate models, it is only necessary to respecify (5.53) in accordance with the outline presented in ¢ 5.1.2 or ¢ 5.1.3, respectively. 18 Alternative simplifying approaches to estimate multivariate GARCH models, which will not find any consideration in this thesis, include the factor multivariate GARCH model by Diebold and Nerlove (1989) where the conditional covariance structure of the observed variables arises from a joint dependence on common factors; the orthogonal GARCH model by Alexander (2000) which is based on univariate GARCH volatilities of a limited number of uncorrelated risk factors obtained from a principal component analysis; and the flexible multivariate GARCH model by Ledoit et al. (2003) who propose to estimate univariate models and to transform the estimated parameters such that the conditional covariance matrix is positive definite.
5.3 Multivariate conditional heteroskedasticity 79 5.3.1.5 The dynamic conditional correlation model The dynamic conditional correlation model (DCC) by Engle (2000) generalizes the CCC model by allowing the conditional correlations to be time-varying: Ht = DtRtDt, (5.57) The N × N matrix Rt contains the conditional correlations, which are estimated as follows: univariate GARCH models are estimated for each of the series in the Ndimensional vector of observations Y t. The resulting conditional standard deviations are used to build Dt and to define the N × 1 vector of standardized residuals: ˜ɛt = D −1 t ɛt. (5.58) These are employed to calculate the N × N matrix of unconditional variances of ˜ɛt: ˜R = 1 T T� t=1 ˜ɛt˜ɛ ′ t , (5.59) which is guaranteed to be positive definite. In the next step, the symmetric N × N covariance matrix of the standardized residuals, Q t, can be estimated as Q t = (1 − α − β) ˜ R + α(˜ɛt−1˜ɛ ′ t−1) + βQ t−1. (5.60) Finally, the matrix of conditional correlations is obtained by standardization of the elements in Qt: � Rt = diag q −1/2 11,t . . . q −1/2 � � NN,t Qt diag q −1/2 11,t . . . q −1/2 � NN,t . (5.61) As long as the sum of the nonnegative scalars α and β is smaller than one, this process will be mean-reverting. For α + β = 1, the process induced by Q t will be integrated, and (5.60) reduces to an exponential smoothing process: Q t = (1 − λ)(˜ɛt−1˜ɛ ′ t−1) + λQ t−1, (5.62) with 0 < λ < 1. According to Engle (2000) this leads to one of the simplest and most successful parameterizations of Rt, in which the individual elements can alternatively be estimated as a geometrically weighted average of the standardized residuals: 19 ρij,t = qij,t √ qii,tqjj,t = �t−1 o=1 λo˜ɛi,t−o˜ɛj,t−o � �t−1 o=1 λo ˜ɛ 2 i,t−o �t−1 o=1 λo˜ɛ 2 . (5.63) j,t−o With respect to forecasting dynamic conditional correlations, Engle (2000) developed the following forecasting expression for an l-step ahead forecast. It is based on an 19 This specification is also used by RiskMetrics , a well known risk management system developed by J.P. Morgan, with the smoothing parameter λ set to 0.94 for daily data (cf. Riskmetrics Group 1996).
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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 60 and 61: 28 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
Page 109: 5.3 Multivariate conditional hetero
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Page 116 and 117: 84 6 Time-varying market beta risk
Page 155 and 156: Chapter 7 A Kalman filter based con
Page 157 and 158: 7.1 Factor modeling 125 uncondition
Page 159 and 160: 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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