Applications of state space models in finance

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14 2 Some stylized facts of weekly sector return series AC(R t) 2 AC(Rt ) 0.2 0.1 0.0 −0.1 (a) Broad −0.2 0 5 10 15 20 25 30 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 30 AC(R t) 2 AC(Rt ) 0.2 0.1 0.0 −0.1 (b) Insurance −0.2 0 5 10 15 20 25 30 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 30 AC(R t) 2 AC(Rt ) 0.2 0.1 0.0 −0.1 (c) Food & Beverages −0.2 0 5 10 15 20 25 30 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 30 Figure 2.3: Autocorrelation functions (first 30 lags) of the broad market, the Insurance sector and Food & Beverages. The top row displays correlograms of excess returns, the bottom row displays correlograms of the corresponding squares. 2.3 Implications The last column of Table 2.2 reports the unconditional correlation between the chosen market proxy and sector i for i = 1, . . . , 18, denoted as ρ0i. Correlations between an asset and the overall market play an important role in financial markets. They are widely used in portfolio and risk management applications. For the sample under consideration, all estimated values of ρ0i are higher than 0.65. This indicates a strong linear association between the respective sectors and the market index over the entire sample. However, given the empirical properties described above, especially with regard to the observed time-variation of volatilities and covariances, it is reasonable to assume that the true degree of correlation is not constant but changing over time. This is illustrated by considering the following regression relationship between sector i and the market proxy: Ri,t = αi + βiR0,t + ɛi,t, t = 1, . . . , T. (2.3) The ordinary least squares (OLS) estimator of the slope coefficient in this regression represents the usual estimator of an asset’s beta in the context of the CAPM, which will be dealt with in more detail in the empirical part of this thesis. Under the assumption that the true value of beta is constant, it is defined as βi = Cov(R0, Ri) , (2.4) V ar(R0) where Cov(R0, Ri) is the unconditional covariance of the return of the market proxy with the return of sector i, and V ar(R0) is the unconditional variance of market returns. Taking the observed stylized facts of volatility clustering and volatility co-movements
2.3 Implications 15 CUSUMSQ test 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 (a) Insurance 1990 1995 2000 2005 CUSUMSQ test 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 (b) Food & Beverages 1990 1995 2000 2005 Figure 2.4: CUSUMSQ tests with 5% confidence intervals for the excess return series of (a) Insurance and (b) Food & Beverages. into account, the numerator and denominator of (2.4) should be replaced by the conditional covariance and conditional variance, respectively, to allow the true beta to be time-varying. Based on the cumulative sum of squares (CUSUMSQ) of recursive residuals, Figure 2.4 demonstrates the instability of beta at the example of the two sectors Insurance and Food & Beverages. The CUSUMSQ test, as proposed by Brown et al. (1975), detects instability in the regression coefficients caused by time-varying second moments if the cumulative sum of squares moves outside two critical lines. For both sectors, the signal line moves outside the lower critical bound. This indicates instability in the relationship between the respective sector and the overall market during the sample period. Similar observations leading to the same conclusion have been made for all sectors (not reported here). The test statistics and the significance lines have been computed using the objectoriented matrix programming language Ox 3.30 by Doornik (2001) together with the package SsfPack 2.3 by Koopman et al. (1999). The findings above contradict the assumption of beta constancy and further motivate the scope of this thesis to deal with time-varying sensitivities in financial markets. Given previous evidence in the literature and economic arguments, which contradict the assumption of beta constancy, the focus throughout this thesis will be on the modeling of change, and not on testing the paradigm of beta stability. For an overview of alternative parameter stability tests in the context of beta, see, for example, Wells (1996, ¢ 2). With regard to the modeling of conditional relationships in financial markets, the literature distinguishes between two different approaches: time-varying sensitivities can be modeled as linear functions of observable state variables as proposed, for example, by Shanken (1990). However, as it is not clear which instrumental variables should be included, any choice may exclude relevant conditioning information. An alternative procedure is to rely on advanced time series techniques to model time-varying betas as stochastic, possibly hidden processes. As demonstrated by Leusner et al. (1996) a stochastic process may capture the impact of the complete set of potential determinants of systematic risk, thus avoiding the omitted variables problem. Following this route,
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 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
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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
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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
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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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