Applications of state space models in finance

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118 6 Time-varying market beta risk of pan-European sectors probability density probability density probability density 1.2 1.0 0.8 0.6 0.4 0.2 0.0 (a) OLS Median = 0.249 F(0) = 30% −1.0 −0.5 0.0 0.5 1.0 1.2 (d) RW Median = 0.298 1.0 0.8 0.6 0.4 0.2 F(0) = 25% 0.0 −1.0 −0.5 0.0 0.5 1.0 2.0 1.5 1.0 0.5 (g) GRW Median = 0.254 F(0) = 14% 0.0 −1.0 −0.5 0.0 0.5 1.0 probability density probability density probability density 1.2 (b) tG Median = 0.278 1.0 0.8 0.6 0.4 0.2 F(0) = 24% 0.0 −1.0 −0.5 0.0 0.5 1.0 1.2 (e) MR Median = 0.260 1.0 0.8 0.6 0.4 0.2 F(0) = 25% 0.0 −1.0 −0.5 0.0 0.5 1.0 (h) MSM 1.2 Median = 0.244 1.0 0.8 0.6 0.4 0.2 F(0) = 27% 0.0 −1.0 −0.5 0.0 0.5 1.0 probability density probability density probability density 1.2 (c) SV Median = 0.289 1.0 0.8 0.6 0.4 0.2 F(0) = 28% 0.0 −1.0 −0.5 0.0 0.5 1.0 1.2 1.0 0.8 (f) MMR Median = 0.295 F(0) = 23% 0.6 0.4 0.2 0.0 −1.0 −0.5 0.0 0.5 1.0 (i) MS 1.4 1.2 Median = 0.250 1.0 0.8 0.6 0.4 0.2 F(0) = 29% 0.0 −1.0 −0.5 0.0 0.5 1.0 Figure 6.13: Histograms of Spearman’s out-of-sample rank correlations (100 samples). correlations are smaller than zero. This means that the risk of generating a misleading signal is reduced. When evaluating a model, a common way to take the risk related to a forecast explicitly into account is to calculate an information ratio (IR). Alexander (2001, p. 445) defines an information ratio as “the mean prediction error divided by the standard deviation of the prediction error”. In the context of the cross-sectional analysis using rank correlations, an information ratio for a given modeling techniques n can be defined as IRn = E(ρS � n) , (6.44) V ar(ρS n) where ρS n = {ρS n,1, ρS n,2, . . . , ρS n,T }. The computed information ratios are reported in Table 6.8. For the chosen out-of-sample period, the GRW model is confirmed to offer the best risk-adjusted forecasting performance, followed by the MMR and the RW model. To check which of these three models yields the best forecasting performance over a more representative period of time, an out-of-sample period of ten years based on 520 samples Table 6.8: Information criteria of out-of-sample rank correlations. OLS tG SV RW MR MMR GRW MSM MS 0.63 0.65 0.64 0.70 0.64 0.72 1.42 0.62 0.65
6.4 Concluding remarks 119 of 150 weekly observations is considered in the second step of the proposed evaluation procedure. 6.3.3.2 Step II: Out-of-sample period of ten years to identify the overall best modeling approach Table 6.9 summarizes the calculated mean absolute and squared errors for the RW, GRW and MMR model, respectively, by reporting their averages across all sectors; for a comprehensive sectoral breakdown, see Table C.6 in the appendix. According to the considered average mean errors, the out-of-sample forecasting performance of all three models over the last ten years of the given sample is very similar. Even though the average MAE is slightly lower for the MMR model and the average MSE is lowest for the RW model, based on these results neither approach significantly stands out. This is also confirmed by looking at the relative ranks, which indicate that the different modeling approaches yield an average rank of around two. Table 6.9: Average out-of-sample MAE and MSE across sectors (520 samples). Mean absolute error (×10 2 ) Mean squared error (×10 4 ) RW GRW MMR RW GRW MMR Average error 1.311 1.310 1.309 3.584 3.600 3.589 Average rank 2.33 2.11 1.94 1.94 2.22 2.17 When Spearman’s rank correlation coefficient is employed to evaluate the forecasting performance in a cross-sectional context, the RW model has a small advantage over its competitors. While Figure 6.14 illustrates that the realized rank correlations are similarly distributed for all three models, the median rank correlation as well as the information criteria of the RW model are slightly higher. Overall, neither the less parsimonious moving mean reverting nor the generalized random walk model, which are both motivated by their respective capability to capture volatility clusters and outliers, yield a forecasting advantage over the random walk specification. This result suggests that heteroskedasticity and outliers can be considered as being “third-order” problems in the context of applying the Kalman filter to model the time-varying behavior of systematic risk for pan-European sector indices. 6.4 Concluding remarks Despite the considerable empirical evidence that systematic risk is not constant over time, only few studies deal with the explicit modeling of the time-varying behavior of betas. Previous studies with a focus on Australia, India, New Zealand, the U.S. and the UK primarily employed Kalman filter and GARCH based techniques. The empirical analysis presented in this chapter contributes an investigation of time-varying betas for pan-European industry portfolios. The spectrum of modeling techniques is extended by (i) incorporating two Markov switching approaches, whose capabilities to model time-
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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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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
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
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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
Page 161 and 162: 7.1 Factor modeling 129 usefulness,
Page 163 and 164: 7.2 Specification of a conditional
Page 165 and 166: 7.3 The risk factors 133 Table 7.1:
Page 167 and 168: 7.3 The risk factors 135 introduced
Page 169 and 170: 7.3 The risk factors 137 auxiliary
Page 171 and 172: 7.4 Empirical results 139 Table 7.4
Page 173 and 174: 7.4 Empirical results 141 are consi
Page 177 and 178: 7.4 Empirical results 145 β TS 0.4
Page 185 and 186: 7.4 Empirical results 153 cumulativ
Page 187 and 188: Chapter 8 Conclusion and outlook Th
Page 189 and 190: Conclusion and outlook 157 cannot o
Page 191 and 192: Appendix A Brief review of asset pr
Page 193 and 194: A.3 Alternative asset pricing model
Page 195 and 196: Appendix B Figures
Page 197 and 198: Figures 165 β t β t β t 2.0 1.5
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Figures 169 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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