Applications of state space models in finance

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90 6 Time-varying market beta risk of pan-European sectors Table 6.3: Comparison of different GARCH(1,1) specifications. To decide whether (i) the innovations zt in (6.7) should be modeled by a normal or a standardized Student-t distribution, (ii) asymmetric effects should be explicitly taken into account, and (iii) a nonzero constant should be included in the mean equation in (6.6), eight different GARCH(1,1) specifications have been analyzed. This table reports the estimated values of the corresponding log-likelihood functions and the Bayesian information criteria. Sector symmetric GARCH asymmetric GARCH [zt ∼ N, µ �= 0] [zt ∼ N, µ = 0] [zt ∼ t, µ �= 0] [zt ∼ t, µ = 0] [zt ∼ N, µ �= 0] [zt ∼ N, µ = 0] [zt ∼ t, µ �= 0] [zt ∼ t, µ = 0] log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC Broad 2215.9 −4.91 2209.6 −4.90 2228.5 −4.93 2220.1 −4.92 2226.2 −4.93 2223.5 −4.93 2234.6 −4.94 2229.5 −4.93 Automobiles 1871.9 −4.14 1870.7 −4.15 1892.5 −4.18 1890.6 −4.19 1879.9 −4.15 1879.7 −4.16 1895.0 −4.18 1893.9 −4.18 Banks 2112.8 −4.68 2108.0 −4.68 2132.4 −4.72 2124.1 −4.71 2119.3 −4.69 2116.7 −4.69 2136.0 −4.72 2129.9 −4.71 Basics 1988.2 −4.40 1985.5 −4.40 2013.8 −4.45 2009.6 −4.45 1995.7 −4.41 1993.8 −4.42 2017.7 −4.45 2014.6 −4.45 Chemicals 2110.1 −4.67 2106.3 −4.67 2117.5 −4.68 2112.2 −4.68 2120.7 −4.69 2119.3 −4.69 2125.5 −4.69 2122.8 −4.70 Construction 2112.9 −4.68 2109.8 −4.68 2129.7 −4.71 2124.3 −4.71 2126.0 −4.70 2124.6 −4.71 2137.6 −4.72 2134.5 −4.72 Financials 2144.4 −4.75 2137.4 −4.74 2175.7 −4.81 2168.1 −4.80 2156.2 −4.77 2152.0 −4.77 2181.6 −4.82 2175.3 −4.81 Food 2240.5 −4.97 2238.1 −4.97 2257.3 −5.00 2252.1 −4.99 2247.5 −4.97 2246.3 −4.98 2261.8 −5.00 2258.0 −5.00 Healthcare 2083.3 −4.61 2076.7 −4.61 2090.4 −4.62 2083.8 −4.62 2088.4 −4.62 2084.6 −4.62 2093.1 −4.62 2088.4 −4.62 Industrials 2144.2 −4.75 2137.8 −4.74 2158.9 −4.78 2150.9 −4.77 2154.8 −4.77 2151.3 −4.77 2164.3 −4.78 2158.9 −4.78 Insurance 1973.8 −4.37 1971.3 −4.37 2006.3 −4.44 2003.1 −4.44 1981.3 −4.38 1980.4 −4.39 2008.6 −4.43 2006.2 −4.44 Media 1927.4 −4.27 1925.1 −4.27 1941.8 −4.29 1937.5 −4.29 1930.7 −4.27 1929.8 −4.27 1943.5 −4.29 1940.7 −4.29 Oil & Gas 2048.7 −4.54 2044.0 −4.53 2056.8 −4.55 2052.6 −4.55 2061.0 −4.56 2059.4 −4.56 2066.7 −4.56 2064.9 −4.57 Personal 1570.8 −4.56 1567.7 −4.56 1577.9 −4.57 1574.2 −4.57 1582.9 −4.59 1582.3 −4.60 1587.4 −4.59 1586.2 −4.60 Retail 1466.1 −4.25 1464.6 −4.26 1521.2 −4.41 1517.6 −4.41 1478.8 −4.28 1478.5 −4.29 1526.2 −4.41 1524.1 −4.42 Technology 1780.0 −3.94 1776.9 −3.94 1793.5 −3.96 1788.3 −3.96 1781.8 −3.93 1779.5 −3.94 1794.3 −3.96 1790.2 −3.95 Telecom 1863.2 −4.12 1858.5 −4.12 1866.1 −4.12 1861.0 −4.12 1864.9 −4.12 1861.3 −4.12 1867.4 −4.12 1863.3 −4.12 Travel 1640.8 −4.77 1637.4 −4.77 1652.3 −4.79 1649.7 −4.79 1653.5 −4.79 1652.9 −4.80 FTC a FTC FTC FTC Utilities 2270.9 −5.03 2265.7 −5.03 2284.4 −5.06 2276.1 −5.04 2273.4 −5.03 2269.7 −5.03 2285.6 −5.05 2278.6 −5.04 Mean BIC −4.55 −4.55 −4.58 −4.58 −4.56 −4.56 −4.57 −4.57 a Failed to converge.
6.2 Modeling conditional betas 91 Regarding asymmetric effects, symmetric GARCH models with t-distributed innovations are superior to the alternative GJR-counterparts. These results are in line with Bollerslev et al. (1992, ¢ 3.3). They argue that leverage effects are often attributable to a few outliers, which can be better captured by employing a fat-tailed distribution than by explicitly allowing for asymmetric effects. The inclusion of a constant term generally improves the values of the corresponding log-likelihood without leading to higher BIC. These results suggest to model ARCH-based conditional betas by a bivariate GARCH framework, in which univariate t-GARCH(1,1) models with nonzero constants in the mean equation are fitted to the excess returns of each sector and the overall market. The estimation results are summarized in Table 6.4. The corresponding beta series are denoted as β tG i,t . Table 6.4: Parameter estimates for t-GARCH(1,1) models. This table reports the estimated parameters for the t-GARCH(1,1) models for the eighteen DJ STOXX sectors and the DJ Stoxx Broad as market index; *** means significance at the 1% level (**: 5%, *: 10%). Sector µ × 10 2 ω × 10 4 γ δ DF a JB b Q(12) c Broad 0.25 *** 0.16 ** 0.13 *** 0.84 *** 9.29 *** 127.12 *** 16.96 Automobiles 0.17* 0.25** 0.10*** 0.88*** 7.03*** 223.82*** 17.46 Banks 0.26*** 0.12** 0.12*** 0.87*** 7.88*** 245.82*** 13.14 Basics 0.22*** 0.05 0.06*** 0.94*** 6.63*** 580.94*** 22.99** Chemicals 0.22*** 0.27** 0.14*** 0.82*** 9.48*** 44.56*** 9.70 Construction 0.23*** 0.24** 0.11*** 0.85*** 8.26*** 279.77*** 18.71* Financials 0.25*** 0.16* 0.14*** 0.84*** 6.27*** 485.41*** 37.67*** Food 0.20*** 0.23** 0.11*** 0.84*** 7.79*** 144.74*** 12.76 Healthcare 0.27*** 0.23* 0.10*** 0.86*** 10.54*** 27.956*** 13.70 Industrials 0.26*** 0.16** 0.14*** 0.84*** 8.26*** 139.28*** 34.59*** Insurance 0.18** 0.14** 0.09*** 0.90*** 6.01*** 648.31*** 14.16 Media 0.23*** 0.18** 0.10*** 0.88*** 7.76*** 88.42*** 10.92 Oil & Gas 0.22*** 0.20 0.08*** 0.89*** 9.43*** 29.39*** 9.38 Personal 0.23*** 0.51** 0.16*** 0.76*** 12.10*** 69.55*** 18.74* Retail 0.24*** 0.67** 0.14*** 0.78*** 6.17*** 14986.00*** 10.97 Technology 0.29*** 0.14** 0.11*** 0.89*** 8.70*** 137.26*** 17.75 Telecom 0.29*** 0.14* 0.09*** 0.90*** 15.42*** 8.94** 19.20* Travel 0.17** 0.54** 0.15*** 0.75*** 8.43*** 90.72*** 6.26 Utilities 0.24*** 0.38** 0.15*** 0.76*** 7.76*** 90.24*** 19.19* a DF denotes the number of degrees of freedom of the Student-t distribution, which has been estimated along with the other parameters of the t-GARCH(1,1) models. b JB is the Jarque-Bera statistic for testing normality. The relevant critical values at the 95% (99%) level are 5.99 (9.21). c Q(12) is the test statistic of the Ljung-Box portmanteau test for the null hypothesis of no autocorrelation in the errors up to order 12. The critical values at the 95% (99%) level are 21.03 (26.22).
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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 72 and 73: 40 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
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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
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7.4 Empirical results 141 are consi
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7.4 Empirical results 143 Table 7.6
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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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