Applications of state space models in finance

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96 6 Time-varying market beta risk of pan-European sectors null hypothesis of normality can be generally rejected, signs of autocorrelation in the residuals are observed at the 5% level for twelve out of eighteen sectors. Even though the ARCH-test is highly significant for all sectors, for eleven sectors the corresponding LMstatistics are lower than in case of OLS. It can be concluded that the RW specification is able to capture the heteroskedasticity in the resulting residuals at least partially. With regard to the goodness of fit, the R 2 is always significantly higher than in case of OLS. 6.2.3.2 The mean reverting model As an alternative to the random walk, the dynamic process of beta can be modeled as being mean reverting. In the MR model, an autoregressive process of order one, AR(1), with a constant mean is used for the evolution of beta: ˆβ MR i,t+1 − ¯ βi = φi(βi,t − ¯ βi) + ηi,t. (6.18) In order to use the Kalman filter for estimating the MR model, the mean beta coefficient is augmented into the state vector. This leads to the following state space model: � βi,t+1 − ¯ � � � � βi φi 0 βi,t − = ¯βi 0 1 ¯ � � � � � βi 1 0 ηi,t + , (6.19) ¯βi 0 1 0 yt = � � xt xt � βi,t − ¯ � βi + ɛt, (6.20) ¯βi where overall four parameters (σ 2 ɛi , σ2 ηi , ¯ βi, φi) have to be estimated. As outlined in ¢ 3.5.2.3, the AR(1) parameter φi should be restricted to values between zero and unity. This constraint is implemented in the estimation procedure by application of the parameter transformation (3.67). The estimated variances of observation and state disturbances are significant at the 1% level for every sector. Compared to the RW model above, the estimated variance for the observation equation, σ2 i , is generally lower in case of the MR parameterization; the opposite holds with respect to σ2 ηi , the variance of the dynamic process of the timevarying beta. For the MR model, two additional parameters have been estimated. The value for ¯ βi, which compares well to the estimated OLS betas as reported in Table 6.1, is always significant at the 1% level. The estimates for the second extra parameter can be clustered across all sectors into three groups. In the first group, φi is close to unity. The closer to unity the transition parameter gets, the more the conditional beta resembles its RW counterpart. In case of Food & Beverages, Healthcare and Personal & Household Goods, the MR betas literally follow a random walk. In the second group with values for φi around 0.5, the conditional betas return faster to their individual means, which implies more noisy series of conditional betas. In the third group, where φi is close to zero, the resulting beta series follow a random coefficient model as defined in ¢ 3.5.2.1. Note that the estimated speed parameters in the last group (Industrials and Travel & Leisure) are not statistically significant. According to the fit statistics, the residuals are generally non-normal. With the exception of Healthcare and Retail, the null of no autocorrelation can be rejected for all sectors. According to the conducted ARCH-test, the null of no heteroskedasticity cannot
6.2 Modeling conditional betas 97 Table 6.6: Parameter estimates for Kalman filter models. sectors. All estimates for σ 2 , σ 2 η, This table reports the estimated parameters for the Kalman filter based models for the eighteen DJ Stoxx σ 2 ζ and ¯ β are significant at the 1% level. For φ and the test statistics that refer to the observation errors, *** means significance at the 1% level (**: 5%, *: 10%). Sector Model σ 2 × 10 4 σ 2 η × 10 2 σ 2 ζ × 10 2 φ β¯ 2 log L BIC R JB a Q(12) b LM(6) c Automobiles RW 3.20 0.80 2289.9 −5.09 0.72 39.65*** 13.63 23.15*** MR 2.61 16.61 0.55*** 1.15 2304.5 −5.11 0.81 84.93*** 13.79* 13.02** MMR 2.50 19.96 0.22 0.01 2309.5 −5.12 0.82 115.11*** 14.26** 10.03 GWR 0.02 0.25 4668.2 −10.39 0.69 9.60*** 14.01 4.30 Banks RW 1.06 0.19 2792.9 −6.21 0.86 368.83*** 13.31 97.17*** MR 0.75 9.38 0.41*** 1.03 2808.8 −6.23 0.92 1186.10*** 25.16*** 15.65** MMR 0.75 7.44 0.10 0.00 2825.1 −6.27 0.92 1052.70*** 21.14*** 13.50** GWR 0.02 0.07 4664.6 −10.39 0.85 186.35*** 17.28** 1.10 Basics RW 3.22 0.47 2297.7 −5.11 0.62 417.28*** 25.31*** 116.97*** MR 3.07 2.08 0.92*** 0.96 2303.6 −5.11 0.65 525.94*** 22.91*** 112.63*** MMR 2.77 11.51 0.23 0.00 2308.5 −5.12 0.71 759.67*** 27.08*** 80.00*** GWR 0.02 0.13 4665.4 −10.39 0.70 20.71*** 23.04*** 4.18 Chemicals RW 1.85 0.23 2547.9 −5.67 0.73 287.48*** 16.34* 70.59*** MR 1.78 0.89 0.94*** 0.91 2551.5 −5.66 0.75 360.87*** 15.81** 63.37*** MMR 1.62 5.40 0.10 0.40* 2555.4 −5.67 0.79 578.40*** 20.31*** 52.85*** GWR 0.02 0.05 4653.5 −10.36 0.73 28.35*** 19.46** 8.76 Continued on the next page. . . a JB is the Jarque-Bera statistic for testing normality. The relevant critical values at the 95% (99%) level are 5.99 (9.21). b Q(l) is the test statistic of the Ljung-Box portmanteau test for the null hypothesis of no autocorrelation in the errors up to order l. In a structural model, the Q-statistic is asymptotically χ 2 distributed with l − w − 1 degrees of freedom, where w denotes the total number of estimated parameters (Harvey 1989, p. 259). c LM(l) is the LM-statistic of Engle’s ARCH test for the null hypothesis of no ARCH effects up to order l. The test statistic is asymptotically χ 2 distributed with l degrees of freedom. The relevant critical values at the 95% (99%) level are 12.59 (16.81).
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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
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
Page 173 and 174: 7.4 Empirical results 141 are consi
Page 175 and 176: 7.4 Empirical results 143 Table 7.6
Page 177 and 178: 7.4 Empirical results 145 β TS 0.4
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7.4 Empirical results 147 Table 7.7
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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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