Applications of state space models in finance

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142 7 A Kalman filter based conditional multifactor pricing model 7.4.1 Estimation of factor loadings The estimation results for the fully specified Kalman filter based model in (7.18)–(7.19) are provided in Table 7.6. The estimated hyperparameters are broadly significant. While the market factor is significant for all sectors except for Travel & Leisure, both fundamental factors are significant at the 1% level for all sectors. With regard to the set of macroeconomic factors it can be observed that not all factors are relevant for all sectors. The term structure factor, which is supposed to pick up systematic risks that are related to a changing slope of the yield curve, turns out to be significant for thirteen out of eighteen sectors. The estimated sensitivities to changes in the oil price are significantly different from zero for twelve sectors. Surprisingly, the oil price does not represent a systematic risk factor for the energy-related Chemicals, Utilities and Travel & Leisure sectors. This leaves the latter as the only sector with only four significant factors. The overall most important macroeconomic factor in explaining the sector return series over time is the exchange rate. With the exception of Healthcare, Personal & Household Goods and Telecommunications, the dollar is significant for all sectors. As measured by the adjusted coefficient of determination, the proposed specification with the selected set of explanatory variables explains between 65% and 93% of the variance of sectors. The lowest values of ¯ R 2 are observed for Travel & Leisure and the defensive sectors Food & Beverages, Healthcare, Oil & Gas and Utilities. The model is found to be more appropriate for financials and non-financial cyclicals than for defensives and TMT. Regarding the estimation results of the alternative least squares based multiple factor specifications given by (7.20), the finding that the market and fundamental factors are more significant than the macroeconomic factors is only partially maintained. For the RLS specification, both the market and the size factor are significant for all eighteen sectors. With the exception of Construction and Healthcare, the value-growth spread is significant at least at the 10% level for all sectors. In contrast to the observation made above, the term structure factor also represents a systematic risk factor for sixteen sectors. The exchange rate factor is only significantly different from zero for Automobiles. Changes in the oil price represent the second most important macroeconomic factor. As expected the explanatory power is lower than in case of the more flexible Kalman filter based model: as measured by the adjusted R 2 between 49% (Food & Beverages) and 78% (Banks) of the sector variance can be explained by movements of the chosen systematic risk factors. The estimation results for the rolling RR1 and RR5 specifications are summarized by reporting the respective ranges of estimated coefficients. It can be seen that all factors are significant for each sector at some stage. This supports the hypothesis of time-varying factor sensitivities. For a detailed summary of the estimated parameters of the alternative least squares based multifactor models, see Table C.7 in the appendix. Examples of the paths of Kalman filter based factor loadings are illustrated in the following set of figures. All charts show filtered state estimates so that large outliers may occur at the beginning of the respective series of factor loadings. The two panels in Figure 7.1 display the respective time-varying exposure of Technology and Food & Beverages to the value-growth spread. Technology’s value-growth beta became clearly negative during the bubble and its subsequent aftermath, which reflects the growth
7.4 Empirical results 143 Table 7.6: Parameter estimates for multifactor Kalman filter models. sectors. Columns This table reports the estimated parameters for the Kalman filter based multifactor specification for the eighteen DJ Stoxx 3–8 report the estimated state disturbance variance terms σ 2 ηk for k = 1, . . . , 6, with k referring to the respective regressor (1: BMR, 2: VGS, 3: SIZ, 4: TS, 5: OIL, 6: FX); *** means significance at the 1% level (**: 5%, *: 10%). The last column shows adjusted coefficients of determination. Sector σ 2 × 10 4 σ 2 η1 × 10 4 σ 2 η2 × 10 4 σ 2 η3 × 10 4 σ 2 η4 × 10 4 σ 2 η5 × 10 4 σ 2 η6 × 10 4 log L BIC R¯ 2 Automobiles 3.14*** 8.80*** 43.37*** 19.61*** 3.03*** 0.11*** 42.13*** 1692.1 −4.89 0.77 Banks 0.78*** 1.68*** 29.82*** 96.09*** 2.63*** 0.00 3.32*** 2105.7 −6.10 0.93 Basics 3.03*** 6.07*** 68.37*** 8.24*** 1.66*** 0.10*** 25.96*** 1716.0 −4.96 0.70 Chemicals 1.92*** 2.52*** 16.46*** 36.45*** 0.90*** 0.00 60.08*** 1861.0 −5.38 0.77 Construction 1.32*** 2.31*** 18.37*** 24.25*** 1.06*** 0.13*** 27.40*** 1979.0 −5.73 0.81 Financials 1.04*** 1.11*** 11.78*** 59.83*** 2.36*** 1.12*** 13.26*** 2025.7 −5.86 0.89 Food 1.84*** 5.66*** 25.15*** 18.80*** 0.22*** 0.01* 6.08*** 1900.2 −5.50 0.65 Healthcare 2.57*** 0.79*** 61.77*** 39.41*** 0.00 0.03*** 0.00 1802.7 −5.21 0.65 Industrials 0.79*** 2.59*** 43.62*** 29.01*** 1.52*** 0.00** 24.84*** 2113.8 −6.12 0.91 Insurance 1.79*** 31.85*** 0.00*** 85.91*** 3.87*** 0.00 8.54*** 1855.6 −5.37 0.89 Media 2.42*** 22.02*** 64.54*** 30.86*** 1.64*** 0.56*** 5.13*** 1775.1 −5.13 0.85 Oil & Gas 2.86*** 0.40*** 12.48*** 48.60*** 0.00 0.91*** 31.06*** 1758.9 −5.08 0.67 Personal 1.53*** 6.73*** 46.08*** 19.53*** 0.71*** 0.00 0.00 1946.7 −5.63 0.79 Retail 1.69*** 2.62*** 9.74*** 31.96*** 0.22*** 0.20*** 16.79*** 1923.5 −5.57 0.72 Technology 3.04*** 36.69*** 165.38*** 28.17*** 4.10*** 0.12*** 11.59*** 1680.2 −4.85 0.88 Telecom 3.21*** 2.99*** 22.99*** 20.19*** 0.00 0.03*** 0.00 1738.8 −5.02 0.78 Travel 3.14*** 0.00 1.56*** 47.01*** 0.00 0.00 43.06*** 1737.6 −5.02 0.66 Utilities 1.52*** 4.65*** 21.57*** 29.40*** 0.00 0.00 4.10*** 1970.7 −5.70 0.69
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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
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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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Page 124 and 125: 92 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: 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
Page 209 and 210: Appendix C Tables
Page 211 and 212: Tables 179 Table C.1 — continued
Page 213 and 214: Tables 181 Table C.3: In-sample mea
Page 215 and 216: Tables 183 Table C.5: Out-of-sample
Page 217 and 218: Tables 185 Table C.7: Parameter est
Page 219 and 220: Tables 187 Table C.7 — continued
Page 221 and 222: Tables 189 Table C.8: Out-of-sample
Page 223 and 224: 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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