Applications of state space models in finance

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148 7 A Kalman filter based conditional multifactor pricing model following form: Rt = λ1,t ˆβ 1,t−1 + λ2,t ˆβ 2,t−1 + λ3,t ˆβ 3,t−1 + λ4,t ˆβ 4,t−1 + λ5,t ˆβ 5,t−1 λ6,t ˆ β 6,t−1 + νt, for (7.24) for t = 164, . . . , 683, with Rt and ˆ β 1,t−1, . . . , ˆ β 6,t−1 denoting 18 × 1 vectors of excess sector returns and factor loadings, respectively. The factor loadings used as independent variables have been estimated from one of the alternative time series regressions (KF, RLS, RR5 or RR1) of excess sector returns on the set of explanatory variables given by (7.18)–(7.20). Estimation of (7.24) for each date t of the out-of-sample period yields a time series of associated risk premia λk,t, k = 1, . . . , 6, each of length 520. 24 In accordance with (7.12)–(7.14) the significance of the chosen risk factors is tested by employing a t-test to the time series means of estimated risk premia. The cross-sectional regression results are summarized in Table 7.8, where the coefficient averages together with the corresponding Fama-MacBeth t-statistics are reported. To ascertain whether the proposed Kalman filter based conditional factor loadings are relatively better in explaining sector pricing, the cross-sectional regressions have also been conducted for the betas estimated by recursive least squares and by rolling OLS. Table 7.8 is divided into four panels: Panel A covers the complete out-of-sample period from 22 February 1995 to 2 February 2005; Panels B to D refer to the three subperiods of observed bull market, bear market and market recovery regimes. Overall, there is only little evidence that the multifactor model is better able to describe the cross-section of returns when conditional Kalman filter based factor loadings are used as independent variables. Over the entire estimation period, the average ¯ R 2 is only slightly higher than in case of the three alternative least squares specifications. The four different models explain between 28.9% (KF ) and 28.0% (RR1) of the total cross-sectional variation of excess sector returns. The only risk premium that is significantly different from zero over the complete sample is estimated for the Kalman filter based T S factor. Dividing the sample under consideration into three market-regime dependent subperiods reveals that both the average ability to explain the cross-sectional variability and the significance of estimated risk premia differ across subperiods. For the bull market period (Panel B) the risk premium of the SIZ factor is significantly positive for all models. The sign on SIZ reflects the outperformance of large caps over small caps during the dotcom bubble. Besides, for the KF and RR1 based factor loadings the risk premium of the T S factor is significantly negative at the 10% level. This means that during 24 For each cross-sectional regression at a given date t, six parameters have to be estimated with the dependent variable consisting of 18 data points only. Similar settings with comparable degrees of freedom are commonly employed in the literature, see, for example, Chen et al. (1986) who use 20 portfolios as dependent variables to estimate up to seven parameters in the cross-section. Nevertheless, in order to check whether the conclusion to be drawn in this subsection is sensitive to the dimension of the vector of dependent variables, (7.24) has also been estimated using 37 sectors instead of the 18 supersectors employed throughout this thesis. As the employment of a finer market segmentation leads to the same conclusion with respect to the practical relevance of time-varying factor loadings, only the case with 18 supersectors is discussed here; results of the Fama-MacBeth procedure using 37 sectors are available on request.
7.4 Empirical results 149 Table 7.8: Fama-MacBeth regression results I. This table presents the estimation results for the Fama-MacBeth regressions where the regressors are the out-of-sample factor loadings on the proposed set of explanatory variables. The factor loadings have been estimated from one of the time series regressions of excess sector returns on the set of explanatory variables as presented in 7.4.1. Average coefficients are expressed as percent per week ×10. The number of cross-sectional regressions is 520 for each model; t-statistics are reported in parentheses with the relevant critical value at the 95% (90%, 99%) level being −1.96 (−1.65, −2.59). Significant average coefficients are printed in boldface. The last column reports average adjusted coefficients of determination. Model BMR V GS SIZ T S OIL F X ¯ R 2 A. Complete out-of-sample period (22 February 1995–2 February 2005) KF 0.214 0.114 0.993 −12.041 0.321 −0.284 0.289 (0.186) (0.128) (1.268) (−1.694) (0.062) (−0.237) RLS 1.030 0.152 1.099 19.764 3.356 0.492 0.284 (0.543) (0.099) (1.059) (0.519) (0.381) (0.180) RR5 1.214 0.993 1.427 16.663 1.700 0.980 0.282 (0.755) (0.754) (1.448) (0.781) (0.223) (0.475) RR1 −0.675 0.524 0.912 −7.340 −1.292 −0.316 0.280 (−0.553) (0.610) (1.152) (−0.813) (−0.331) (−0.273) B. Bull market subperiod (22 February 1995–8 March 2000) KF 0.730 −2.000 2.706 −18.968 0.401 −1.132 0.209 (0.490) (−1.464) (2.346) (−1.829) (0.052) (−0.624) RLS 3.113 −3.797 3.755 58.609 21.553 3.575 0.196 (1.254) (−1.503) (2.680) (1.229) (1.488) (0.857) RR5 1.754 −1.375 4.358 48.533 12.872 2.164 0.215 (0.769) (−0.625) (3.205) (1.320) (0.985) (0.635) RR1 −0.483 −1.538 3.056 −23.658 −1.853 −1.515 0.189 (−0.324) (−1.234) (2.632) (−1.756) (−0.323) (−0.930) C. Bear market subperiod (15 March 2000–12 March 2003) KF −2.923 3.322 −2.012 −6.410 −1.153 1.093 0.420 (−1.127) (1.907) (−1.360) (−0.434) (−0.126) (0.489) RLS −6.521 6.264 −2.647 4.862 −25.850 −5.049 0.441 (−1.534) (2.399) (−1.160) (0.055) (−1.723) (−0.953) RR5 −3.093 5.020 −2.569 −12.368 −14.200 −0.227 0.419 (−0.938) (2.256) (−1.210) (−0.406) (−1.217) (−0.067) RR1 −3.658 3.948 −2.742 6.247 0.181 −0.970 0.406 (−1.290) (2.137) (−1.774) (0.365) (0.026) (−0.446) D. Market recovery subperiod (19 March 2003–2 February 2005) KF 3.813 0.661 1.187 −2.502 2.444 −0.205 0.296 (2.026) (0.897) (0.891) (−0.276) (0.239) (−0.104) RLS 7.449 0.989 −0.044 −60.188 1.151 1.057 0.268 (2.479) (0.992) (−0.029) (−0.971) (0.141) (0.306) RR5 6.603 0.919 −0.053 −22.285 −2.879 −0.264 0.244 (2.563) (1.116) (−0.039) (−0.921) (−0.411) (−0.104) RR1 3.546 0.591 0.989 14.628 −2.134 3.920 0.319 (1.613) (0.838) (0.841) (0.998) (−0.273) (1.600)
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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 130 and 131: 98 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 179: 7.4 Empirical results 147 Table 7.7
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
Page 225 and 226: References 193 Bulla, J. (2006). Ap
Page 227 and 228: References 195 Fabozzi, F. J. and J
Page 229 and 230: 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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