Applications of state space models in finance

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140 7 A Kalman filter based conditional multifactor pricing model Table 7.5: Risk factor cross-correlations. This table reports the estimated contemporaneous correlations between the risk factors for different time periods. Panel A covers to full sample from 8 January 1992 to 2 February 2005. The remaining panels refer to four subperiods: the first break takes place at 15 February 1995 representing the end of the formation period and the beginning of the out-of-sample period (520 obsrevations). The out-of-sample period itself is broken into three subsamples according to different market regimes: Panel C reaches to end of the TMT bubble, Panel D covers the subsequent bear market, and Panel E refers to the market recovery that began thereafter. Symbol BMR V GS SIZ T S OIL F X A. 8 January 1992–2 February 2005 (t = 1, . . . , 683) BMR 1.000 V GS −0.024 1.000 SIZ −0.023 −0.227 1.000 T S −0.011 −0.020 −0.047 1.000 OIL 0.035 0.020 −0.009 −0.007 1.000 F X 0.007 0.036 −0.204 −0.104 −0.001 1.000 B. 8 January 1992–15 February 1995 (t = 1, . . . , 163) BMR 1.000 V GS 0.024 1.000 SIZ 0.093 −0.235 1.000 T S −0.011 −0.097 −0.222 1.000 OIL 0.117 0.105 0.009 −0.012 1.000 F X −0.067 0.022 −0.240 0.037 −0.017 1.000 C. 22 February 1995–8 March 2000 (t = 164, . . . , 427) BMR 1.000 V GS −0.027 1.000 SIZ −0.010 −0.255 1.000 T S −0.007 0.104 −0.196 1.000 OIL 0.036 0.011 0.028 0.000 1.000 F X 0.009 0.055 −0.206 −0.135 0.005 1.000 D. 15 March 2000–12 March 2003 (t = 428, . . . , 584) BMR 1.000 V GS 0.000 1.000 SIZ −0.010 −0.173 1.000 T S 0.039 −0.121 0.286 1.000 OIL 0.021 −0.040 0.072 0.122 1.000 F X 0.006 0.053 −0.155 −0.141 −0.085 1.000 E. 19 March 2003–2 February 2005 (t = 585, . . . , 683) BMR 1.000 V GS −0.156 1.000 SIZ −0.225 0.110 1.000 T S −0.169 0.006 −0.002 1.000 OIL −0.149 0.233 0.018 −0.099 1.000 F X 0.112 −0.370 −0.041 −0.385 0.020 1.000
7.4 Empirical results 141 are considered. Each of the alternative specifications is based on the simple factor model Ri = βi1BMRt + βi2V GSt + βi3SIZt + βi4T St + βi5OILt + βi6F Xt + ɛi,t, ɛi,t ∼ N(0, σ2 i ). (7.20) The first alternative model employs recursive least squares (RLS). The factor model in (7.20) is estimated repeatedly using expanding subsets of the sample. As outlined in ¢ 7.3.4, the first 163 observations represent the formation period, on which the first estimate of the vector of factor loadings, ˆ β i,163, is based. Adding the subsequent observation to the subset of data leads to an estimate of β i,164, adding the next observation yields an estimate of β i,165 , and so on. This procedure is repeated until the number of observations in the subset grows to T = 682. This gives overall 520 estimates of ex-ante factor loadings; the last one is computed on 26 January 2005 based only on information up to this date. The other two alternative models are based on rolling regressions with estimation periods of one and five years, respectively. In the literature, factor models are most commonly estimated using five years of data (cf. Campbell et al. 1997, ¢ 5.1). In this chapter, a one-year rolling period is chosen so that short-term effects can be captured by simple OLS. In case of the one-year rolling regression (RR1) specification, (7.20) is estimated using a sample spanning an estimation period from t − 51 to t. The sample is rolled forward week-by-week for t = 163, . . . , 682 such that the 520-th loading is obtained using data from 4 February 2004 to 26 January 2005. Analogously, for the five-years rolling regression (RR5) setup the sample size is targeted at 260 observations. As before, 520 ex-ante estimates of the vector of factor loadings are generated. As the complete data set contains only 683 weeks, the first 97 ex-ante instruments are generated based on expanding samples consisting of less than 260 data points. The first estimate is available for 15 February 1995 and is based on 163 observations. After 18 December 1996 all factor loadings are based on individual samples that contain the respective previous five years of weekly data. Following the procedures described above, four different estimates of the 18×6 matrix Bt of ex-ante factor loadings, denoted as ˆ B KF t , ˆ B RLS t , ˆ B RR1 t and ˆ B RR5 t , are obtained for t = 163, . . . , 682. In a second step, these estimates will be employed as instruments to estimate the corresponding risk premia in a cross-sectional setting according to (7.11). Unless mentioned otherwise, all estimation procedures in this chapter have been carried out using Ox 3.30 by Doornik (2001). For all Kalman filter based computations, Ox has been used together with the package SsfPack 2.3 by Koopman et al. (1999). The remainder of this section presents the empirical results and is organized as follows: Subsection 7.4.1 summarizes the estimation results and compares the various factor loading series. Subsection 7.4.2 compares the out-of-sample forecasting accuracy of the Kalman filter based sensitivities with that of the proposed alternatives. Subsection 7.4.3 analyzes whether the superior forecasting performance of the Kalman filter based specification can be exploited either from a risk management or from a portfolio management perspective.
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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 122 and 123: 90 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: 7.4 Empirical results 139 Table 7.4
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
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
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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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