Applications of state space models in finance

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102 6 Time-varying market beta risk of pan-European sectors correct for heteroskedasticity is to base wt on a filtered or smoothed estimator of the conditional volatility, calculated by one of the methods discussed in Chapter 5. Even though this does not necessarily give the most efficient estimator in small samples, the feasible GLS estimator is usually more efficient than OLS in the presence of heteroskedasticity (cf. Ghysels et al. 1996). The concept of WLS can be easily adapted to the random walk model discussed above. For the original state space model of the form the weighted transformation is Ri,t = βi,tR0,t + ɛi,t, (6.28) βi,t+1 = βi,t + ηi,t, (6.29) W LS Ri,t/wt = βi,t (R0,t/wt) + ɛi,t/wt, (6.30) W LS LS LS βi,t+1 = βW i,t + ηW i,t , (6.31) with regressand Ri,t/wt and regressor R0,t/wt. Summary statistics are reported in terms of the transformed regressand to ensure orthogonality of the corresponding residuals. In order to estimate the transformed model in (6.30) and (6.31) consistent estimates of the weighting factor are needed. In the following, wt is set equal to the filtered conditional standard deviation estimated from an auxiliary heteroskedastic regression model. The behavior of hi,t is modeled by a t-GARCH(1,1) model: Ri,t = β ∗ i R0,t + ɛ ∗ i,t, (6.32) ɛ ∗ i,t � = zi,t hi,t, (6.33) hi,t = ωi + γiɛ ∗2 i,t−1 + δihi,t−1, (6.34) where zi,t is assumed to be t-distributed. The weighting factor ˆwi,t can now be computed in annualized percentages as ˆwi,t = 100 ∗ � Ahi,t, (6.35) where the annualizing factor A = 52 is equal to the number of weekly returns per year (cf. Alexander 2001, ¢ 1.1). For all t = 1, . . . , T , ˆwt depends on the information set Ωt−1, which only contains relevant information through date t − 1. Therefore, the transformation based on filtered volatility estimated over the full sample is valid both for in-sample and out-of-sample purposes. The second issue addressed above is related to outliers. For many estimation methods outliers can be captured straightforwardly by employing a t-distribution for the disturbance terms. The introduction of t-distributed errors to a Kalman filter model is not trivial as the resulting likelihood cannot be evaluated analytically. One way of handling non-Gaussian state space models is to employ importance sampling techniques as introduced in the context of SV models in ¢ 5.2.3; see Durbin and Koopman (2000, ¢ 11) for details. However, in order to limit the level of complexity, in the following outliers should be treated without relying on simulation-based techniques. A common way to remove the influence of outliers is to truncate the variables that enter the model. According to Granger et al. (2000) outlying observations can be identified as those data points that
6.2 Modeling conditional betas 103 R t (%) 20 10 0 −10 −20 (a) Automobiles returns 1990 1995 2000 2005 R t (%) 15 10 5 0 −5 −10 −15 (b) DJ Stoxx broad returns 1990 1995 2000 2005 Figure 6.2: Weekly excess log-return series of (a) Automobiles and (b) the broad market. are at least four standard deviations away from the mean of regressand and regressors, respectively. They propose to employ outlier-reduced series where any data point above or below ˆµ ± 4ˆσ is replaced by ˆµ ± 4ˆσ; ˆµ and ˆσ are estimated from the original data. In this thesis, more rigid limit lines at three standard deviations are imposed. Any outlying observation in (6.30) is simply set equal to the mean of yt/wt or xt/wt plus — or minus for negative outliers — three standard deviations. Obviously, unless outliers and the standard deviations are determined simultaneously, outliers will distort the standard deviations to be estimated. This leads to inflated sigmas and possibly masks influential observations. However, the objective of the “three sigma”-rule is not to fundamentally remove all outliers, but to approximately remove the biggest distractive effects from the data. Therefore, the chosen procedure can be considered being appropriate in the following. The resulting model will be referred to as the generalized random walk (GRW) model. The estimates of conditional beta series are denoted as ˆ β GRW i,t . Of course, other methods for dealing with non-normality are available. For example, in order to take conditional heteroskedasticity explicitly into account, Harvey et al. (1992) proposed a modified Kalman filter estimated by QML, and Kim (1993) developed a state space models with Markov switching heteroskedasticity. An alternative approach to deal with outliers is discussed by Judge et al. (1985, ¢ 20) who, instead of truncating the independent and dependent variables directly, truncate the residuals from a robust regression. While those techniques may be more sophisticated, in the following the methodology should be kept as simple and as relevant for practical implementation purposes as possible. Hence, the preference will be on the methodology described above. Even though this approach is simple, its relevance can be tested in a straightforward fashion: if the forecasting accuracy is superior in comparison to the standard RW model, then the proposed modifications can be regarded as being justified. Note that all reported diagnostics refer to the trimmed generalized input variables, while the error measures to be used in the next section to evaluate the forecast performances will be based on the original return series. This allows for a fair comparison of the different Kalman filter based models.
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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
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
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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 177 and 178: 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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