Applications of state space models in finance

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84 6 Time-varying market beta risk of pan-European sectors dependent variables are the weekly excess DJ Stoxx sector return series introduced in ¢ 2.1; the DJ Stoxx Broad index serves as market proxy. The main purpose of the present chapter is to analyze the time-varying behavior of market betas for European sectors by applying elaborate modeling techniques. The respective ability of the various approaches to explain sector returns by movements of the overall market is compared. This chapter aims at contributing a comprehensive comparison of modeling techniques to the literature including non-standard procedures such as stochastic volatility, Markov switching and various Kalman filter based approaches. The results are expected to give an indication of how to proceed in the second part of empirical applications: the technique that turns out to model the systematic risk of European sectors most adequately will be employed in the subsequent chapter to analyze the time-varying impact of macroeconomic and fundamental determinants on industry portfolios. The remainder of the chapter is organized as follows. Section 6.1 gives a brief overview of the CAPM and defines an unconditional beta. Section 6.2 reviews the literature on time-varying betas. The theoretical part of this thesis is linked to the empirical analyses to be conducted in this chapter by summarizing the competing modeling techniques together with their respective parameter estimates. Section 6.3 discusses the estimated conditional beta series and evaluates their in-sample and out-of-sample forecasting performances. Section 6.4 concludes. 6.1 The unconditional beta in the CAPM The benchmark for time-varying market beta risk is the unconditional beta in the CAPM, introduced by Sharpe (1964) and Lintner (1965). The CAPM has played a dominant role in the field of asset pricing modeling over the last forty years. Assuming that investors can borrow and lend at a risk-free rate r f , it linearly relates the expectation of the return on asset i, E(ri), to the systematic risk — or beta — of that security. The systematic risk is measured against the market portfolio, which consists of all securities on the market: E(ri) = r f � + βi E(r0) − r f � , (6.1) with r0 being the market return. Using excess returns as defined in (2.2) instead of real returns, yields a more compact version of the Sharpe-Lintner CAPM: E(Ri) = βiE(R0), (6.2) where R0 denotes the realized excess return of the market portfolio and Ri denotes the excess return to sector i for i = 1, . . . , N. As the CAPM is designed as a single-period model a time dimension is missing. The unconditional beta of an asset, βi, is usually estimated by the OLS slope coefficient of the excess-return market model: Ri,t = αi + βiR0,t + ɛi,t, ɛi,t ∼ IID(0, σ 2 i ), (6.3)
6.1 The unconditional beta in the CAPM 85 for period t = 1, . . . , T . In the context of excess returns, αi is expected to be zero. Beta is defined as ˆβ OLS i = Cov(R0, Ri) . (6.4) V ar(R0) The error terms ɛi,t are assumed to be normally and independently distributed over time with zero mean and constant variance σ 2 i ; cf. Campbell et al. (1997, ¢ 5) who provide a review of the fundamental assumptions of the CAPM and their derivations. Table 6.1 summarizes the OLS results of the excess market model for the eighteen DJ Stoxx sector indices estimated over the full sample. As expected, the intercept is not different from zero at the 5% level for any sector. Therefore, if not mentioned otherwise, αi will be assumed to be zero in the following. The estimated betas are all significant at the 1% level. Over the entire sample, the lowest beta was estimated for Table 6.1: OLS estimates of the excess market model. This table presents summary statistics for OLS estimation of the excess market model; *** means significance at the 1% level (**: 5%, *: 10%). Sector α × 10 2 β R 2 JB a Q(12) b LM(6) c Automobiles −0.09 1.15*** 0.64 168.43*** 11.75 53.38*** Banks 0.03 1.06*** 0.82 2360.85*** 27.67*** 44.81*** Basics 0.03 0.90*** 0.54 358.16*** 20.31* 171.64*** Chemicals 0.00 0.91*** 0.66 500.13*** 19.48* 86.82*** Construction −0.01 0.89*** 0.69 369.89*** 17.70 47.81*** Financials −0.03 1.00*** 0.79 599.60*** 19.44* 79.92*** Food 0.04 0.65*** 0.50 1485.44*** 55.02*** 184.76*** Healthcare 0.09 0.78*** 0.50 324.62*** 12.25 58.89*** Industrials −0.03 0.98*** 0.83 930.30*** 13.41 58.00*** Insurance −0.09 1.27*** 0.77 1397.30*** 22.58** 74.91*** Media −0.05 1.22*** 0.67 4508.10*** 51.20*** 74.82*** Oil & Gas 0.07 0.76*** 0.43 279.22*** 36.97*** 114.59*** Personal 0.00 0.91*** 0.74 1055.98*** 21.01** 95.63*** Retail −0.05 0.95*** 0.61 964.95*** 19.48* 7.26 Technology −0.08 1.49*** 0.66 1079.03*** 10.21 98.73*** Telecom 0.01 1.19*** 0.64 464.12*** 24.50** 65.18*** Travel −0.01 0.77*** 0.65 598.81*** 18.62* 38.80*** Utilities 0.08* 0.69*** 0.62 38.38*** 15.19 36.20*** 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(12) is the test statistic of the Ljung-Box portmanteau test for the null hypothesis of no autocorrelation in the errors up to order 12. In case of OLS, the Q-statistic is asymptotically χ 2 distributed with 12 degrees of freedom. The critical values at the 95% (99%) level are 21.03 (26.22). c LM(6) is the LM-statistic of Engle’s ARCH test for the null hypothesis of no ARCH effects up to order 6. The test statistic is asymptotically χ 2 distributed with 6 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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Page 66 and 67: 34 3 Linear Gaussian state space mo
Page 75 and 76: 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 118 and 119: 86 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:
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7.3 The risk factors 135 introduced
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7.3 The risk factors 137 auxiliary
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7.4 Empirical results 139 Table 7.4
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7.4 Empirical results 141 are consi
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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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