Applications of state space models in finance

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132 7 A Kalman filter based conditional multifactor pricing model with mean and standard error ˆσ 2 λk = ˆλk = 1 T 1 T (T − 1) T� t=1 ˆλk,t, (7.13) T� ( ˆ λk,t − ˆ λk) 2 , (7.14) where w( ˆ λk) is Student-t distributed with T − 1 degrees of freedom (cf. Campbell et al. 1997, ¢ 5.8). 7.2.2.2 Econometric issues Several problems related to the methodology may arise. The conditional betas in (7.11) are unknown and have to be estimated in a first step based on a time series regression. This induces an errors-in-variables problem. Secondly, even if the true betas were known, the errors of the proposed cross-sectional regression would be likely to be heteroskedastic and correlated, because the stock returns that are used as dependent variables are usually correlated for a given date t. As a consequence, the estimated standard errors might be unreliable (cf. Ferson and Harvey 1991). The empirical evidence presented by Ferson and Harvey (1999) suggests that the main results are robust to errors-in-variables. Nevertheless, different ways to explicitly cope with the issue have been proposed. Shanken (1992) presents a general procedure that corrects for errors in the estimates of conditional betas by using adjusted standard errors. Alternatively, better estimates of the conditional betas can be produced by using returns of portfolios rather than of single stocks (cf. Chen et al. 1986). As this thesis exclusively deals with industry portfolios, the impact of a potential errors-in-variables problem can be assumed to be of limited nature. The second problem of heteroskedastic and correlated errors is especially alarming in small samples where the t-ratios might be biased. One way to address this is to employ an efficient generalized least squares estimator where the T estimates are weighted according their reciprocal variances; see Ferson and Harvey (1999) for details. In the following, the analysis will be conducted based on ten years of weekly data. The sample is sufficiently large to assume that the corresponding t-ratios will be unbiased. Therefore, throughout this chapter, the widely used unadjusted standard errors of the Fama-MacBeth approach will be relied upon. 7.3 The risk factors The weekly excess returns on the eighteen DJ Stoxx sector indices introduced in ¢ 2.1 constitute the set of dependent variables to be studied. This section describes the risk factors to be used in the context of the proposed conditional multifactor model. According to the literature review provided in ¢ 7.1, various fundamental factors, in particular size and valuation, as well as different proxies for macroeconomic risks, which can be thought of having an impact on equity returns, have been previously analyzed. As this thesis focuses on weekly data, the spectrum of available macroeconomic series is limited. Many factors that are typically employed in empirical research, such as growth t=1
7.3 The risk factors 133 Table 7.1: Glossary and definition of risk factors. Symbol Definition Mnemonic a T S Change in the euro term structure defined as the BDBRYLD; GERMDRQ; difference between the yield of a 10-year German government bond and a money market rate (assembled series: until 18.11.92 German 3-month rate; thereafter Euro 3-month rate) EMINT3M OIL Log-return of the crude oil-brent 1-month forward contract OILBRNI F X Log-return of the synthetic US-dollar to euro exchange rate USEURWD V GS Difference between log-returns of the chosen value FRUS1VA; FRUS1GR; and growth indices (assembled series: until 2.7.97 Frank Russell 1000 Value and Growth indices; DJTSVAE; DJTSGRE thereafter DJ Stoxx indices) TMI Value and Growth SIZ Difference between log-returns of the DJ Stoxx DJSLARG; DJSSMAL TMI Large 200 index and the DJ Stoxx Small 200 index TMI BMR Residual excess benchmark return calculated from a time-varying regression of the log-return of the DJSTOXX; FIBOR3M DJ Stoxx Broad return index in excess of the riskfree rate, calculated from the 3-month FIBOR, on the set of macroeconomic and fundamental factors a This column provides the codes used to access the corresponding data series via Thomson Financial Datastream. in gross domestic product, unexpected inflation or consumer and industrial confidence indicators, are only surveyed monthly or quarterly. Table 7.1 lists the variables to be employed as risk factors in this study. They are all assumed to capture underlying exogenous influences that have an impact on all sectors. The focus in this chapter is on the modeling of risk using conditional factor loadings and not on factor selection procedures. Hence the factors have been chosen from widely tested variables that can be justified theoretically and that have been successfully employed in the literature in a similar context, as reviewed in ¢ 7.1.1. Following Ferson and Harvey (1991) who used a similar set of risk factors to study predictable components of stock and bond returns, it cannot be claimed that these factors uniquely pick up all relevant fundamental and macroeconomic risks. These variables could well represent a joint approximation for a set of unobserved variables that influence asset returns. However, the chosen factors are theoretically appealing. They each have been previously documented to capture different aspects of systematic risk. In order to work with a set of independent variables of the same length, the beginning of the sample period is set to the earliest date for which historical pricing data for all eighteen DJ Stoxx sectors is available (as outlined in ¢ 2.1, Stoxx Ltd. redefined its
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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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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: 7.2 Specification of a conditional
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 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
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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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