Applications of state space models in finance

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124 7 A Kalman filter based conditional multifactor pricing model switching framework. In this thesis, time-varying sensitivities will be modeled directly as continuous stochastic random walk processes based on the state space framework introduced in Chapter 3. The research goal of this chapter is to study the practical relevance of time-varying factor loadings in a multifactor pricing framework. Based on the conditional multiple beta series to be estimated via the Kalman filter, it will be analyzed whether an explicit consideration of the time-varying impact of macroeconomics and fundamentals on European industry portfolios can be exploited in a profitable way, either from a risk management or from a portfolio management perspective. The present chapter contributes to the literature a synthesis of the classical cross-sectional regression approach by Fama and MacBeth (1973) and conditional factor loadings. The chapter is organized as follows. After briefly summarizing the concept of factor modeling, Section 7.1 reviews the anomalies literature with a focus on macroeconomic and fundamental factors. Section 7.2 outlines the conditional multifactor methodology. It will be differentiated between time series and cross-sectional regressions. Section 7.3 introduces the set of common risk factors. Section 7.4 discusses the empirical results and evaluates the relative out-of-sample forecasting ability of the proposed conditional multifactor specification. It will be analyzed whether the statistical superiority of the Kalman filter based model can be exploited in practice. Section 7.5 concludes. 7.1 Factor modeling As outlined in ¢ 6.1 the most widely used factor model is the CAPM, in which the excess return of the overall market is chosen as the single common factor. The measure of systematic risk is referred to as market beta. Despite the CAPM’s popularity, various studies over the last three decades have suggested that a single beta model, while describing a large portion of the common variation in returns, may not be sufficient to explain the cross-section of returns. The biggest challenge to the CAPM includes the empirical evidence that macroeconomic sources of risk and company-specific characteristics are priced beyond market risk. In accordance with these well-documented findings, the empirical deficiencies of the CAPM are most commonly explained by missing risk factors or by a misapproximation of the total wealth portfolio. 22 This leads to multifactor pricing models as motivated by the Intertemporal Capital Asset Pricing Model (ICAPM), introduced by Merton (1973), or the Arbitrage Pricing Theory (APT), developed by Ross (1976). The basic idea of a multifactor pricing model is that the common variation in asset returns can be accounted for by multiple common components, or risk factors. 23 Without explicitly differentiating between the ICAPM and the APT in the following, 22 Alternatively, instead of explaining the violations of the CAPM using risk-based arguments, nonrisk-based explanations have been proposed. They include, among others, data-snooping biases, the existence of market frictions, transaction costs and liquidity effects. As nonriskbased explanations will not find any consideration in this chapter, the reader is referred to MacKinlay (1995) for more details. 23 An introduction to the basic conception of asset pricing theory is provided in Appendix A. As the theory on multifactor models is well-established, it is not intended to derive the underlying assumptions of the various multifactor pricing models in this thesis; for any details and
7.1 Factor modeling 125 unconditional multifactor models, which describe the return generating process for asset i by linking returns to common sources of risk, can be specified in the general form of a beta pricing model: ri,t = αi + K� k=1 βikfk,t + ɛi,t, (7.1) where βik is referred to as factor loading or exposure, i.e. the sensitivity of the i-th asset on the k-th risk factor, denoted by fk,t for k = 1, . . . , K. The asset-specific intercept is given by αi. The idiosyncratic components of the return series, captured by the disturbances ɛi,t, are assumed to be uncorrelated with zero mean for i = 1, . . . , N, and to be uncorrelated with the set of risk factors, each for all t. Whenever the set of common factors does not have mean zero, it is usually convenient to construct zero mean factors as ˜f t := f t − E(f), (7.2) where f t is the K × 1 vector of risk factors (cf. Cochrane 2005, ¢ 9.4). In a world with existing risk-free assets, the constant term is usually assumed to be equal to the risk-free interest rate and (7.1) can be written in terms of excess returns, Ri,t, as defined in (2.2). With αi set to zero, this can be represented in matrix notation as Ri,t = β ′ if t + ɛi,t, (7.3) where β i is a K × 1 vector of factor loadings of asset i to the set of common factors. A central derivation of the APT theorem is that in the absence of arbitrage, expected excess returns conditioned by the information set at date t − 1, Ωt−1, depend approximately linear on the factor loadings: E(Rt|Ωt−1) ≈ Bλt, (7.4) where Rt is an N × 1 vector of excess returns, B is an N × K matrix of factor loadings and λt is a K × 1 vector of factor risk premia. The risk premium can be interpreted as the compensation for owning one unit of the k-th factor risk. In the following, we align ourselves with the majority of the literature and assume that (7.4) holds exactly: E(Rt|Ωt−1) = Bλt. (7.5) By testing whether risk premia are significantly different from zero, one can determine which risk factors are priced by the market. As indicated by the chosen notation, Rt and λt are usually considered for a given date t, while B is assumed to remain constant over time. Despite its intuitive appeal, the theory on multifactor models does not prescribe which and how many factors should be included. Answers to these questions can be derived from the so-called anomalies literature. It has emerged in the course of empirical works further references, see one of the many excellent texts on the subject, for example, Fama and French (1996), Campbell et al. (1997, 6) or Cochrane (2005, 9).
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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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Page 116 and 117: 84 6 Time-varying market beta risk
Page 155: Chapter 7 A Kalman filter based con
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 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
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Page 197 and 198: Figures 165 β t β t β t 2.0 1.5
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Figures 175 β t β t β t 1.2 1.1
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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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