Applications of state space models in finance

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xxiv Used abbreviations and symbols Q β t block matrix determining the nature of regression coefficients Qsq Box-Ljung test statistic based on squared (excess) returns q row dimension of diffuse vector δ; the order of an ARCH process Rt system matrix in state equation; multivariate vector of excess returns R0 selection matrix ˜R matrix of unconditional variances of standardized residuals R0,t excess log-return of the market proxy excess log-return of sector i Ri,t R2 coefficient of determination ¯R 2 adjusted coefficient of determination ri,t log-return of sector i rt vector of weighted sum of future innovations r f t risk-free interest rate r column dimension of the system matrix Rt S − t−1 , S+ t−1 measures of asymmetry in tests for asymmetric GARCH effects St Markov chain S (t) history of the Markov chain St up to time t s2 p.e.v. for models with time-varying regression coefficients s2 ∗ ML estimator of σ2 ∗ depending on generalized recursive residuals SIZt size factor sk skewness of a series T t system matrix in state equation T number of dates included in the sample T St term structure factor ut smoothing error V t smoothed state variance matrix vt one-step ahead prediction error v † t ˜v recursive residuals † t v generalized recursive residuals + t ˜v least squares residuals + t generalized least squares residuals V GSt value growth spread factor wt sum of forward probabilities; weighting factor in the context of WLS w dimension of ψ sequence of observations Xt X (t) history of the sequence of observations Xt up to time t X (−u) sequence of random variables Xt with Xu being excluded x (−u) observations xt with xu being excluded xt realized observation at time t; asset payoff at time t Y t set of observations up to time t
Used abbreviations and symbols xxv y t multivariate time series vector of observations y stacked vector of observations with y = (y ′ 1, . . . , y ′ T )′ ¯y T +l minimum MSE forecast of yT +l given y yt time series of observations ¯y unconditional mean of yt y∗ t mean-corrected univariate time series zt IID process with mean zero and variance unity system matrix in observation equation Zt Mathematical symbols Corr(X, Y ) correlation between X and Y Cov(X, Y ) covariance between X and Y diag(·) diagonal matrix operator exp(x) the number e raised to the x power F (·) empirical cumulative distribution function E(X) expectation of X f(·) probability density function g gradient vector g(·) conditional Gaussian density function ⊙ Hadamard product (element-by-element multiplication) H Hessian matrix ∞ infinity In n-dimensional identity matrix I(·) indicator function L(·) likelihood function; lag operator with L(xt) := xt−1 Lc(·) concentrated likelihood function Ld(·) diffuse likelihood function log(x) natural logarithm of x N(·) normal density function P (A) probability that event A occurs p(·) probability function R the set of all real numbers S discrete random variable U(·), u(·) utility function V ar(X) variance of X vech(·) vech operator X random variable (discrete or stochastic) X ′ transpose of the matrix X (X)•j j-th column of the matrix X
Page 1: Sascha Mergner Applications of Stat
Page 4 and 5: erschienen im Universitätsverlag G
Page 6 and 7: Bibliographische Information der De
Page 9 and 10: Contents List of figures . . . . .
Page 11 and 12: Contents ix 4.5 Model selection and
Page 13: Contents xi 7.5 Concluding remarks
Page 16 and 17: xiv List of figures 6.14 Histograms
Page 19: Notation and conventions The outlin
Page 22 and 23: xx Used abbreviations and symbols L
Page 24 and 25: xxii Used abbreviations and symbols
Page 29: Preface The work on this book began
Page 33 and 34: Chapter 1 Introduction “Economist
Page 35 and 36: 1.2 Research objectives 3 1.2 Resea
Page 37: 1.3 Organization of the thesis 5 to
Page 40 and 41: 8 2 Some stylized facts of weekly s
Page 42 and 43: 10 2 Some stylized facts of weekly
Page 50 and 51: 18 3 Linear Gaussian state space mo
Page 75 and 76: 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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Chapter 7 A Kalman filter based con
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7.1 Factor modeling 125 uncondition
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7.1 Factor modeling 127 predictabil
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7.1 Factor modeling 129 usefulness,
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7.2 Specification of a conditional
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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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Applications of state space models in finance

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