Applications of state space models in finance

More documents

Recommendations

Info

48 4 Markov regime switching While γij(l) describes the conditional probability of being in state j at time t + l, with the Markov chain starting from state i at time t, it does not provide the marginal probability of being in state i at a given time t. With the probability distribution of the initial state, π(1) := (π1(1), . . . , πm(1)) = (P (S1 = 1), . . . , P (S1 = m)), the probability function of the state at time t is given by π(t) := (P (St = 1), . . . , P (St = m)) = π(t)Γ l−1 . (4.10) For a homogeneous and irreducible Markov chain, π(t) can be shown to converge to a fixed vector πs := (π1, . . . , πm) for large t. This unique vector of dimension m satisfies πs = πsΓ, (4.11) and is called the vector of stationary transition probabilities. For existing πs, a Markov chain is referred to as being stationary if πs describes the marginal distribution of the states for all t = 1, . . . , T . For more details on the well developed theory of Markov chains and further references, see, for example, Hamilton (1994b, ¢ 22). 4.2 The basic hidden Markov model The sequence of observations and hidden states in an independent mixture model are independent by definition. Any potential correlation between the states cannot be captured by an independent mixture as it does not take into account the respective information. One method of modeling serially correlated time series is to use an unobserved Markov chain to select the parameters. This yields the hidden Markov model as a special dependent mixture model. With {Xt} = {Xt, t = 1, . . . , T } denoting a sequence of observations and {St} = {St, t = 1, . . . , T } denoting a Markov chain in the state space {1, . . . , m}, their respective histories up to time t can be written as X (t) := {X1, . . . , Xt}, (4.12) S (t) := {S1, . . . , St}. (4.13) Consider a stochastic process that consists of two elements: (i) an underlying and unobserved parameter process {St} for which the Markov property (4.7) holds, and (ii) a state-dependent observation process {Xt}, which fulfills the conditional independence property P (Xt = xt|X (t−1) = x (t−1) , S (t) = s (t) ) = P (Xt = xt | St = st). (4.14) This means that with known St, Xt only depends on St and not on the history of states or observations. The pair of stochastic processes {Xt} and {St} is referred to as an m-state hidden Markov model whose basic structure is illustrated in Figure 4.3, which is taken from Bulla (2006, ¢ 2). Generally, different distributions are imposed for the various states. In this thesis, the Markov chain with transition probability matrix Γ will be assumed to be homogeneous
4.3 Parameter estimation 49 X 1 X 2 . . . S S S . . . 1 2 3 X 3 Observations Markov chain (unobserved) Figure 4.3: Basic structure of a hidden Markov model (source: Bulla 2006, p. 17). and irreducible and to have a unique stationary distribution πs (cf. ¢ 4.1.2). In practical applications, the underlying state process {St} is hidden and only the state-dependent sequence of observations {Xt} is known. Usually, the unobservable states can be interpreted in a natural way: for example, a hidden Markov model with two states could be employed in the context of the weekly return series of the Technology sector considered in ¢ 4.1.1: the observations in times of high volatility would be referred to by one state, while the other state would refer to low volatility markets. For more details on the basic hidden Markov model, including a derivation of its moments and marginal distributions, see, for example, MacDonald and Zucchini (1997, ¢ 2). 4.3 Parameter estimation The parameters of a hidden Markov model are generally estimated via the maximum likelihood principle. As the estimates represent a solution to a system of nonlinear equations, it is usually impossible to estimate the unknown parameters analytically. Instead, one has to refer either to direct numerical maximization procedures or to the EM algorithm, which was briefly mentioned in ¢ 3.4.3. In this thesis, the focus will be on parameter estimation by means of direct numerical maximization procedures. Provided that the initial values to be employed are adequately accurate, numerical maximization leads to faster convergence than the EM algorithm. Direct numerical maximization, which avoids the derivation of the required formula for the EM algorithm, can be used both for non-stationary and for stationary Markov chains; cf. Bulla and Berzel (2006) who compare the competing estimation procedures based on a simulation experiment. For details on the implementation of the EM algorithm with regard to HMMs, the reader is referred to Baum et al. (1970) and MacDonald and Zucchini (1997, ¢ 2). The likelihood function of the basic hidden Markov model is presented in ¢ 4.3.1. Subsection 4.3.2 briefly overviews parameter estimation by numerically maximizing the likelihood.
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 26 and 27:
xxiv 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
Page 77 and 78: 4.1 Basic concepts 45 R t (%) 20 10
Page 79: 4.1 Basic concepts 47 probability d
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
Page 113: 5.3 Multivariate conditional hetero
Page 116 and 117: 84 6 Time-varying market beta risk
Page 130 and 131:
98 6 Time-varying market beta risk
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
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 175 and 176:
Page 177 and 178:
7.4 Empirical results 145 β TS 0.4
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
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 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
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
Page 215 and 216:
Tables 183 Table C.5: Out-of-sample
Page 217 and 218:
Tables 185 Table C.7: Parameter est
Page 219 and 220:
Tables 187 Table C.7 — continued
Page 221 and 222:
Tables 189 Table C.8: Out-of-sample
Page 223 and 224:
References Abell, J. D. and T. M. K
Page 225 and 226:
References 193 Bulla, J. (2006). Ap
Page 227 and 228:
References 195 Fabozzi, F. J. and J
Page 229 and 230:
References 197 Harvey, A. C., E. Ru
Page 231 and 232:
References 199 Lie, F., R. Brooks,
Page 233 and 234:
References 201 Shephard, N. (1993).
Page 235:
State space models play a key role
show all

Applications of state space models in finance

Create successful ePaper yourself

Delete template?

Save as template?