Applications of state space models in finance

More documents

Recommendations

Info

52 4 Markov regime switching Taking the logarithm of LT gives the loglikelihood function as log LT = T� t=1 � � wt log . (4.24) wt−1 It can be shown that the ratio of the scaling factors can be obtained as wt/wt−1 = φ t−1Bt1 ′ . An efficient algorithm for recursively evaluating the loglikelihood based on the scaled forward probabilities can be derived with starting equations and updating equations log L0 = 0, (4.25) φ 0 = πs, (4.26) vt = φ t−1Bt, (4.27) ut = vt1 ′ , (4.28) log Lt = log Lt−1 + log ut, (4.29) φ t = vtu −1 t . (4.30) To obtain the loglikelihood function, the loop is repeated for t = 1, . . . , T (cf. Zucchini et al. 2006, ¢ 4.3). Together with a statistical software package that offers functions for numerical maximization (or minimization), this algorithm can be employed to estimate the unknown parameters of a hidden Markov model by means of direct numerical maximization. In this thesis, the ML parameters of HMMs are estimated using the R-functions nlm() and optim(), which can be employed to minimize a negative loglikelihood. 4.3.3 Standard errors of ML estimates In the context of HMMs it is not trivial to determine the accuracy of the ML parameters estimated by one of the methods described above. As the elements of the vector of estimated parameters can be shown to be correlated, it is not possible to compute the standard error of the overall model directly. One possibility to obtain distributional properties of the parameter estimates is to apply parametric bootstrap methods. As these procedures are beyond the scope of this thesis, in the following the quality of a hidden Markov model will be evaluated based on its forecast performance rather than using classical statistical fit statistics. For details on how parametric bootstrap procedures can be employed to analyze the distributional properties of parameter estimates, the reader is referred to Zucchini et al. (2006, ¢ 4.4) and the references given therein. 4.4 Forecasting and decoding Based on Zucchini et al. (2006, ¢ 5) this section briefly summarizes how HMMs can be used to produce forecasts ( ¢ 4.4.1), and how information with regard to the unobservable states of the Markov chain can be obtained ( ¢ 4.4.2). The inference regarding the hidden states is commonly referred to as decoding.
4.4 Forecasting and decoding 53 4.4.1 Forecast distributions Before a forecast distribution for HMMs can be derived, it is necessary to compute the corresponding conditional distribution. Let X −u denote the sequence of random variables Xt for t = 1, . . . , T with Xu being excluded, and let x −u denote the observations xt for t = 1, . . . , T with xu being excluded: X (−u) = {X1, . . . , Xu−1, Xu+1, . . . , XT }, (4.31) x (−u) = {x1, . . . , xu−1, xu+1, . . . , xT }. (4.32) Together with the likelihood function in (4.15) and the forward-backward probabilities defined in ¢ 4.3.2.1, the conditional distribution of Xu for u = 1, . . . , T , given all other observations, can be derived as � P Xu = x|X (−u) = x (−u)� = αu−1ΓP (x)β ′ u αu−1Γβ ′ . (4.33) u The numerator can be regarded as the likelihood of the observed series with x being substituted for xu. The denominator can be interpreted as the likelihood of the series with xu treated as being missing. Alternatively, the conditional probability can be interpreted as a mixture of state-dependent probability distributions. The forecast distribution of a hidden Markov model, which is needed to compute the probability of an observation occurring l steps in the future, represents a special type of a conditional distribution. It can be derived as � P XT +l = x|X (T ) (T = x )� = αT Γ l P (x)1 ′ αT 1 ′ = φT Γ l P (x)1 ′ . (4.34) The scaling algorithm introduced in ¢ 4.3.2.2 can be employed to avoid problems of numerical underflow. For increasing l the forecast distribution converges to πsP (x)1 ′ , the stationary distribution. 4.4.2 Decoding Decoding refers to the determination of the most probable states of the Markov chain for an estimated hidden Markov model. One distinguishes between local decoding and global decoding. The former refers to the derivation of the most likely state at date t, which can also be used to generate state predictions as will be shown in ¢ 4.4.2.1. Global decoding looks for the most probable sequence of states ( ¢ 4.4.2.3). 4.4.2.1 Local decoding With the forward-backward probabilities introduced above, the joint probability of the observations X (T ) = x (t) and the Markov chain St being in state i at date t, can be shown to be equal to αt(i)βt(i) = P � X (T ) = x (T ) � , St = i , (4.35)
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 and 80: 4.1 Basic concepts 47 probability d
Page 81 and 82: 4.3 Parameter estimation 49 X 1 X 2
Page 83: 4.3 Parameter estimation 51 4.3.2.1
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 134 and 135:
102 6 Time-varying market beta risk
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?