Application of Hidden Markov Models and Hidden Semi-Markov Models to Financial Time Series
Bulla, Jan
Georg-August-Universit¨at G¨ottingen
2006
Pages 157

Contents
1 Introduction 1
2 Hidden Markov Models 6
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Independent Mixture Distributions . . . . . . . . . . . . 7
2.1.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 The basic Hidden Markov Model . . . . . . . . . . . . . 16
2.2.2 The Likelihood of a Hidden Markov Model . . . . . . . . 18
3 Parameter Estimation for Hidden Markov Models 20
3.1 Estimation Algorithms for
Stationary Hidden Markov Models . . . . . . . . . . . . . . . . 21
3.1.1 Direct Numerical Maximization . . . . . . . . . . . . . . 21
3.1.2 The Stationary EM Algorithm . . . . . . . . . . . . . . . 23
3.1.3 The Hybrid Algorithm . . . . . . . . . . . . . . . . . . . 25
3.2 A simulation experiment . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Study design . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Results for different parameterizations . . . . . . . . . . 26
3.2.3 Performance of the hybrid algorithm . . . . . . . . . . . 29
3.2.4 Coverage probability of confidence intervals . . . . . . . 32
3.3 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Markov Switching Approaches to Model Time-Varying Betas 38
4.1 The Unconditional Beta in the CAPM . . . . . . . . . . . . . . 40
4.2 The Markov Switching Approach . . . . . . . . . . . . . . . . . 41
4.3 Data and Preliminary Analysis . . . . . . . . . . . . . . . . . . 43
4.3.1 Data Series . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.2 Univariate Statistics . . . . . . . . . . . . . . . . . . . . 44
4.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.1 Unconditional Beta Estimates . . . . . . . . . . . . . . . 45
4.4.2 Modeling Conditional Betas . . . . . . . . . . . . . . . . 46
4.4.3 Comparison of Conditional Beta Estimates . . . . . . . . 46
4.4.4 In-Sample and Out-Of-Sample Forecasting Accuracy . . 48
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Hidden Semi-Markov Models 57
5.1 The Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1.1 Semi-Markov Chains . . . . . . . . . . . . . . . . . . . . 59
5.1.2 Hidden Semi-Markov Models . . . . . . . . . . . . . . . . 61
5.2 The Likelihood Function of a Hidden
Semi-Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 The Partial Likelihood Estimator . . . . . . . . . . . . . 64
5.2.2 The Complete Likelihood Estimator . . . . . . . . . . . . 66
5.3 The EM Algorithm for Hidden
Semi-Markov Models . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 The Q-Function . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.2 The Forward-Backward Algorithm . . . . . . . . . . . . 72
5.3.2.1 The Forward Iteration . . . . . . . . . . . . . . 75
5.3.2.2 The Backward Iteration . . . . . . . . . . . . . 77
5.3.3 The Sojourn Time Distribution . . . . . . . . . . . . . . 80
5.3.3.1 The Q-Function based on the Full Likelihood
Estimator . . . . . . . . . . . . . . . . . . . . . 81

5.3.3.2 The Q-Function based on the Partial Likelihood
Estimator . . . . . . . . . . . . . . . . . . 83
5.3.4 Parameter Re-estimation . . . . . . . . . . . . . . . . . . 84
5.3.4.1 The Initial Parameters . . . . . . . . . . . . . . 85
5.3.4.2 The Transition Probabilities . . . . . . . . . . . 85
5.3.4.3 The State Occupancy Distribution . . . . . . . 86
5.3.4.4 The Observation Component . . . . . . . . . . 94
5.4 Asymptotic properties of the maximum
likelihood estimators . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5 Stationary Hidden Semi-Markov Models . . . . . . . . . . . . . 105
6 Stylized Facts of Daily Return Series and Hidden Semi-Markov
Models 107
6.1 Modeling Daily Return Series . . . . . . . . . . . . . . . . . . . 108
6.2 The Data Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Conclusion and Future Work 124
A The EM Algorithm 126
A.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.2 Implementation of the EM Algorithm . . . . . . . . . . . . . . . 128
A.3 Convergence properties of the EM Algorithm . . . . . . . . . . . 129
B The Forward-Backward Algorithm 131
C Source Code for the Estimation Procedures 135
D Notational Conventions and Abbreviations 136