1 Introduction 1
2 The Term Structure of Interest Rates 5
2.1 Notation and Basic Interest Rate Relationships 5
2.2 Data Set and Some Stylized Facts 7
3 Discrete-Time Models of the Term Structure 13
3.1 Arbitrage, the Pricing Kernel and the Term Structure 13
3.2 One-Factor Models 21
3.2.1 The One-Factor Vasicek Model 21
3.2.2 The Gaussian Mixture Distribution 25
3.2.3 A One-Factor Model with Mixture Innovations 31
3.2.4 Comparison of the One-Factor Models 34
3.2.5 Moments of the One-Factor Models 36
3.3 Affine Multifactor Gaussian Mixture Models 39
3.3.1 Model Structure and Derivation of Arbitrage-Free Yields 40
3.3.2 Canonical Representation 44
3.3.3 Moments of Yields 50
4 Continuous-Time Models of the Term Structure 55
4.1 The Martingale Approach to Bond Pricing 55
4.1.1 One-Factor Models of the Short Rate 58
4.1.2 Comments on the Market Price of Risk 60
4.1.3 Multifactor Models of the Short Rate 61
4.1.4 Martingale Modeling 62
4.2 The Exponential-Affine Class 62
4.2.1 Model Structure 62
4.2.2 Specific Models 64
4.3 The Heath-Jarrow-Morton Class 66

VIII Contents
5 State Space Models 69
5.1 Structure of the Model 69
5.2 Filtering, Prediction, Smoothing, and Parameter Estimation .. 71
5.3 Linear Gaussian Models 74
5.3.1 Model Structure 74
5.3.2 The Kalman Filter 74
5.3.3 Maximum Likelihood Estimation 79
6 State Space Models with a Gaussian Mixture 83
6.1 The Model 83
6.2 The Exact Filter 86
6.3 The Approximate Filter AMF(fc) 93
6.4 Related Literature 97
7 Simulation Results for the Mixture Model 101
7.1 Sampling from a Unimodal Gaussian Mixture 102
7.1.1 Data Generating Process 102
7.1.2 Filtering and Prediction for Short Time Series 104
7.1.3 Filtering and Prediction for Longer Time Series 107
7.1.4 Estimation of Hyperparameters 112
7.2 Sampling from a Bimodal Gaussian Mixture 117
7.2.1 Data Generating Process 117
7.2.2 Filtering and Prediction for Short Time Series 118
7.2.3 Filtering and Prediction for Longer Time Series 120
7.2.4 Estimation of Hyperparameters 121
7.3 Sampling from a Student t Distribution 126
7.3.1 Data Generating Process 126
7.3.2 Estimation of Hyperparameters 127
7.4 Summary and Discussion of Simulation Results 131
8 Estimation of Term Structure Models in a State Space
Framework 135
8.1 Setting up the State Space Model 137
8.1.1 Discrete-Time Models from the AMGM Class 137
8.1.2 Continuous-Time Models 139
8.1.3 General Form of the Measurement Equation 143
8.2 A Survey of the Literature 144
8.3 Estimation Techniques 146
8.4 Model Adequacy and Interpretation of Results 149
9 An Empirical Application 153
9.1 Models and Estimation Approach 153
9.2 Estimation Results 160
9.3 Conclusion and Extensions 174
10 Summary and Outlook 179

Contents IX
A Properties of the Normal Distribution 181
B Higher Order Stationarity of a VAR(l) 185
C Derivations for the One-Factor Models in Discrete Time .. . 189
C.l Sharpe Ratios for the One-Factor Models 189
C.2 The Kurtosis Increases in the Variance Ratio 191
C.3 Derivation of Formula (3.53) 192
C.4 Moments of Factors 192
C.5 Skewness and Kurtosis of Yields 193
C.6 Moments of Differenced Factors 194
C.7 Moments of Differenced Yields 195
D A Note on Scaling 197
E Derivations for the Multifactor Models in Discrete Time .. 201
E.l Properties of Factor Innovations 201
E.2 Moments of Factors 202
E.3 Moments of Differenced Factors 204
E.4 Moments of Differenced Yields 205
F Proof of Theorem 6.3 209
G Random Draws from a Gaussian Mixture Distribution 213
References 215
List of Figures 221
List of Tables 223