Term Structure Modeling and Estimation in a State Space Framework (eBook)

Preface 6
Contents 7
1 Introduction 10
2 The Term Structure of Interest Rates 14
2.1 Notation and Basic Interest Rate Relationships 14
2.2 Data Set and Some Stylized Facts 16
3 Discrete-Time Models of the Term Structure 22
3.1 Arbitrage, the Pricing Kernel and the Term Structure 22
3.2 One-Factor Models 30
3.2.1 The One-Factor Vasicek Model 30
3.2.2 The Gaussian Mixture Distribution 34
3.2.3 A One-Factor Model with Mixture Innovations 40
3.2.4 Comparison of the One-Factor Models 43
3.2.5 Moments of the One-Factor Models 45
3.3 Affine Multifactor Gaussian Mixture Models 48
3.3.1 Model Structure and Derivation of Arbitrage-Free Yields 49
3.3.2 Canonical Representation 53
3.3.3 Moments of Yields 59
4 Continuous-Time Models of the Term Structure 64
4.1 The Martingale Approach to Bond Pricing 64
4.1.1 One-Factor Models of the Short Rate 67
4.1.2 Comments on the Market Price of Risk 69
4.1.3 Multifactor Models of the Short Rate 70
4.1.4 Martingale Modeling 71
4.2 The Exponential-AfRne Class 71
4.2.1 Model Structure 71
4.2.2 Specific Models 73
4.3 The Heath-Jarrow-Morton Class 75
5 State Space Models 78
5.1 Structure of the Model 78
5.2 Filtering, Prediction, Smoothing, and Parameter Estimation 80
5.3 Linear Gaussian Models 83
5.3.1 Model Structure 83
5.3.2 The Kalman Filter 83
5.3.3 Maximum Likelihood Estimation 88
6 State Space Models with a Gaussian Mixture 92
6.1 The Model 92
6.2 The Exact Filter 95
6.3 The Approximate Filter AMF(fc) 102
6.4 Related Literature 106
7 Simulation Results for the Mixture Model 110
7.1 Sampling from a Unimodal Gaussian Mixture 111
7.1.1 Data Generating Process 111
7.1.2 Filtering and Prediction for Short Time Series 113
7.1.3 Filtering and Prediction for Longer Time Series 116
7.1.4 Estimation of Hyperparameters 121
7.2 Sampling from a Bimodal Gaussian Mixture 126
7.2.1 Data Generating Process 126
7.2.2 Filtering and Prediction for Short Time Series 127
7.2.3 Filtering and Prediction for Longer Time Series 129
7.2.4 Estimation of Hyperparameters 130
7.3 Sampling from a Student t Distribution 135
7.3.1 Data Generating Process 135
7.3.2 Estimation of Hyperparameters 136
7.4 Summary and Discussion of Simulation Results 140
8 Estimation of Term Structure Models in a State Space Framework 144
8.1 Setting up the State Space Model 146
8.1.1 Discrete-Time Models from the AMGM Class 146
8.1.2 Continuous-Time Models 148
8.1.3 General Form of the Measurement Equation 152
8.2 A Survey of the Literature 153
8.3 Estimation Techniques 155
8.4 Model Adequacy and Interpretation of Results 158
9 An Empirical Application 162
9.1 Models and Estimation Approach 162
9.2 Estimation Results 169
9.3 Conclusion and Extensions 183
10 Summary and Outlook 188
A Properties of the Normal Distribution 190
B Higher Order Stationarity of a VAR(1) 194
C Derivations for the One-Factor Models in Discrete Time 198
C.l Sharpe Ratios for the One-Factor Models 198
C.2 The Kurtosis Increases in the Variance Ratio 200
C.3 Derivation of Formula (3.53) 201
C.4 Moments of Factors 201
C.5 Skewness and Kurtosis of Yields 202
C.6 Moments of Differenced Factors 203
C.7 Moments of Differenced Yields 204
D A Note on Scaling 206
E Derivations for the Multifactor Models in Discrete Time 210
E.l Properties of Factor Innovations 210
E.2 Moments of Factors 211
E.3 Moments of Differenced Factors 213
E.4 Moments of Differenced Yields 214
F Proof of Theorem 6.3 218
G Random Draws from a Gaussian Mixture Distribution 222
References 224
List of Figures 230
List of Tables 232

7 Simulation Results for the Mixture Model (p.101)

After discussing mixture state space models in the previous section, we now present a small simulation study that aims to explore the properties of the AMF(fc) with respect to filtering, prediction and parameter estimation. There are three groups of simulations, where each group is characterized by the distribution of the state innovation. We examine a Gaussian mixture with two components having both mean zero but different variances; a Gaussian mixture with two components having the same variance but different means; and a Student t distribution with three degrees of freedom.

A first objective of the simulations is an assessment of how close the results of the AMF and the Kalman Filter come to those of the exact filter. Since the number of components of the exact filter increases exponentially with time, such comparisons can only be undertaken for small time series. For time series of length T = 10 we explore the discrepancy between the AMF and the Kalman filter on one side and the exact filter on the other side. Such a comparison is carried out with respect to the mean and variance of the filtered and predicted state as well as for the prediction and filtering density at T = 10.

Second, for longer time series of length T = 350 we compare the performance of the AMF and the Kalman filter in filtering and prediction. Of course, for such long time series the exact filter cannot be employed anymore. We also assess how the relative performance of the two algorithms depends on the parameterization of the data generating state space model. For this type of simulation we treat the hyperparameters of the state space models under consideration as known.

Third, again for time series of length T = 350, a subset of the hyperparameters constituting the state space model is treated as unknown. The parameters are estimated using the likelihood methods based on the Kalman filter and the AMF. The distributions of the estimated parameters, obtained from using different algorithms, are compared to each other.

For the group of simulations that uses the Student t distribution, the exact filter algorithm is not known and thus cannot be compared to the AMF and the Kalman filter. However, hyperparameters are estimated even for this case: the state innovation is falsely assumed to be distributed as a normal or a normal mixture and is then estimated using the Kalman filter and the AMF, respectively.

All of the simulations are conducted using GAUSS 3.6. Standard Gaussian pseudo random variables and pseudo random variables from the uniform distribution over [0,1] are generated using GAUSS's rndnO and rnduO function respectively. For generating pseudo random variables from a Student t distribution, the function rstudentO from the DISTRIB library by Rainer Schlittgen and Thomas Noack is used. The appendix shows how random draws from the Gaussian Mixture are generated.

Numerical maximization of the likelihoods is performed using the BFGS algorithm as implemented in GAUSS's MAXLIK Ubrary. The convergence criterion for the gradient is set to 0.5 • 10 ^-6. Gradients of the likelihood are computed numerically.