Lemke Term Structure Modeling and Estimation in a State Space Framework

1. Auflage 2005
ISBN: 978-3-540-28344-7
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 565, 226 Seiten

Reihe: Lecture Notes in Economics and Mathematical Systems

This book has been prepared during my work as a research assistant at the Institute for Statistics and Econometrics of the Economics Department at the University of Bielefeld, Germany. It was accepted as a Ph.D. thesis titled 'Term Structure Modeling and Estimation in a State Space Framework' at the Department of Economics of the University of Bielefeld in November 2004. It is a pleasure for me to thank all those people who have been helpful in one way or another during the completion of this work. First of all, I would like to express my gratitude to my advisor Professor Joachim Frohn, not only for his guidance and advice throughout the com­ pletion of my thesis but also for letting me have four very enjoyable years teaching and researching at the Institute for Statistics and Econometrics. I am also grateful to my second advisor Professor Willi Semmler. The project I worked on in one of his seminars in 1999 can really be seen as a starting point for my research on state space models. I thank Professor Thomas Braun for joining the committee for my oral examination.

Autoren/Hrsg.

Lemke, Wolfgang

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;1 Introduction;10
4;2 The Term Structure of Interest Rates;14
4.1;2.1 Notation and Basic Interest Rate Relationships;14
4.2;2.2 Data Set and Some Stylized Facts;16
5;3 Discrete-Time Models of the Term Structure;22
5.1;3.1 Arbitrage, the Pricing Kernel and the Term Structure;22
5.2;3.2 One-Factor Models;30
5.2.1;3.2.1 The One-Factor Vasicek Model;30
5.2.2;3.2.2 The Gaussian Mixture Distribution;34
5.2.3;3.2.3 A One-Factor Model with Mixture Innovations;40
5.2.4;3.2.4 Comparison of the One-Factor Models;43
5.2.5;3.2.5 Moments of the One-Factor Models;45
5.3;3.3 Affine Multifactor Gaussian Mixture Models;48
5.3.1;3.3.1 Model Structure and Derivation of Arbitrage-Free Yields;49
5.3.2;3.3.2 Canonical Representation;53
5.3.3;3.3.3 Moments of Yields;59
6;4 Continuous-Time Models of the Term Structure;64
6.1;4.1 The Martingale Approach to Bond Pricing;64
6.1.1;4.1.1 One-Factor Models of the Short Rate;67
6.1.2;4.1.2 Comments on the Market Price of Risk;69
6.1.3;4.1.3 Multifactor Models of the Short Rate;70
6.1.4;4.1.4 Martingale Modeling;71
6.2;4.2 The Exponential-AfRne Class;71
6.2.1;4.2.1 Model Structure;71
6.2.2;4.2.2 Specific Models;73
6.3;4.3 The Heath-Jarrow-Morton Class;75
7;5 State Space Models;78
7.1;5.1 Structure of the Model;78
7.2;5.2 Filtering, Prediction, Smoothing, and Parameter Estimation;80
7.3;5.3 Linear Gaussian Models;83
7.3.1;5.3.1 Model Structure;83
7.3.2;5.3.2 The Kalman Filter;83
7.3.3;5.3.3 Maximum Likelihood Estimation;88
8;6 State Space Models with a Gaussian Mixture;92
8.1;6.1 The Model;92
8.2;6.2 The Exact Filter;95
8.3;6.3 The Approximate Filter AMF(fc);102
8.4;6.4 Related Literature;106
9;7 Simulation Results for the Mixture Model;110
9.1;7.1 Sampling from a Unimodal Gaussian Mixture;111
9.1.1;7.1.1 Data Generating Process;111
9.1.2;7.1.2 Filtering and Prediction for Short Time Series;113
9.1.3;7.1.3 Filtering and Prediction for Longer Time Series;116
9.1.4;7.1.4 Estimation of Hyperparameters;121
9.2;7.2 Sampling from a Bimodal Gaussian Mixture;126
9.2.1;7.2.1 Data Generating Process;126
9.2.2;7.2.2 Filtering and Prediction for Short Time Series;127
9.2.3;7.2.3 Filtering and Prediction for Longer Time Series;129
9.2.4;7.2.4 Estimation of Hyperparameters;130
9.3;7.3 Sampling from a Student t Distribution;135
9.3.1;7.3.1 Data Generating Process;135
9.3.2;7.3.2 Estimation of Hyperparameters;136
9.4;7.4 Summary and Discussion of Simulation Results;140
10;8 Estimation of Term Structure Models in a State Space Framework;144
10.1;8.1 Setting up the State Space Model;146
10.1.1;8.1.1 Discrete-Time Models from the AMGM Class;146
10.1.2;8.1.2 Continuous-Time Models;148
10.1.3;8.1.3 General Form of the Measurement Equation;152
10.2;8.2 A Survey of the Literature;153
10.3;8.3 Estimation Techniques;155
10.4;8.4 Model Adequacy and Interpretation of Results;158
11;9 An Empirical Application;162
11.1;9.1 Models and Estimation Approach;162
11.2;9.2 Estimation Results;169
11.3;9.3 Conclusion and Extensions;183
12;10 Summary and Outlook;188
13;A Properties of the Normal Distribution;190
14;B Higher Order Stationarity of a VAR(1);194
15;C Derivations for the One-Factor Models in Discrete Time;198
15.1;C.l Sharpe Ratios for the One-Factor Models;198
15.2;C.2 The Kurtosis Increases in the Variance Ratio;200
15.3;C.3 Derivation of Formula (3.53);201
15.4;C.4 Moments of Factors;201
15.5;C.5 Skewness and Kurtosis of Yields;202
15.6;C.6 Moments of Differenced Factors;203
15.7;C.7 Moments of Differenced Yields;204
16;D A Note on Scaling;206
17;E Derivations for the Multifactor Models in Discrete Time;210
17.1;E.l Properties of Factor Innovations;210
17.2;E.2 Moments of Factors;211
17.3;E.3 Moments of Differenced Factors;213
17.4;E.4 Moments of Differenced Yields;214
18;F Proof of Theorem 6.3;218
19;G Random Draws from a Gaussian Mixture Distribution;222
20;References;224
21;List of Figures;230
22;List of Tables;232

Leseproben

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.