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Parameter Estimation in Stochastic Volatility Models with Missing Data Using Particle Methods and the Em Algorithm

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Parameter Estimation in Stochastic Volatility Models with Missing Data Using Particle Methods and the Em Algorithm PARAMETER ESTIMATION IN STOCHASTIC VOLATILITY MODELS WITH MISSING DATA USING PARTICLE METHODS AND THE EM ALGORITHM by Jeongeun Kim BS, Seoul National University, 1998 MS, Seoul National University, 2000 Submitted to the Graduate Faculty of the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2005 UNIVERSITY OF PITTSBURGH DEPARTMENT OF STATISTICS This dissertation was presented by Jeongeun Kim It was defended on April 22, 2005 and approved by David S. Stoffer, Professor Ori Rosen, Professor Wesley Thompson, Professor Anthony Brockwell, Professor Dissertation Director: David S. Stoffer, Professor ii Copyright c by Jeongeun Kim 2005 iii PARAMETER ESTIMATION IN STOCHASTIC VOLATILITY MODELS WITH MISSING DATA USING PARTICLE METHODS AND THE EM ALGORITHM Jeongeun Kim, PhD University of Pittsburgh, 2005 The main concern of financial time series analysis is how to forecast future values of financial variables, based on all available information. One of the special features of financial variables, such as stock prices and exchange rates, is that they show changes in volatility, or variance, over time. Several statistical models have been suggested to explain volatility in data, and among them Stochastic Volatility models or SV models have been commonly and successfully used. Another feature of financial variables I want to consider is the existence of several missing data. For example, there is no stock price data available for regular holidays, such as Christmas, Thanksgiving, and so on. Furthermore, even though the chance is small, stretches of data may not available for many reasons. I believe that if this feature is brought into the model, it will produce more precise results. The goal of my research is to develop a new technique for estimating parameters of SV models when some parts of data are missing. By estimating parameters, the dynamics of the process can be fully specified, and future values can be estimated from them. SV models have become increasingly popular in recent years, and their popularity has resulted in several different approaches proposed regarding the problem of estimating the parameters of the SV models. However, as of yet there is no consensus on this problem. In addition there has been no serious consideration of the missing data problem. A new statistical approach based on the EM algorithm and particle filters is presented. Moreover, I expand the scope of application of SV models by introducing a slight modification of the models. iv TABLE OF CONTENTS PREFACE ......................................... x 1.0 INTRODUCTION ................................. 1 1.1 Problem Statement ............................... 1 1.2 Stochastic Volatility Model ........................... 3 1.2.1 Stochastic Volatility Model ....................... 3 1.2.2 ARCH/GARCH vs. SV Model ..................... 4 1.3 Thesis Outline .................................. 6 1.4 Contribution of Thesis ............................. 6 2.0 PARAMETER ESTIMATION IN SV MODEL ............... 8 2.1 Literature Review ............................... 8 2.1.1 Brief Review of EM Algorithm ..................... 9 2.1.2 Brief Review of Particle Filters and Smoothers ............ 10 2.1.2.1 State-Space Models and Related Concepts .......... 10 2.1.2.2 Particle Filters and Smoothers ................. 11 2.1.2.3 Particle Filtering Algorithm .................. 15 2.1.2.4 Particle Smoothing Algorithm ................. 17 2.2 Overview of the Estimation Algorithm .................... 18 2.3 Details of the Procedure ............................ 20 2.3.1 Filtering Step ............................... 20 2.3.2 Smoothing Step ............................. 20 2.3.3 Estimation Step ............................. 21 2.4 Other Issues ................................... 24 v 2.4.1 Initial Parameter Selection ....................... 24 2.4.2 Relative Likelihood ........................... 25 2.4.3 Stopping Rule and Selection of Particle Size .............. 26 2.4.4 Standard Deviation of Parameter Estimates .............. 27 2.5 Summary of Methods from Other Authors .................. 29 2.5.1 Parameter Estimation of SV Models Without Missing Data ..... 29 2.5.2 Stochastic EM Algorithm ........................ 31 2.5.3 MCEM Algorithm ............................ 32 3.0 SLIGHT MODIFICATION - NORMAL MIXTURES ........... 34 3.1 Normal Mixtures as an Observation Noise .................. 34 3.1.1 Model Structures ............................. 34 3.1.2 Advantages Over the Regular Model .................. 36 3.2 Modification for the Normal Mixture Model ................. 36 3.2.1 Overview of the Algorithm for Normal Mixture Cases ........ 37 3.2.2 Estimation Step ............................. 37 3.2.3 Filtering Step ............................... 40 3.2.4 Smoothing Step ............................. 41 3.3 Other Issues ................................... 43 3.3.1 Initial Parameter Selection ....................... 43 4.0 MISSING DATA PROBLEM .......................... 44 4.1 General Idea and Model Structure ....................... 44 4.2 Details of the Procedure ............................ 45 4.2.1 Overview of the Estimation Algorithm ................. 45 4.2.2 Data-Completion Step .......................... 46 4.2.3 Filtering Step ............................... 46 4.2.4 Smoothing Step ............................. 47 4.2.5 Estimation Step ............................. 48 5.0 SIMULATION STUDY AND DATA ANALYSIS ............. 50 5.1 Simulation Studies ............................... 50 5.1.1 Data A .................................. 51 vi 5.1.1.1 Method in Chapter 2: With Algorithm A ........... 51 5.1.1.2 Method in Chapter 3: With Algorithm B ........... 53 5.1.1.3 Method in Chapter 4: Missing Data Case ........... 53 5.1.2 Data B .................................. 55 5.1.2.1 Method in Chapter 2: With Algorithm A ........... 55 5.1.2.2 Method in Chapter 3: With Algorithm B ........... 59 5.1.2.3 Method in Chapter 4: Missing Data Case ........... 60 5.2 Pound and Dollar Daily Exchange Rates ................... 60 5.2.1 Estimates in the References ....................... 61 5.2.2 Estimates Using Algorithm A ...................... 63 5.2.3 Estimates Using Algorithm B ...................... 63 5.3 Pound and Dollar Daily Exchange Rates With Missing Values ....... 63 6.0 CONCLUSIONS AND FUTURE WORK .................. 70 6.1 Summary and Contributions .......................... 70 6.2 Future Work ................................... 72 APPENDIX A. NOTATION ............................. 73 APPENDIX B. CALCULATIONS FOR THE STANDARD DEVIATION . 76 B.1 Algorithm A For Chapter 2 .......................... 76 B.2 Algorithm B For Chapter 3 .......................... 77 APPENDIX C. PRACTICAL PROBLEM IN CALCULATION OF STAN- DARD DEVIATIONS ............................... 80 C.1 Practical Problems and a Possible Solution .................. 80 C.2 Example : Simple State-Space Model ..................... 81 APPENDIX D. MATLAB FUNCTIONS ..................... 86 D.1 Simulation of Data from the SV Models ................... 86 D.2 Algorithm A: Parameter Estimation for Chapter 2 .............. 87 D.3 Algorithm B: Parameter Estimation for Chapter 3 and Chapter 4 ..... 92 BIBLIOGRAPHY .................................... 101 vii LIST OF TABLES 5.1 Estimation results for Data A: Algorithm A ................... 52 5.2 Estimation results for Data A: Algorithm B ................... 54 5.3 Estimation results for Data A: Missing data case ................ 56 5.4 Estimation results for Data B: Algorithm A ................... 57 5.5 Estimation results for Data B: Algorithm B ................... 60 5.6 Estimation results for Data B: Missing data case ................ 61 5.7 Parameter estimates in the reference papers ................... 62 5.8 Estimation results for the pound/dollar exchange rates ............ 65 5.9 Estimation results for the pound/dollar exchange rates with missing values . 68 C.1 Variance-covariance matrices for Data A (2 tries) ................ 82 C.2 Variance-covariance matrices calculated using trimmed mean ......... 84 C.3 Variance-covariance matrix for Example in Section C.2 ............ 85 viii LIST OF FIGURES 1.1 Log returns of the pound/dollar exchange rates. ................ 2 2.1 SIS particle filter. ................................. 13 2.2 Problem of the SIS particle filter: degeneracy. ................. 14 2.3 Resampling. .................................... 14 5.1 Plot and histogram of data A .......................... 51 5.2 Estimation result for Data A: Algorithm A. ................... 54 5.3 Relative likelihood of Data A: Algorithm B. ................... 55 5.4 Plot and histogram of data B. .......................... 56 5.5 Estimation result for Data B: Algorithm A. ................... 58 5.6 Relative likelihood: Data B. ........................... 59 5.7 Pound/dollar exchange rates. ........................... 62 5.8 Estimation results for the pound/dollar exchange rates: Algorithm A. .... 64 5.9 Estimation results for the pound/dollar exchange rates: Algorithm B. .... 65 5.10 Pound/dollar daily exchange rates with missing values. ............ 67 5.11 Estimation results for the pound/dollar exchange rates with missing values. 69 C.1 Histogram of Term2: M=1000. .......................... 83 ix PREFACE I would like to acknowledge the advice, support, and friendship of a number of people who helped me during the writing of this thesis and my time as a graduate student. First, I would like to thank Professor David
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