Multivariate Time Series Analysis E-Book

Ruey S. Tsay

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 666

Veröffentlichungsjahr: 2013

Beschreibung

An accessible guide to the multivariate time series tools used in numerous real-world applications Multivariate Time Series Analysis: With R and Financial Applications is the much anticipated sequel coming from one of the most influential and prominent experts on the topic of time series. Through a fundamental balance of theory and methodology, the book supplies readers with a comprehensible approach to financial econometric models and their applications to real-world empirical research. Differing from the traditional approach to multivariate time series, the book focuses on reader comprehension by emphasizing structural specification, which results in simplified parsimonious VAR MA modeling. Multivariate Time Series Analysis: With R and Financial Applications utilizes the freely available R software package to explore complex data and illustrate related computation and analyses. Featuring the techniques and methodology of multivariate linear time series, stationary VAR models, VAR MA time series and models, unitroot process, factor models, and factor-augmented VAR models, the book includes: * Over 300 examples and exercises to reinforce the presented content * User-friendly R subroutines and research presented throughout to demonstrate modern applications * Numerous datasets and subroutines to provide readers with a deeper understanding of the material Multivariate Time Series Analysis is an ideal textbook for graduate-level courses on time series and quantitative finance and upper-undergraduate level statistics courses in time series. The book is also an indispensable reference for researchers and practitioners in business, finance, and econometrics.

Leseprobe

Contents

Preface

Acknowledgements

CHAPTER 1: Multivariate Linear Time Series

1.1 INTRODUCTION

1.2 SOME BASIC CONCEPTS

1.3 CROSS-COVARIANCE AND CORRELATION MATRICES

1.4 SAMPLE CCM

1.5 TESTING ZERO CROSS-CORRELATIONS

1.6 FORECASTING

1.7 MODEL REPRESENTATIONS

1.8 OUTLINE OF THE BOOK

1.9 SOFTWARE

EXERCISES

REFERENCES

CHAPTER 2: Stationary Vector Autoregressive Time Series

2.1 INTRODUCTION

2.2 VAR(1) MODELS

2.3 VAR(2) MODELS

2.4 VAR(p) MODELS

2.5 ESTIMATION

2.6 ORDER SELECTION

2.7 MODEL CHECKING

2.8 LINEAR CONSTRAINTS

2.9 FORECASTING

2.10 IMPULSE RESPONSE FUNCTIONS

2.11 FORECAST ERROR VARIANCE DECOMPOSITION

2.12 PROOFS

EXERCISES

REFERENCES

CHAPTER 3: Vector Autoregressive Moving-Average Time Series

3.1 VECTOR MA MODELS

3.2 SPECIFYING VMA ORDER

3.3 ESTIMATION OF VMA MODELS

3.4 FORECASTING OF VMA MODELS

3.5 VARMA MODELS

3.6 IMPLICATIONS OF VARMA MODELS

3.7 LINEAR TRANSFORMS OF VARMA PROCESSES

3.8 TEMPORAL AGGREGATION OF VARMA PROCESSES

3.9 LIKELIHOOD FUNCTION OF A VARMA MODEL

3.10 INNOVATIONS APPROACH TO EXACT LIKELIHOOD FUNCTION

3.11 ASYMPTOTIC DISTRIBUTION OF MAXIMUM LIKELIHOOD ESTIMATES

3.12 MODEL CHECKING OF FITTED VARMA MODELS

3.13 FORECASTING OF VARMA MODELS

3.14 TENTATIVE ORDER IDENTIFICATION

3.15 EMPIRICAL ANALYSIS OF VARMA MODELS

3.16 APPENDIX

EXERCISES

REFERENCES

CHAPTER 4: Structural Specification of VARMA Models

4.1 THE KRONECKER INDEX APPROACH

4.2 THE SCALAR COMPONENT APPROACH

4.3 STATISTICS FOR ORDER SPECIFICATION

4.4 FINDING KRONECKER INDICES

4.5 FINDING SCALAR COMPONENT MODELS

4.6 ESTIMATION

4.7 AN EXAMPLE

4.8 APPENDIX: CANONICAL CORRELATION ANALYSIS

EXERCISES

REFERENCES

CHAPTER 5: Unit-Root Nonstationary Processes

5.1 UNIVARIATE UNIT-ROOT PROCESSES

5.2 MULTIVARIATE UNIT-ROOT PROCESSES

5.3 SPURIOUS REGRESSIONS

5.4 MULTIVARIATE EXPONENTIAL SMOOTHING

5.5 COINTEGRATION

5.6 AN ERROR-CORRECTION FORM

5.7 IMPLICATIONS OF COINTEGRATING VECTORS

5.8 PARAMETERIZATION OF COINTEGRATING VECTORS

5.9 COINTEGRATION TESTS

5.10 ESTIMATION OF ERROR-CORRECTION MODELS

5.11 APPLICATIONS

5.12 DISCUSSION

5.13 APPENDIX

EXERCISES

REFERENCES

CHAPTER 6: Factor Models and Selected Topics

6.1 SEASONAL MODELS

6.2 PRINCIPAL COMPONENT ANALYSIS

6.3 USE OF EXOGENOUS VARIABLES

6.4 MISSING VALUES

6.5 FACTOR MODELS

6.6 CLASSIFICATION AND CLUSTERING ANALYSIS

EXERCISES

REFERENCES

CHAPTER 7: Multivariate Volatility Models

7.1 TESTING CONDITIONAL HETEROSCEDASTICITY

7.2 ESTIMATION OF MULTIVARIATE VOLATILITY MODELS

7.3 DIAGNOSTIC CHECKS OF VOLATILITY MODELS

7.4 EXPONENTIALLY WEIGHTED MOVING AVERAGE

7.5 BEKK MODELS

7.6 CHOLESKY DECOMPOSITION AND VOLATILITY MODELING

7.7 DYNAMIC CONDITIONAL CORRELATION MODELS

7.8 ORTHOGONAL TRANSFORMATION

7.9 COPULA-BASED MODELS

7.10 PRINCIPAL VOLATILITY COMPONENTS

EXERCISES

REFERENCES

APPENDIX A: Review of Mathematics and Statistics

A.1 REVIEW OF VECTORS AND MATRICES

A.2 LEAST-SQUARES ESTIMATION

A.3 MULTIVARIATE NORMAL DISTRIBUTIONS

A.4 MULTIVARIATE STUDENT-t DISTRIBUTION

A.5 WISHART AND INVERTED WISHART DISTRIBUTIONS

A.6 VECTOR AND MATRIX DIFFERENTIALS

REFERENCES

Index

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Library of Congress Cataloging-in-Publication Data:

Tsay, Ruey S., 1951– Multivariate time series analysis: with R and financial applications / Ruey S. Tsay, Booth School of Business, University of Chicago, Chicago, IL.  pages cm Includes bibliographical references and index. ISBN 978-1-118-61790-8 (hardback)1. Time-series analysis. 2. R (Computer program language) 3. Econometric models. I. Title. QA280.T73 2014 519.5′5–dc23               2014009453

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To my teacher and friend George

Preface

This book is based on my experience in teaching and research on multivariate time series analysis over the past 30 years. It summarizes the basic concepts and ideas of analyzing multivariate dependent data, provides econometric and statistical models useful for describing the dynamic dependence between variables, discusses the identifiability problem when the models become too flexible, introduces ways to search for simplifying structure hidden in high-dimensional time series, addresses the applicabilities and limitations of multivariate time series methods, and, equally important, develops a software package for readers to apply the methods and models discussed in the book.

Multivariate time series analysis provides useful tools and methods for processing information embedded in multiple measurements that have temporal and cross-sectional dependence. The goal of the analysis is to provide a better understanding of the dynamic relationship between variables and to improve the accuracy in forecasting. The models built can also be used in policy simulation or in making inference. The book focuses mainly on linear models as they are easier to comprehend and widely applicable. I tried to draw a balance between theory and applications and kept the notation as consistent as possible. I also tried to make the book self-contained. However, given the complexity of the subject, the level of coverage on selected topics may vary throughout the book. This reflects in part my own preference and understanding of the topics and in part my desire to keep the book at a reasonable length.

The field of high-dimensional data analysis is still under rapid developments, especially for dependent data. Omission of some important topics or methods is not avoidable for a book like this one. For instance, nonlinear models are not discussed, nor the categorical time series. Readers are advised to consult recent articles or journals for further development.

The book starts with some general concepts of multivariate time series in Chapter 1, including assessing and quantifying temporal and cross-sectional dependence. As the dimension increases, the difficulty in presenting multivariate data quickly becomes evident. I tried to keep the presentation in a compact form if possible. In some cases, scalar summary statistics are given. Chapter 2 focuses on vector autoregressive (VAR) models as they are, arguably, the most widely used multivariate time series models. My goal is to make the chapter as comprehensive as possible for readers who are interested in VAR models. Both Bayesian and classical analyses of VAR models are included. Chapter 3 studies stationary vector autoregressive moving-average (VARMA) models. It begins with properties and estimation of vector moving-average (VMA) models. The issue of identifiability of VARMA models is investigated and properties of the models are given. Chapter 4 investigates the structural specification of a multivariate time series. Two methods are introduced to seek the simplifying structure hidden in a vector time series. These methods enable users to discover the skeleton of a linear multivariate time series. Chapter 5 deals with unit-root nonstationarity and cointegration. It includes the basic theory for understanding unit-root time series and some applications. In Chapter 6, I discuss factor models and some selected topics of multivariate time series. Both the classical and approximate factor models are studied. My goal is to cover all factor models currently available in the literature and to provide the relationship between them. Chapter 7 focuses on multivariate volatility modeling. It covers volatility models that are relatively easy to use and produce positive-definite volatility matrices. The chapter also discusses ways to detect conditional heteroscedasticity in a vector time series and methods for checking a fitted multivariate volatility model. Throughout the book, real examples are used to demonstrate the analysis. Every chapter contains some exercises that analyze empirical vector time series.

Software is an integral part of multivariate time series analysis. Without software packages, multivariate time series becomes a pure theoretical exercise. I have tried my best to write R programs that enable readers to apply all methods and models discussed in the book. These programs are included in the MTS package available in R. Readers can duplicate all the analyses shown in the book with the package and some existing R packages. Not a professional programmer, I am certain that many of the codes in MTS are not as efficient as they can be and are likely to have bugs. I would appreciate any suggestions and/or corrections to both the package and the book.

RUEY S. TSAY

Chicago, IllinoisSeptember 2014

Acknowledgements

This book would not have been written without the great teachers I have had. In particular, I would like to express my sincere thanks to Professor George C. Tiao who taught me time series analysis and statistical research. His insightful view of empirical time series and his continual encouragements are invaluable. I would like to thank Professor Tea-Yuan Hwang who introduced me to statistics and has remained a close friend over the past four decades. I would also like to thank Mr. Sung-Nan Chen, my junior high school teacher. Without his foresight, I would not have pursued my college education. I would like to thank many other teachers, including late Professor George E. P. Box and late Professor Gregory Reinsel of University of Wisconsin, and friends, including Dr. David F. Findley, Professors Daniel Peña, Manny Parzen, Buddy Gray, and Howell Tong, and late Professor Hirotugu Akaike, for their support of my research in time series analysis. Dr. David Matteson and Mr. Yongning Wang kindly allowed me to use their programs and Yongning has read over the draft carefully. I appreciate their help. I would also like to thank many students who asked informative questions both in and outside the classrooms. I wish to express my sincere thanks to Stephen Quigley and Sari Friedman for their support in preparing this book. I also wish to acknowledge the financial support of Chicago Booth. Finally, I would like to thank my parents who sacrificed so much to support me and for their unconditional love. As always, my children are my inspiration and sources of energy. Finally, I would like to express my sincere thanks to my wife for her love and constant encouragement. In particular, she has always put my career ahead of her own.

The web page of the book is http://faculty.chicagobooth.edu/ruey.tsay/teaching/mtsbk.

R. S. T.

CHAPTER 1

Multivariate Linear Time Series

1.1 INTRODUCTION

Multivariate time series analysis considers simultaneously multiple time series. It is a branch of multivariate statistical analysis but deals specifically with dependent data. It is, in general, much more complicated than the univariate time series analysis, especially when the number of series considered is large. We study this more complicated statistical analysis in this book because in real life decisions often involve multiple inter-related factors or variables. Understanding the relationships between those factors and providing accurate predictions of those variables are valuable in decision making. The objectives of multivariate time series analysis thus include

1. To study the dynamic relationships between variables

2. To improve the accuracy of prediction

FIGURE 1.1 Time plots of U.S. quarterly real GDP (in logarithm) and unemployment rate from 1948 to 2011. The data are seasonally adjusted.

FIGURE 1.2 Time plots of the growth rate of U.S. quarterly real GDP (in logarithm) and the change series of unemployment rate from 1948 to 2011. The data are seasonally adjusted.

FIGURE 1.3 Scatter plot of the changes in quarterly U.S. unemployment rate versus the growth rate of quarterly real GDP (in logarithm) from the second quarter of 1948 to the last quarter of 2011. The data are seasonally adjusted.

FIGURE 1.4 Time plots of the monthly housing starts for the New England, Middle Atlantic, and Pacific divisions of the United States from January 1995 to June 2011. The data are not seasonally adjusted.

FIGURE 1.5 Time plots of the monthly unemployment rates of the 50 states in the United States from January 1976 to September 2011. The data are seasonally adjusted.

In this book, we refer to {zit} as the ith component of the multivariate time series zt. The objectives of the analysis discussed in this book include (a) to investigate the dynamic relationships between the components of zt and (b) to improve the prediction of zit using information in all components of zt.

Suppose we are interested in predicting zT+1 based on the data {z1,…,zT}. To this end, we may entertain the model

where denotes a prediction of zT+1 and g(.) is some suitable function. The goal of multivariate time series analysis is to specify the function g(.) based on the available data. In many applications, g(.) is a smooth, differentiable function and can be well approximated by a linear function, say,

under the linearity assumption.

To build a solid foundation for making prediction described in the previous paragraph, we need sound statistical theories and methods. The goal of this book is to provide some useful statistical models and methods for analyzing multivariate time series. To begin with, we start with some basic concepts of multivariate time series.

1.2 SOME BASIC CONCEPTS