CONTENTS

Cover

Wiley Series in Probability and Statistics

Title Page

Copyright

Dedication

Preface to the Fifth Edition

Preface to the Fourth Edition

Preface to the Third Edition

Chapter 1: Introduction

1.1 Five Important Practical Problems

1.2 Stochastic and Deterministic Dynamic Mathematical Models

1.3 Basic Ideas in Model Building

Appendix A1.1 Use Of The R Software

Exercises

Part One: Stochastic Models and Their Forecasting

Chapter 2: Autocorrelation Function and Spectrum of Stationary Processes

2.1 Autocorrelation Properties of Stationary Models

2.2 Spectral Properties of Stationary Models

Appendix A2.1 Link Between the Sample Spectrum and Autocovariance Function Estimate

Exercises

Chapter 3: Linear Stationary Models

3.1 General Linear Process

3.2 Autoregressive Processes

3.3 Moving Average Processes

3.4 Mixed Autoregressive–Moving Average Processes

Appendix A3.1 Autocovariances, Autocovariance Generating Function, and Stationarity Conditions for a General Linear Process

Appendix A3.2 Recursive Method for Calculating Estimates of Autoregressive Parameters

Exercises

Chapter 4: Linear Nonstationary Models

4.1 Autoregressive Integrated Moving Average Processes

4.2 Three Explicit Forms for the Arima Model

4.3 Integrated Moving Average Processes

Appendix A4.1 Linear Difference Equations

Appendix A4.2 IMA(0, 1, 1) Process with Deterministic Drift

Appendix A4.3 Arima Processes with Added Noise

Exercises

Chapter 5: Forecasting

5.1 Minimum Mean Square Error Forecasts and Their Properties

5.2 Calculating Forecasts and Probability Limits

5.3 Forecast Function and Forecast Weights

5.4 Examples of Forecast Functions and Their Updating

5.5 Use of State-Space Model Formulation for Exact Forecasting

5.6 Summary

Appendix A5.1 Correlation Between Forecast Errors

Appendix A5.2 Forecast Weights for Any Lead Time

Appendix A5.3 Forecasting in Terms of the General Integrated Form

Exercises

Part Two: Stochastic Model Building

Chapter 6: Model Identification

6.1 Objectives of Identification

6.2 Identification Techniques

6.3 Initial Estimates for the Parameters

6.4 Model Multiplicity

Appendix A6.1 Expected Behavior of the Estimated Autocorrelation Function for a Nonstationary Process

Exercises

Chapter 7: Parameter Estimation

7.1 Study of the Likelihood and Sum-of-Squares Functions

7.2 Nonlinear Estimation

7.3 Some Estimation Results for Specific Models

7.4 Likelihood Function Based on the State-Space Model

7.5 Estimation Using Bayes' Theorem

Appendix A7.1 Review of Normal Distribution Theory

Appendix A7.2 Review of Linear Least-Squares Theory

Appendix A7.3 Exact Likelihood Function for Moving Average and Mixed Processes

Appendix A7.4 Exact Likelihood Function for an Autoregressive Process

Appendix A7.5 Asymptotic Distribution of Estimators for Autoregressive Models

Appendix A7.6 Examples of the Effect of Parameter Estimation Errors on Variances of Forecast Errors and Probability Limits for Forecasts

Appendix A7.7 Special Note on Estimation of Moving Average Parameters

Exercises

Chapter 8: Model Diagnostic Checking

8.1 Checking the Stochastic Model

8.2 Diagnostic Checks Applied to Residuals

8.3 Use of Residuals to Modify the Model

Exercises

Chapter 9: Analysis of Seasonal Time Series

9.1 Parsimonious Models for Seasonal Time Series

9.2 Representation of the Airline Data by a Multiplicative (0, 1, 1) × (0, 1, 1)12 Model

9.3 Some Aspects of More General Seasonal Arima Models

9.4 Structural Component Models and Deterministic Seasonal Components

9.5 Regression Models with Time Series Error Terms

Appendix A9.1 Autocovariances for some Seasonal Models

Exercises

Chapter 10: Additional Topics and Extensions

10.1 Tests for Unit Roots in Arima Models

10.2 Conditional Heteroscedastic Models

10.3 Nonlinear Time Series Models

10.4 Long Memory Time Series Processes

Exercises

Part Three: Transfer Function and Multivariate Model Building

Chapter 11: Transfer Function Models

11.1 Linear Transfer Function Models

11.2 Discrete Dynamic Models Represented by Difference Equations

11.3 Relation Between Discrete and Continuous Models

Appendix A11.1 Continuous Models with Pulsed Inputs

Appendix A11.2 Nonlinear Transfer Functions and Linearization

Exercises

Chapter 12: Identification, Fitting, and Checking of Transfer Function Models

12.1 Cross-Correlation Function

12.2 Identification of Transfer Function Models

12.3 Fitting and Checking Transfer Function Models

12.4 Some Examples of Fitting and Checking Transfer Function Models

12.5 Forecasting with Transfer Function Models Using Leading Indicators

12.6 Some Aspects of the Design of Experiments to Estimate Transfer Functions

Appendix A12.1 Use of Cross-Spectral Analysis for Transfer Function Model Identification

Appendix A12.2 Choice of Input to Provide Optimal Parameter Estimates

Exercises

Chapter 13: Intervention Analysis, Outlier Detection, and Missing Values

13.1 Intervention Analysis Methods

13.2 Outlier Analysis for Time Series

13.3 Estimation for ARMA Models with Missing Values

Exercises

Chapter 14: Multivariate Time Series Analysis

14.1 Stationary Multivariate Time Series

14.2 Vector Autoregressive Models

14.3 Vector Moving Average Models

14.4 Vector Autoregressive–Moving Average Models

14.5 Forecasting for Vector Autoregressive–Moving Average Processes

14.6 State-Space form of the Varma Model

14.7 Further Discussion of Varma Model Specification

14.8 Nonstationarity and Cointegration

Appendix A14.1 Spectral Characteristics and Linear Filtering Relations for Stationary Multivariate Processes

Exercises

Part Four: Design of Discrete Control Schemes

Chapter 15: Aspects of Process Control

15.1 Process Monitoring and Process Adjustment

15.2 Process Adjustment Using Feedback Control

15.3 Excessive Adjustment Sometimes Required by MMSE Control

15.4 Minimum Cost Control with Fixed Costs of Adjustment and Monitoring

15.5 Feedforward Control

15.6 Monitoring Values of Parameters of Forecasting and Feedback Adjustment Schemes

Appendix A15.1 Feedback Control Schemes Where the Adjustment Variance is Restricted

Appendix A15.2 Choice of the Sampling Interval

Exercises

Part Five: Charts and Tables

Collection of Tables and Charts

Collection of Time Series Used for Examples in the Text and in Exercises

References

Index

End User License Agreement

List of Tables

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 3.1

Table 3.2

Table 4.1

Table 4.2

Table 5.1

Table 5.2

Table 5.3

Table A5.1

Table A5.2

Table A5.3

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 7.6

Table 8.1

Table 8.2

Table 8.3

Table 9.1

Table 9.2

Table 9.3

Table A9.1

Table 10.1

Table 11.1

Table 11.2

Table 11.3

Table 11.4

Table 12.1

Table 12.2

Table 12.3

Table 12.4

Table 12.5

Table 12.6

Table 12.7

Table 12.8

Table 12.9

Table 12.10

Table 13.1

Table 14.1

Table 14.2

Table 14.3

Table 15.1

Table 15.2

Table 15.3

Table A15.1

Table A15.2

Table A

Table E

Table F

Table G

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure A5.1

Figure A5.2

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure A6.1

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure A7.1

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 9.8

Figure 9.9

Figure 9.10

Figure 9.11

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 11.8

Figure 11.10

Figure 11.9

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure 12.5

Figure 12.6

Figure 12.7

Figure 12.8

Figure 12.9

Figure 12.10

Figure 12.11

Figure 12.12

Figure 13.1

Figure 13.2

Figure 14.1

Figure 14.2

Figure 14.3

Figure 14.4

Figure 14.5

Figure 14.6

Figure 15.1

Figure 15.2

Figure 15.3

Figure 15.4

Figure 15.5

Figure 15.6

Figure 15.7

Figure 15.8

Figure 15.9

Figure 15.10

Figure 15.11

Figure 15.12

Figure 15.13

Figure 15.14

Figure 15.15

Figure 15.16

Figure 15.17

Figure A15.1

Chart B

Chart C

Chart D

Guide

Cover

Table of Contents

Begin Reading

Begin Reading

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Wiley Series in Probability and Statistics

Established by Walter A. Shewhart and Samuel S. Wilks

Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Geof H. Givens, Harvey Goldstein, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg

Editors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

Time Series Analysis

Forecasting and Control

Fifth Edition

George E. P. Box

Gwilym M. Jenkins

Gregory C. Reinsel

Greta M. Ljung

Copyright 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

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Library of Congress Cataloging-in-Publication Data:

Box, George E. P.

Time series analysis : forecasting and control. -Fifth edition / George E.P. Box, Gwilym M. Jenkins,

Gregory C. Reinsel, Greta M. Ljung.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-67502-1 (cloth : alk. paper) 1. Time-series analysis. 2. Prediction theory. 3. Transfer functions. 4. Feedback control systems-Mathematical models. I. Jenkins, Gwilym M. II. Reinsel, Gregory C. III. Ljung, Greta M., 1941-IV. Title.

QA280.B67 2016

519.5’5-dc23

2015015492

To the Memory of

George E. P. Box

Gwilym M. Jenkins

Gregory C. Reinsel

Preface to the Fifth Edition

This book describes statistical models and methods for analyzing discrete time series and presents important applications of the methodology. The models considered include the class of autoregressive integrated moving average (ARIMA) models and various extensions of these models. The properties of the models are examined and statistical methods for model specification, parameter estimation, and model checking are presented. Applications to forecasting nonseasonal as well as seasonal time series are described. Extensions of the methodology to transfer function modeling of dynamic relationships between two or more time series, modeling the effects of intervention events, multivariate time series modeling, and process control are discussed. Topics such as state-space and structural modeling, nonlinear models, long-memory models, and conditionally heteroscedastic models are also covered. The goal has been to provide a text that is practical and of value to both academicians and practitioners.

The first edition of this book appeared in 1970 and around that time there was a great upsurge in research on time series analysis and forecasting. This generated a large influx of new ideas, modifications, and improvements by many authors. For example, several new research directions began to emerge in econometrics around that time, leading to what is now known as time series econometrics. Many of these developments were reflected in the fourth edition of this book and have been further elaborated upon in this new edition.

The main goals of preparing a new edition have been to expand and update earlier material, incorporate new literature, enhance and update numerical illustrations through the use of R, and increase the number of exercises in the book. Some of the chapters in the previous edition have been reorganized. For example, Chapter 14 on multivariate time series analysis has been reorganized and expanded, placing more emphasis on vector autoregressive (VAR) models. The VAR models are by far the most widely used multivariate time series models in applied work. This edition provides an expanded treatment of these models that includes software demonstrations.

Chapter 10 has also been expanded and updated. This chapter covers selected topics in time series analysis that either extend or supplement material discussed in earlier chapters. This includes unit roots testing, modeling of conditional heteroscedasticity, nonlinear models, and long memory models. A section of unit root testing that appeared in Chapter 7 of the previous edition has been expanded and moved to Section 10.1 in this edition. Section 10.2 deals with autoregressive conditionally heteroscedastic models, such as the ARCH and GARCH models. These models focus on the variability in a time series and are useful for modeling the volatility or variability in economic and financial series, in particular. The treatment of the ARCH and GARCH models has been expanded and several extensions have been added.

Elsewhere in the text, the exposition has been enhanced by revising, modifying, and omitting text as appropriate. Several tables have either been edited or replaced by graphs to make the presentation more effective. The number of exercises has been increased throughout the text and they now appear at the end of each chapter.

A further enhancement to this edition is the use of the statistical software R for model building and forecasting. The R package is available as a free download from the R Project for Statistical Computing at www.r-project.org. A brief description of the software is given in Appendix A1.1 of Chapter 1. Graphs generated using R now appear in many of the chapters along with R code that will help the reader reconstruct the graphs. The software is also used for numerical illustration in many of the examples in the text.

The fourth edition of this book was published by Wiley in 2008. Plans for a new edition began during the fall of 2012. I was deeply honored when George Box asked me to help him with this update. George was my Ph.D. advisor at the University of Wisconsin-Madison and remained a dear friend to me over the years as he did to all his students. Sadly, he was rather ill when the plans for this new edition were finalized towards the end of 2012. He did not have a chance to see the project completed as he passed away in March of 2013. I am deeply grateful for the opportunity to work with him and for the confidence he showed in assigning me this task. The book is dedicated to his memory and to the memory of his distinguished co-authors Gwilym Jenkins and Gregory Reinsel. Their contributions were many and they are all missed.

I also want to express my gratitude to several friends and colleagues in the time series community who have read the manuscript and provided helpful comments and suggestions. These include Ruey Tsay, William Wei, Sung Ahn, and Raja Velu who have read Chapter 14 on multivariate time series analysis, and David Dickey, Johannes Ledolter, Timo Teräsvirta, and Niels Haldrup who have read Chapter 10 on special topics. Their constructive comments and suggestions are much appreciated. Assistance and support from Paul Lindholm in Finland is also gratefully acknowledged. The use of R in this edition includes packages developed for existing books on time series analysis such as Cryer and Chan (2010), Shumway and Stoffer (2011), and Tsay (2014). We commend these authors for making their code and datasets available for public use through the R Project.

Research for the original version of this book was supported by the Air Force Office of Scientific Research and by the British Science Research Council. Research incorporated in the third edition was partially supported by the Alfred P. Sloan Foundation and by the National Aeronautics and Space Administration. Permission to reprint selected tables from Biometrika Tables for Statisticians, Vol. 1, edited by E. S. Pearson and H. O. Hartley is also acknowledged. On behalf of my co-authors, I would like to thank George Tiao, David Mayne, David Pierce, Granville Tunnicliffe Wilson, Donald Watts, John Hampton, Elaine Hodkinson, Patricia Blant, Dean Wichern, David Bacon, Paul Newbold, Hiro Kanemasu, Larry Haugh, John MacGregor, Bovas Abraham, Johannes Ledolter, Gina Chen, Raja Velu, Sung Ahn, Michael Wincek, Carole Leigh, Mary Esser, Sandy Reinsel, and Meg Jenkins, for their help, in many different ways, in preparing the earlier editions. A very special thanks is extended to Claire Box for her long-time help and support.

The guidance and editorial support of Jon Gurstelle and Sari Friedman at Wiley is gratefully acknowledged. We also thank Stephen Quigley for his help in setting up the project, and Katrina Maceda and Shikha Pahuja for their help with the production.

Finally, I want to express my gratitude to my husband Bert Beander for his encouragement and support during the preparation of this revision.

Greta M. Ljung

Lexington, MA May 2015

Preface to the Fourth Edition

It may be of interest to briefly recount how this book came to be written. Gwilym Jenkins and I first became friends in the late 1950s. We were intrigued by an idea that a chemical reactor could be designed that optimized itself automatically and could follow a moving maximum. We both believed that many advances in statistical theory came about as a result of interaction with researchers who were working on real scientific problems. Helping to design and build such a reactor would present an opportunity to further demonstrate this concept.

When Gwilym Jenkins came to visit Madison for a year, we discussed the idea with the famous chemical engineer Olaf Hougen, then in his eighties. He was enthusiastic and suggested that we form a small team in a joint project to build such a system. The National Science Foundation later supported this project. It took 3 years, but suffice it to say, that after many experiments, several setbacks, and some successes the reactor was built and it worked.

As expected, this investigation taught us a lot. In particular, we acquired proficiency in the manipulation of difference equations that were needed to characterize the dynamics of the system. It also gave us a better understanding of nonstationary time series required for realistic modeling of system noise. This was a happy time. We were doing what we most enjoyed doing: interacting with experimenters in the evolution of ideas and the solution of real problems, with real apparatus and real data.

Later there was fallout in other contexts, for example, advances in time series analysis, in forecasting for business and economics, and also developments in statistical process control (SPC) using some notions learned from the engineers.

Originally Gwilym came for a year. After that I spent each summer with him in England at his home in Lancaster. For the rest of the year, we corresponded using small reel-to-reel tape recorders. We wrote a number of technical reports and published some papers but eventually realized we needed a book. The first two editions of this book were written during a period in which Gwilym was, with extraordinary courage, fighting a debilitating illness to which he succumbed sometime after the book had been completed.

Later Gregory Reinsel, who had profound knowledge of the subject, helped to complete the third edition. Also in this fourth edition, produced after his untimely death, the new material is almost entirely his. In addition to a complete revision and updating, this fourth edition resulted in two new chapters: Chapter 10 on nonlinear and long memory models and Chapter 12 on multivariate time series.

This book should be regarded as a tribute to Gwilym and Gregory.

I was especially blessed to work with two such gifted colleagues.

George E. P. Box

Madison, Wisconsin March 2008

Lesen Sie weiter in der vollständigen Ausgabe!

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Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!