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,

Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith,

Ruey S. Tsay, Sanford Weisberg

Editors Emeriti: Vic Barnett, J. Stuart Hunter, Jozef L. Teugels

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

WILEY SERIES IN PROBABILITY AND STATISTICS

ESTABLISHED BY WALTER A. SHEWHART AND SAMUEL S. WlLKS

Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J. Stuart Hunter, JozefL. Teugels

The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods.

Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches.

This series provides essential and invaluable reading for all statisticians, whether in aca-demia, industry, government, or research.

† ABRAHAM and LEDOLTER • Statistical Methods for Forecasting

AGRESTI • Analysis of Ordinal Categorical Data

AGRESTI • An Introduction to Categorical Data Analysis, Second Edition AGRESTI • Categorical Data Analysis, Second Edition

ALTMAN, GILL, and

McDONALD • Numerical Issues in Statistical Computing for the Social Scientist

AMARATUNGA and CABRERA • Exploration and Analysis of DNA Microarray and Protein Array Data

ANDL • Mathematics of Chance ANDERSON • An Introduction to Multivariate Statistical Analysis, Third Edition

* ANDERSON • The Statistical Analysis of Time Series

ANDERSON, AUQUIER, HAUCK, OAKES, VANDAELE, and WEISBERG • Statistical Methods for Comparative Studies

ANDERSON and

LOYNES • The Teaching of Practical Statistics

ARMITAGE and DAVID (editors) • Advances in Biometry ARNOLD, BALAKRISHNAN, andNAGARAJA • Records

* ARTHANARI and DODGE • Mathematical Programming in Statistics

* BAILEY • The Elements of Stochastic Processes with Applications to the Natural Sciences

BALAKRISHNAN and KOUTRAS • Runs and Scans with Applications

BALAKRISHNAN and NG • Precedence-Type Tests and Applications

BARNETT • Comparative Statistical Inference, Third Edition

BARNETT • Environmental Statistics

BARNETT and LEWIS • Outliers in Statistical Data, Third Edition BARTOSZYNSKI and

NIEWIADOMSKA-BUGAJ • Probability and Statistical Inference

BASILEVSKY • Statistical Factor Analysis and Related Methods: Theory and Applications

BASU and RIGDON • Statistical Methods for the Reliability of Repairable Systems

BATES and WATTS • Nonlinear Regression Analysis and Its Applications BECHHOFER, SANTNER, and GOLDSMAN • Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons

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BELSLEY • Conditioning Diagnostics: Collinearity and Weak Data in Regression

† BELSLEY, KUH, and WELSCH • Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

BEND AT and PIERSOL • Random Data: Analysis and Measurement Procedures, Third Edition

BERRY, CHALONER, and GEWEKE • Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner

BERNARDO and SMITH • Bayesian Theory

BHAT and MILLER • Elements of Applied Stochastic Processes, Third Edition BHATTACHARYA and WAYMIRE • Stochastic Processes with Applications

BILLINGSLEY • Convergence of Probability Measures, Second Edition BILLINGSLEY • Probability and Measure, Third Edition

BIRKES and DODGE • Alternative Methods of Regression

BISWAS, DATTA, FINE, and SEGAL • Statistical Advances in the Biomedical Sciences: Clinical Trials, Epidemiology, Survival Analysis, and Bioinformatics

BLISCHKE AND MURTHY (editors) • Case Studies in Reliability and Maintenance

BLISCHKE AND MURTHY • Reliability: Modeling, Prediction, and Optimization

BLOOMFIELD • Fourier Analysis of Time Series: An Introduction, Second Edition

BOLLEN • Structural Equations with Latent Variables

BOLLEN and CURRAN • Latent Curve Models: A Structural Equation Perspective

BOROVKOV • Ergodicity and Stability of Stochastic Processes BOULEAU • Numerical Methods for Stochastic Processes

BOX • Bayesian Inference in Statistical Analysis

BOX • R. A. Fisher, the Life of a Scientist

BOX and DRAPER • Response Surfaces, Mixtures, and Ridge Analyses, Second Edition

* BOX and DRAPER • Evolutionary Operation: A Statistical Method for Process Improvement

BOX and FRIENDS • Improving Almost Anything, Revised Edition BOX, HUNTER, and HUNTER • Statistics for Experimenters: Design, Innovation, and Discovery, Second Editon

BOX, JENKINS, and REINSEL • Time Series Analysis: Forcasting and Control, Fourth Edition

BOX and LUCENO • Statistical Control by Monitoring and Feedback Adjustment

BRANDIMARTE • Numerical Methods in Finance: A MATLAB-Based Introduction t

†BROWN and HOLLANDER • Statistics: A Biomedical Introduction

BRUNNER, DOMHOF, and LANGER • Nonparametric Analysis of Longitudinal Data in Factorial Experiments

BUCKLEW • Large Deviation Techniques in Decision, Simulation, and Estimation

CAIROLI and DALANG • Sequential Stochastic Optimization CASTILLO, HADI, BALAKRISHNAN, and SARABIA • Extreme Value and Related Models with Applications in Engineering and Science

CHAN • Time Series: Applications to Finance

CHARALAMBIDES • Combinatorial Methods in Discrete Distributions

CHATTERJEE and HADI • Regression Analysis by Example, Fourth Edition

CHATTERJEE and HADI • Sensitivity Analysis in Linear Regression

CHERNICK • Bootstrap Methods: A Guide for Practitioners and Researchers,Second Edition

CHERNICK and FRIIS • Introductory Biostatistics for the Health Sciences

CHILÈS and DELFINER • Geostatistics: Modeling Spatial Uncertainty CHOW and LIU • Design and Analysis of Clinical Trials: Concepts and Methodologies, Second Edition

CLARKE and DISNEY • Probability and Random Processes: A First Course with Applications, Second Edition

* COCHRAN and COX • Experimental Designs, Second Edition

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CONGDON • Applied Bayesian Modelling

CONGDON • Bayesian Models for Categorical Data

CONGDON • Bayesian Statistical Modelling

CONOVER • Practical Nonparametric Statistics, Third Edition

COOK • Regression Graphics

COOK and WEISBERG • Applied Regression Including Computing and Graphics

COOK and WEISBERG • An Introduction to Regression Graphics

CORNELL • Experiments with Mixtures, Designs, Models, and the Analysis of Mixture Data, Third Edition

COVER and THOMAS • Elements of Information Theory COX • A Handbook of Introductory Statistical Methods

* COX • Planning of Experiments

CRESSIE • Statistics for Spatial Data, Revised Edition

CSÖRG and HORVÁTH • Limit Theorems in Change Point Analysis

DANIEL • Applications of Statistics to Industrial Experimentation

DANIEL • Biostatistics: A Foundation for Analysis in the Health Sciences, Eighth Edition

* DANIEL • Fitting Equations to Data: Computer Analysis of Multifactor Data, Second Edition

DASU and JOHNSON • Exploratory Data Mining and Data Cleaning

DAVID and NAGARAJA • Order Statistics, Third Edition

* DEGROOT, FIENBERG, and KADANE • Statistics and the Law

DEL CASTILLO • Statistical Process Adjustment for Quality Control

DEMARIS • Regression with Social Data: Modeling Continuous and Limited Response Variables

DEMIDENKO • Mixed Models: Theory and Applications DENISON, HOLMES, MALLICK and SMITH • Bayesian Methods for Nonlinear Classification and Regression

DETTE and STUDDEN • The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis

DEY and MUKERJEE • Fractional Factorial Plans

DILLON and GOLDSTEIN • Multivariate Analysis: Methods and Applications

DODGE • Alternative Methods of Regression

* DODGE and ROMIG • Sampling Inspection Tables, Second Edition

* DOOB • Stochastic Processes

DOWDY, WEARDEN, and CHILKO • Statistics for Research, Third Edition

DRAPER and SMITH • Applied Regression Analysis, Third Edition

DRYDEN and MARDIA • Statistical Shape Analysis

DUDEWICZ and MISHRA • Modern Mathematical Statistics

DUNN and CLARK • Basic Statistics: A Primer for the Biomedical Sciences, Third Edition

DUPUIS and ELLIS • A Weak Convergence Approach to the Theory of Large Deviations

EDLER and KITSOS • Recent Advances in Quantitative Methods in Cancer and Human Health Risk Assessment

* ELANDT-JOHNSON and JOHNSON • Survival Models and Data Analysis

ENDERS • Applied Econometric Time Series

† ETHIER and KURTZ • Markov Processes: Characterization and Convergence

EVANS, HASTINGS, and PEACOCK • Statistical Distributions, Third Edition

FELLER • An Introduction to Probability Theory and Its Applications, Volume I, Third Edition, Revised; Volume II, Second Edition

FISHER and VAN BELLE • Biostatistics: A Methodology for the Health Sciences

FITZMAURICE, LAIRD, and WARE • Applied Longitudinal Analysis

* FLEISS • The Design and Analysis of Clinical Experiments FLEISS • Statistical Methods for Rates and Proportions, Third Edition

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† FLEMING and HARRINGTON • Counting Processes and Survival Analysis

FULLER • Introduction to Statistical Time Series, Second Edition

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GALLANT • Nonlinear Statistical Models

GEISSER • Modes of Parametric Statistical Inference

GELMAN and MENG • Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives

GEWEKE • Contemporary Bayesian Econometrics and Statistics GHOSH, MUKHOPADHYAY, and SEN • Sequential Estimation GIESBRECHT and

GUMPERTZ • Planning, Construction, and Statistical Analysis of Comparative Experiments

GIFI • Nonlinear Multivariate Analysis

GIVENS and HOETING • Computational Statistics

GLASSERMAN and YAO • Monotone Structure in Discrete-Event Systems

GNANADESIKAN • Methods for Statistical Data Analysis of Multivariate Observations, Second Edition

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GROSS and HARRIS • Fundamentals of Queueing Theory, Third Edition

* HAHN and SHAPIRO • Statistical Models in Engineering

HAHN and MEEKER • Statistical Intervals: A Guide for Practitioners HALD • A History of Probability and Statistics and their Applications Before 1750

HALD • A History of Mathematical Statistics from 1750 to 1930 ‘ † HAMPEL • Robust Statistics: The Approach Based on Influence Functions HANNAN and DEISTLER • The Statistical Theory of Linear Systems

HEIBERGER • Computation for the Analysis of Designed Experiments

HEDAYAT and SINHA • Design and Inference in Finite Population Sampling

HEDEKER and GIBBONS • Longitudinal Data Analysis

HELLER • MACSYMA for Statisticians

HINKELMANN and KEMPTHORNE • Design and Analysis of Experiments, Volume 1: Introduction to Experimental Design, Second Edition

HINKELMANN and KEMPTHORNE • Design and Analysis of Experiments, Volume 2: Advanced Experimental Design

HOAGLIN, MOSTELLER, and TUKEY • Exploratory Approach to Analysis of Variance

* HOAGLIN, MOSTELLER, and TUKEY • Exploring Data Tables, Trends and Shapes

* HOAGLIN, MOSTELLER, and TUKEY • Understanding Robust and Exploratory Data Analysis

HOCHBERG and TAMHANE • Multiple Comparison Procedures HOCKING • Methods and Applications of Linear Models: Regression and the Analysis of Variance, Second Edition

HOEL • Introduction to Mathematical Statistics, Fifth Edition

HOGG and KLUGMAN • Loss Distributions

HOLLANDER and WOLFE • Nonparametric Statistical Methods, Second Edition

HOSMER and LEMESHOW • Applied Logistic Regression, Second Edition

HOSMER, LEMESHOW, and MAY • Applied Survival Analysis: Regression Modeling of Time-to-Event Data, Second Edition

† HUBER • Robust Statistics

HUBERTY • Applied Discriminant Analysis

HUBERTY and OLEJNIK • Applied MANOVA and Discriminant Analysis, Second Edition

HUNT and KENNEDY • Financial Derivatives in Theory and Practice, Revised Edition

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HURD and MIAMEE • Periodically Correlated Random Sequences: Spectral Theory and Practice

HUSKOVA, BERAN, and DUPAC • Collected Works of Jaroslav Hajek—with Commentary

HUZURBAZAR • Flowgraph Models for Multistate Time-to-Event Data

IMAN and CONOVER • A Modern Approach to Statistics

† JACKSON • A User’s Guide to Principle Components

JOHN • Statistical Methods in Engineering and Quality Assurance

JOHNSON • Multivariate Statistical Simulation

JOHNSON and BALAKRISHNAN • Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz

JOHNSON and BHATTACHARYYA • Statistics: Principles and Methods, Fifth Edition

JOHNSON and KOTZ • Distributions in Statistics

JOHNSON and KOTZ (editors) • Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present

JOHNSON, KOTZ, and BALAKRISHNAN • Continuous Univariate Distributions, Volume 1, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN • Continuous Univariate Distributions, Volume 2, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN • Discrete Multivariate Distributions

JOHNSON, KEMP, and KOTZ • Univariate Discrete Distributions, Third Edition

JUDGE, GRIFFITHS, HILL, LÜTKEPOHL, and LEE • The Theory and Practice of Econometrics, Second Edition

JUREČKOVÁ and SEN • Robust Statistical Procedures: Aymptotics and Interrelations

JUREK and MASON • Operator-Limit Distributions in Probability Theory KADANE • Bayesian Methods and Ethics in a Clinical Trial Design KADANE AND SCHUM • A Probabilistic Analysis of the Sacco and Vanzetti Evidence

KALBFLEISCH and PRENTICE • The Statistical Analysis of Failure Time Data, Second Edition

KARIYA and KURATA • Generalized Least Squares KASS and VOS • Geometrical Foundations of Asymptotic Inference

† KAUFMAN and ROUSSEEUW • Finding Groups in Data: An Introduction to Cluster Analysis

KEDEM and FOKIANOS • Regression Models for Time Series Analysis KENDALL, BARDEN, CARNE, and LE • Shape and Shape Theory KHURI • Advanced Calculus with Applications in Statistics, Second Edition

KHURI, MATHEW, and SINHA • Statistical Tests for Mixed Linear Models

KLEIBER and KOTZ • Statistical Size Distributions in Economics and Actuarial Sciences

KLUGMAN, PANJER, and WILLMOT • Loss Models: From Data to Decisions, Second Edition

KLUGMAN, PANJER, and WILLMOT • Solutions Manual to Accompany Loss Models: From Data to Decisions, Second Edition KOTZ, BALAKRISHNAN, and JOHNSON • Continuous Multivariate Distributions, Volume 1, Second Edition

KOVALENKO, KUZNETZOV, and PEGG • Mathematical Theory of Reliability of Time-Dependent Systems with Practical Applications KOWALSKI and TU • Modern Applied U-Statistics

KROONENBERG • Applied Multiway Data Analysis KVAM and VIDAKOVIC • Nonparametric Statistics with Applications to Science and Engineering

LACHIN • Biostatistical Methods: The Assessment of Relative Risks

LAD • Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction

LAMPERTI • Probability: A Survey of the Mathematical Theory, Second Edition

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LANGE, RYAN, BILLARD, BRILLINGER, CONQUEST, and GREENHOUSE •

Case Studies in Biometry

LARSON • Introduction to Probability Theory and Statistical Inference, Third Edition

LAWLESS • Statistical Models and Methods for Lifetime Data, Second Edition

LAWSON • Statistical Methods in Spatial Epidemiology

LE • Applied Categorical Data Analysis

LE • Applied Survival Analysis

LEE and WANG • Statistical Methods for Survival Data Analysis, Third Edition

LEPAGE and BILLARD • Exploring the Limits of Bootstrap

LEYLAND and GOLDSTEIN (editors) • Multilevel Modelling of Health Statistics

LIAO • Statistical Group Comparison

LINDVALL • Lectures on the Coupling Method

LIN • Introductory Stochastic Analysis for Finance and Insurance LINHART and ZUCCHINI • Model Selection

LITTLE and RUBIN • Statistical Analysis with Missing Data, Second Edition

LLOYD • The Statistical Analysis of Categorical Data

LOWEN and TEICH • Fractal-Based Point Processes

MAGNUS and NEUDECKER • Matrix Differential Calculus with Applications in Statistics and Econometrics, Revised Edition

MALLER and ZHOU • Survival Analysis with Long Term Survivors MALLOWS • Design, Data, and Analysis by Some Friends of Cuthbert Daniel

MANN, SCHAFER, and SINGPURWALLA • Methods for Statistical Analysis of Reliability and Life Data

MANTON, WOODBURY, and TOLLEY • Statistical Applications Using Fuzzy Sets

MARCHETTE • Random Graphs for Statistical Pattern Recognition MARDIA and JUPP • Directional Statistics

MASON, GUNST, and HESS • Statistical Design and Analysis of Experiments with Applications to Engineering and Science, Second Edition

McCULLOCH, SEARLE, andNEUHAUS • Generalized, Linear, and Mixed Models, Second Edition

McFADDEN • Management of Data in Clinical Trials, Second Edition

* McLACHLAN • Discriminant Analysis and Statistical Pattern Recognition McLACHLAN, DO, and AMBROISE • Analyzing Microarray Gene Expression Data

McLACHLAN and KRISHNAN • The EM Algorithm and Extensions, Second Edition

McLACHLAN and PEEL • Finite Mixture Models

McNEIL • Epidemiological Research Methods

MEEKER and ESCOBAR • Statistical Methods for Reliability Data

MEERSCHAERT and SCHEFFLER • Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice MICKEY, DUNN, and CLARK • Applied Statistics: Analysis of Variance and Regression, Third Edition

* MILLER • Survival Analysis, Second Edition

MONTGOMERY, JENNINGS, and KULAHCI • Introduction to Time Series Analysis and Forecasting

MONTGOMERY, PECK, and VINING • Introduction to Linear Regression Analysis, Fourth Edition

MORGENTHALER and TUKEY • Configural Poly sampling: A Route to Practical Robustness

MUIRHEAD • Aspects of Multivariate Statistical Theory

MULLER and STOYAN • Comparison Methods for Stochastic Models and Risks

MURRAY • X-STAT 2.0 Statistical Experimentation, Design Data Analysis, and Nonlinear Optimization

MURTHY, XIE, and JIANG • Weibull Models

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MYERS and MONTGOMERY • Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Second Edition

MYERS, MONTGOMERY, and VINING • Generalized Linear Models. With Applications in Engineering and the Sciences

† NELSON • Accelerated Testing, Statistical Models, Test Plans, and Data Analyses

† NELSON • Applied Life Data Analysis

NEWMAN • Biostatistical Methods in Epidemiology

OCHI • Applied Probability and Stochastic Processes in Engineering and Physical Sciences

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OLIVER and SMITH • Influence Diagrams, Belief Nets and Decision Analysis

PALTA • Quantitative Methods in Population Health: Extensions of Ordinary Regressions

PANJER • Operational Risk: Modeling and Analytics

PANKRATZ • Forecasting with Dynamic Regression Models PANKRATZ • Forecasting with Univariate Box-Jenkins Models: Concepts and Cases

* PARZEN • Modern Probability Theory and Its Applications

* PEÑA, TIAO, and TSAY • A Course in Time Series Analysis

* PIANTADOSI • Clinical Trials: A Methodologic Perspective

PORT • Theoretical Probability for Applications

POURAHMADI • Foundations of Time Series Analysis and Prediction Theory

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† PUTERMAN • Markov Decision Processes: Discrete Stochastic Dynamic Programming QIU • Image Processing and Jump Regression Analysis

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* RIPLEY • Stochastic Simulation

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RYAN • Modern Regression Methods

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* SCHEFFE • The Analysis of Variance

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Library of Congress Cataloging-in-Publication Data:

Box, George E. P.

Time series analysis : forecasting and control / George E.P. Box, Gwilym M. Jenkins, Gregory C. Reinsel. —4th ed.

p. cm.

Includes index.

ISBN-978-0-470-27284-8 (cloth)

1. Time-series analysis. 2. Predicition theory. 3. Transfer functions. 4. Feedback control systems—Mathematical models. I. Jenkins, Gwilym M. II. Reinsel, Gregory C. III. Title QA280.B67 2008 519.5'5—dc22

2007044569

To the memory of

Gwilym M. Jenkins

Gregory C. Reinsel

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 three 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

ACKNOWLEDGEMENTS

We are grateful to our editors Steve Quigley and Lisa Van Horn for their encouragement and help in producing this fourth edition and to Surendar Narasimhan for help with the final preparation of this manuscript.

G. E. P. B.

Preface to the Third Edition

This book is concerned with the building of stochastic (statistical) models for time series and their use in important areas of application. This includes the topics of forecasting, model specification, estimation, and checking, transfer function modeling of dynamic relationships, modeling the effects of intervention events, and process control. Coincident with the first publication of Time Series Analysis: Forecasting and Control, there was a great upsurge in research in these topics. Thus, while the fundamental principles of the kind of time series analysis presented in that edition have remained the same, there has been a great influx of new ideas, modifications, and improvements provided by many authors.

The earlier editions of this book were written during a period in which Gwilym Jenkins was, with extraordinary courage, fighting a slowly debilitating illness. In the present revision, dedicated to his memory, we have preserved the general structure of the original book while revising, modifying, and omitting text where appropriate. In particular, Chapter 7 on estimation of ARMA models has been considerably modified. In addition, we have introduced entirely new sections on some important topics that have evolved since the first edition. These include presentations on various more recently developed methods for model specification, such as canonical correlation analysis and the use of model selection criteria, results on testing for unit root nonstationarity in ARIMA processes, the state space representation of ARMA models and its use for likelihood estimation and forecasting, score tests for model checking, structural components, and deterministic components in time series models and their estimation based on regression-time series model methods. A new chapter (12) has been developed on the important topic of intervention and outlier analysis, reflecting the substantial interest and research in this topic since the earlier editions.

Over the last few years, the new emphasis on industrial quality improvement has strongly focused attention on the role of control both in process monitoring as well as in process adjustment. The control section of this book has, therefore, been completely rewritten to serve as an introduction to these important topics and to provide a better understanding of their relationship.

The objective of this book is to provide practical techniques that will be available to most of the wide audience who could benefit form their use. While we have tried to remove the inadequacies of earlier editions, we have not attempted to produce here a rigorous mathematical treatment of the subject.

We wish to acknowledge our indebtedness to Meg (Margaret) Jenkins and to our wives, Claire and Sandy, for their continuing support and assistance throughout the long period of preparation of this revision.

Research on which the original book was based 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. We are grateful to Professor E. S. Pearson and the Biometrika Trustees for permission to reprint condensed and adapted forms of Tables 1, 8, and 12 of Biometrika Tables for Statisticians, Vol. 1, edited by E. S. Pearson and H. O. Hartley, to Dr. Casimer Stralkowski for permission to reproduce and adapt three figures from his doctoral thesis, and to George Tiao, David Mayne, Emanuel Parzen, David Pierce, Granville Wilson, Donald Watts, John Hampton, Elaine Hodkinson, Patricia Blant, Dean Wichern, David Bacon, Paul Newbold, Hiro Kanemasu, Larry Haugh, John MacGregor, Bovas Abraham, Gina Chen, Johannes Ledolter, Greta Ljung, Carole Leigh, Mary Esser, and Meg Jenkins for their help, in many different ways, in preparing the earlier editions.

GEORGE BOX AND GREGORY REINSEL

CHAPTER ONE

Introduction

A time series is a sequence of observations taken sequentially in time. Many sets of data appear as time series: a monthly sequence of the quantity of goods shipped from a factory, a weekly series of the number of road accidents, hourly observations made on the yield of a chemical process, and so on. Examples of time series abound in such fields as economics, business, engineering, the natural sciences (especially geophysics and meteorology), and the social sciences. Examples of data of the kind that we will be concerned with are displayed as time series plots in Figure 4.1. An intrinsic feature of a time series is that, typically, adjacent observations are dependent. The nature of this dependence among observations of a time series is of considerable practical interest. Time series analysis is concerned with techniques for the analysis of this dependence. This requires the development of stochastic and dynamic models for time series data and the use of such models in important areas of application.

In the subsequent chapters of this book we present methods for building, identifying, fitting, and checking models for time series and dynamic systems. The methods discussed are appropriate for discrete (sampled-data) systems, where observation of the system occurs at equally spaced intervals of time.

We illustrate the use of these time series and dynamic models in five important areas of application:

1. The forecasting of future values of a time series from current and past values

2. The determination of the transfer function of a system subject to inertia—the determination of a dynamic input–output model that can show the effect on the output of a system of any given series of inputs

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