Time Series Analysis

The late George E. P. Box, PhD, was professor emeritus of statistics at the University of Wisconsin-Madison. He was a Fellow of the American Academy of Arts and Sciences and a recipient of the Samuel S. Wilks Memorial Medal of the American Statistical Association, the Shewhart Medal of the American Society for Quality, and the Guy Medal in Gold of the Royal Statistical Society. Dr. Box was also author of seven Wiley books.The late Gwilym M. Jenkins, PhD, was professor of systems engineering at Lancaster University in the United Kingdom, where he was also founder and managing director of the International Systems Corporation of Lancaster. A Fellow of the Institute of Mathematical Statistics and the Institute of Statisticians, Dr. Jenkins had a prestigious career in both academia and consulting work that included positions at Imperial College London, Stanford University, Princeton University, and the University of Wisconsin-Madison. He was widely known for his work on time series analysis, most notably his groundbreaking work with Dr. Box on the Box-Jenkins models.The late Gregory C. Reinsel, PhD, was professor and former chair of the department of Statistics at the University of Wisconsin-Madison. Dr. Reinsel's expertise was focused on time series analysis and its applications in areas as diverse as economics, ecology, engineering, and meteorology. He authored over seventy refereed articles and three books, and was a Fellow of both the American Statistical Association and the Institute of Mathematical Statistics.Greta M. Ljung, PhD, is a statistical consultant residing in Lexington, MA. She received her doctorate from the University of Wisconsin-Madison where she did her research in time series analysis under the direction of Professor George Box. Dr. Ljung's career includes teaching positions at Boston University and Massachusetts Institute of Technology, and a position as Principal Scientist at AIR Worldwide in Boston. Her many accomplishments include joint work with George Box on a time series goodness of fit test, which is widely applied in econometrics and other fields.

• Preface to the Fifth Edition xixPreface to the Fourth Edition xxiiiPreface to the Third Edition xxv1 Introduction 11.1 Five Important Practical Problems 21.2 Stochastic and Deterministic Dynamic Mathematical Models 61.3 Basic Ideas in Model Building 14Appendix A. 1 Use of the R Software 17Exercises 18Part One Stochastic Models and Their Forecasting 192 Autocorrelation Function and Spectrum of Stationary Processes 212.1 Autocorrelation Properties of Stationary Models 212.2 Spectral Properties of Stationary Models 34Appendix A2. 1 Link Between the Sample Spectrum and Autocovariance Function Estimate 43Exercises 443 Linear Stationary Models 473.1 General Linear Process 473.2 Autoregressive Processes 543.3 Moving Average Processes 683.4 Mixed Autoregressive--Moving Average Processes 75Appendix A3. 1 Autocovariances Autocovariance Generating Function, and Stationarity Conditions for a General Linear Process 82Appendix A3. 2 Recursive Method for Calculating Estimates of Autoregressive Parameters 84Exercises 864 Linear Nonstationary Models 884.1 Autoregressive Integrated Moving Average Processes 884.2 Three Explicit Forms for the ARIMA Model 974.3 Integrated Moving Average Processes 106Appendix A4. 1 Linear Difference Equations 116Appendix A4. 2 IMA(0, 1, 1) Process with Deterministic Drift 121Appendix A4. 3 ARIMA Processes with Added Noise 122Exercises 1265 Forecasting 1295.1 Minimum Mean Square Error Forecasts and Their Properties 1295.2 Calculating Forecasts and Probability Limits 1355.3 Forecast Function and Forecast Weights 1395.4 Examples of Forecast Functions and Their Updating 1445.5 Use of State-Space Model Formulation for Exact Forecasting 1555.6 Summary 162Appendix A5. 1 Correlation Between Forecast Errors 164Appendix A5. 2 Forecast Weights for any Lead Time 166Appendix A5. 3 Forecasting in Terms of the General Integrated Form 168Exercises 174Part Two STOCHASTIC MODEL BUILDING 1776 Model Identification 1796.1 Objectives of Identification 1796.2 Identification Techniques 1806.3 Initial Estimates for the Parameters 1946.4 Model Multiplicity 202Appendix A6. 1 Expected Behavior of the Estimated Autocorrelation Function for a Nonstationary Process 206Exercises 2077 Parameter Estimation 2097.1 Study of the Likelihood and Sum-of-Squares Functions 2097.2 Nonlinear Estimation 2267.3 Some Estimation Results for Specific Models 2367.4 Likelihood Function Based on the State-Space Model 2427.5 Estimation Using Bayes’ Theorem 245Appendix A7. 1 Review of Normal Distribution Theory 251Appendix A7. 2 Review of Linear Least-Squares Theory 256Appendix A7. 3 Exact Likelihood Function for Moving Average and Mixed Processes 259Appendix A7. 4 Exact Likelihood Function for an Autoregressive Process 266Appendix A7. 5 Asymptotic Distribution of Estimators for Autoregressive Models 274Appendix A7. 6 Examples of the Effect of Parameter Estimation Errors on Variances of Forecast Errors and Probability Limits for Forecasts 277Appendix A. 7 Special Note on Estimation of Moving Average Parameters 280Exercises 2808 Model Diagnostic Checking 2848.1 Checking the Stochastic Model 2848.2 Diagnostic Checks Applied to Residuals 2878.3 Use of Residuals to Modify the Model 301Exercises 3039 Analysis of Seasonal Time Series 3059.1 Parsimonious Models for Seasonal Time Series 3059.2 Representation of the Airline Data by a Multiplicative (0, 1, 1) × (0, 1, 1) 12 Model 3109.3 Some Aspects of More General Seasonal ARIMA Models 3259.4 Structural Component Models and Deterministic Seasonal Components 3319.5 Regression Models with Time Series Error Terms 339Appendix A9. 1 Autocovariances for Some Seasonal Models 345Exercises 34910 Additional Topics and Extensions 35210.1 Tests for Unit Roots in ARIMA Models 35310.2 Conditional Heteroscedastic Models 36110.3 Nonlinear Time Series Models 37710.4 Long Memory Time Series Processes 385Exercises 392Part Three Transfer Function and Multivariate Model Building 39511 Transfer Function Models 39711.1 Linear Transfer Function Models 39711.2 Discrete Dynamic Models Represented by Difference Equations 40411.3 Relation Between Discrete and Continuous Models 414Appendix A11. 1 Continuous Models with Pulsed Inputs 420Appendix A11. 2 Nonlinear Transfer Functions and Linearization 424Exercises 42612 Identification, Fitting, and Checking of Transfer Function Models 42812.1 Cross-Correlation Function 42912.2 Identification of Transfer Function Models 43512.3 Fitting and Checking Transfer Function Models 44612.4 Some Examples of Fitting and Checking Transfer Function Models 45312.5 Forecasting with Transfer Function Models Using Leading Indicators 46112.6 Some Aspects of the Design of Experiments to Estimate Transfer Functions 469Appendix A12.1 Use of Cross-Spectral Analysis for Transfer Function Model Identification 471Appendix A12.2 Choice of Input to Provide Optimal Parameter Estimates 473Exercises 47713 Intervention Analysis, Outlier Detection, and Missing Values 48113.1 Intervention Analysis Methods 48113.2 Outlier Analysis for Time Series 48813.3 Estimation for ARMA Models with Missing Values 495Exercises 50214 Multivariate Time Series Analysis 50514.1 Stationary Multivariate Time Series 50614.2 Vector Autoregressive Models 50914.3 Vector Moving Average Models 52414.4 Vector Autoregressive--Moving Average Models 52714.5 Forecasting for Vector Autoregressive--Moving Average Processes 53414.6 State-Space Form of the VARMA Model 53614.7 Further Discussion of VARMA Model Specification 53914.8 Nonstationarity and Cointegration 546Appendix A14. 1 Spectral Characteristics and Linear Filtering Relations for Stationary Multivariate Processes 552Exercises 554Part Four Design of Discrete Control Schemes 55915 Aspects of Process Control 56115.1 Process Monitoring and Process Adjustment 56215.2 Process Adjustment Using Feedback Control 56615.3 Excessive Adjustment Sometimes Required by MMSE Control 58015.4 Minimum Cost Control with Fixed Costs of Adjustment and Monitoring 58215.5 Feedforward Control 58815.6 Monitoring Values of Parameters of Forecasting and Feedback Adjustment Schemes, 599Appendix A5. 1 Feedback Control Schemes Where the Adjustment Variance Is Restricted, 600Appendix A15. 2 Choice of the Sampling Interval 609Exercises 613Part Five Charts and Tables 617Collection of Tables and Charts 619Collection of Time Series Used for Examples in the Text and in Exercises 625References 642Index 659