Applied Time Series Analysis

By Wayne A. Woodward, Henry L. Gray, Alan C Elliott

© 2011 – CRC Press

564 pages | 189 B/W Illus.

Purchasing Options:
Hardback: 9781439818374
pub: 2011-10-25
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About the Book

Virtually any random process developing chronologically can be viewed as a time series. In economics, closing prices of stocks, the cost of money, the jobless rate, and retail sales are just a few examples of many. Developed from course notes and extensively classroom-tested, Applied Time Series Analysis includes examples across a variety of fields, develops theory, and provides software to address time series problems in a broad spectrum of fields. The authors organize the information in such a format that graduate students in applied science, statistics, and economics can satisfactorily navigate their way through the book while maintaining mathematical rigor.

One of the unique features of Applied Time Series Analysis is the associated software, GW-WINKS, designed to help students easily generate realizations from models and explore the associated model and data characteristics. The text explores many important new methodologies that have developed in time series, such as ARCH and GARCH processes, time varying frequencies (TVF), wavelets, and more. Other programs (some written in R and some requiring S-plus) are available on an associated website for performing computations related to the material in the final four chapters.


"There is scarcely a standard technique that the reader will find left out … this book is highly recommended for those requiring a ready introduction to applicable methods in time series and serves as a useful resource for pedagogical purposes."

International Statistical Review (2014), 82

"Current time series theory for practice is well summarized in this book."

—Emmanuel Parzen, Texas A&M University

"What an extraordinary range of topics covered, all very insightfully. I like [the authors’] innovations very much, such as the AR factor table."

—David Findley, U.S. Census Bureau (retired)

"… impressive coverage of the scope of time series analysis in both frequency and time domain … One unique feature of the book is the emphasis on factoring the AR polynomial function and its roots. … I commend the authors for having included a number of topics on nonstationary processes … an excellent textbook to adopt for a class and a good introductory book for a student who wants to embark on dissertation research in time series. … the book provides the reader with very good background material to be able to conduct practical and insightful data analysis and be able to comprehend the more theory-oriented books. There are many very good exercises in this book …"

—Hernando Ombao, Journal of the American Statistical Association, March 2013

"The book contains many illustrative examples, theorems with proofs, and applied and theoretical problems at the end of each chapter with real-life applications. Also, the book looks at generating realisations of the mentioned time series models via software packages such as GW-WINKS and R. The book’s material is very valuable and is well presented, so it represents a good reference at both undergraduate and postgraduate levels, and also a good source for all who are interested in time series analysis."

—Hassan S. Bakouch, Journal of Applied Statistics, 2012

Table of Contents

Stationary Time Series

Time Series

Stationary Time Series

Autocovariance and Autocorrelation Functions for Stationary Time Series

Estimation of the Mean, Autocovariance, and Autocorrelation for Stationary Time Series

Power Spectrum

Estimating the Power Spectrum and Spectral Density for Discrete Time Series

Time Series Examples

Linear Filters

Introduction to Linear Filters

Stationary General Linear Processes

Wold Decomposition Theorem

Filtering Applications

ARMA Time Series Models

Moving Average Processes

Autoregressive Processes

Autoregressive–Moving Average Processes

Visualizing Autoregressive Components

Seasonal ARMA(p,q)x(Ps,Qs)s Models

Generating Realizations from ARMA(p,q) Processes


Other Stationary Time Series Models

Stationary Harmonic Models

ARCH and GARCH Models

Nonstationary Time Series Models

Deterministic Signal-Plus-Noise Models

ARIMA(p,d,q) and ARUMA(p,d,q) Models

Multiplicative Seasonal ARUMA(p,d,q) x (Ps,Ds,Qs)s Model

Random Walk Models

G-Stationary Models for Data with Time-Varying Frequencies


Mean Square Prediction Background

Box–Jenkins Forecasting for ARMA(p,q) Models

Properties of the Best Forecast Xto(l)

pi-Weight Form of the Forecast Function

Forecasting Based on the Difference Equation

Eventual Forecast Function

Probability Limits for Forecasts

Forecasts Using ARUMA(p,d,q) Models

Forecasts Using Multiplicative Seasonal ARUMA Models

Forecasts Based on Signal-plus-Noise Models

Parameter Estimation


Preliminary Estimates

Maximum Likelihood Estimation of ARMA( p,q) Parameters

Backcasting and Estimating σ2a

Asymptotic Properties of Estimators

Estimation Examples Using Data

ARMA Spectral Estimation

ARUMA Spectral Estimation

Model Identification

Preliminary Check for White Noise

Model Identification for Stationary ARMA Models

Model Identification for Nonstationary ARUMA(p,d,q) Models

Model Identification Based on Pattern Recognition

Model Building

Residual Analysis

Stationarity versus Nonstationarity

Signal-plus-Noise versus Purely Autocorrelation-Driven Models

Checking Realization Characteristics

Comprehensive Analysis of Time Series Data: A Summary

Vector-Valued (Multivariate) Time Series

Multivariate Time Series Basics

Stationary Multivariate Time Series

Multivariate (Vector) ARMA Processes

Nonstationary VARMA Processes

Testing for Association between Time Series

State-Space Models

Proof of Kalman Recursion for Prediction and Filtering

Long-Memory Processes

Long Memory

Fractional Difference and FARMA Models

Gegenbauer and GARMA Processes

k-Factor Gegenbauer and GARMA Models

Parameter Estimation and Model Identification

Forecasting Based on the k-Factor GARMA Model

Modeling Atmospheric CO2 Data Using Long-Memory Models


Shortcomings of Traditional Spectral Analysis for TVF Data

Methods That Localize the ‘‘Spectrum’’ in Time

Wavelet Analysis

Wavelet Packets

Concluding Remarks on Wavelets

Appendix: Mathematical Preliminaries for This Chapter

G-Stationary Processes

Generalized-Stationary Processes

M-Stationary Processes

G(λ)-Stationary Processes

Linear Chirp Processes

Concluding Remarks


About the Authors

Henry L. Gray is a C.F. Frensley Professor Emeritus in the Department of Statistical Science at Southern Methodist University in Dallas, Texas.

Wayne A. Woodward is a professor and chair of the Department of Statistical Science at Southern Methodist University in Dallas, Texas.

Alan C. Elliott is a biostatistician in the Department of Clinical Sciences at the University of Texas Southwestern Medical Center in Dallas.

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General