Contents

Preface

1. Introduction to Forecasting

1.1 The Nature and Uses of Forecasts

1.2 Some Examples of Time Series

1.3 The Forecasting Process

1.4 Resources for Forecasting

2. Statistics Background for Forecasting

2.1 Introduction

2.2 Graphical Displays

2.3 Numerical Description of Time Series Data

2.4 Use of Data Transformations and Adjustments

2.5 General Approach to Time Series Modeling and Forecasting

2.6 Evaluating and Monitoring Forecasting Model Performance

3. Regression Analysis and Forecasting

3.1 Introduction

3.2 Least Squares Estimation in Linear Regression Models

3.3 Statistical Inference in Linear Regression

3.4 Prediction of New Observations

3.5 Model Adequacy Checking

3.6 Variable Selection Methods in Regression

3.7 Generalized and Weighted Least Squares

3.8 Regression Models for General Time Series Data

4. Exponential Smoothing Methods

4.1 Introduction

4.2 First-Order Exponential Smoothing

4.3 Modeling Time Series Data

4.4 Second-Order Exponential Smoothing

4.5 Higher-Order Exponential Smoothing

4.6 Forecasting

4.7 Exponential Smoothing for Seasonal Data

4.8 Exponential Smoothers and ARIMA Models

5. Autoregressive Integrated Moving Average (ARIMA) Models

5.1 Introduction

5.2 Linear Models for Stationary Time Series

5.3 Finite Order Moving Average (MA) Processes

5.4 Finite Order Autoregressive Processes

5.5 Mixed Autoregressive-Moving Average (ARMA) Processes

5.6 Nonstationary Processes

5.7 Time Series Model Building

5.8 Forecasting ARIMA Processes

5.9 Seasonal Processes

5.10 Final Comments

6. Transfer Functions and Intervention Models

6.1 Introduction

6.2 Transfer Function Models

6.3 Transfer Function-Noise Models

6.4 Cross Correlation Function

6.5 Model Specification

6.6 Forecasting with Transfer Function-Noise Models

6.7 Intervention Analysis

7. Survey of Other Forecasting Methods

7.1 Multivariate Time Series Models and Forecasting

7.2 State Space Models

7.3 ARCH and GARCH Models

7.4 Direct Forecasting of Percentiles

7.5 Combining Forecasts to Improve Prediction Performance

7.6 Aggregation and Disaggregation of Forecasts

7.7 Neural Networks and Forecasting

7.8 Some Comments on Practical Implementation and Use of Statistical Forecasting Procedures

Appendix A. Statistical Tables

Appendix B. Data Sets for Exercises

Bibliography

Index

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Library of Congress Cataloging-in-Publication Data:

Montgomery, Douglas C.
Introduction to time series analysis and forecasting / Douglas C. Montgomery,
Cheryl L. Jennings, Murat Kulahci.
p. cm. – (Wiley series in probability and statistics)
Includes bibliographical references and index.
ISBN 978-0-471-65397-4 (cloth)

1. Time-series analysis. 2. Forecasting. I. Jennings, Cheryl L. II. Kulahci, Murat. III. Title.
QA280.M662 2007

519.5’5-dc22
2007019891

Preface

Analyzing time-oriented data and forecasting future values of a time series are among the most important problems that analysts face in many fields, ranging from finance and economics, to managing production operations, to the analysis of political and social policy sessions, to investigating the impact of humans and the policy decisions that they make on the environment. Consequently, there is a large group of people in a variety of fields including finance, economics, science, engineering, statistics, and public policy who need to understand some basic concepts of time series analysis and forecasting. Unfortunately, most basic statistics and operations management books give little if any attention to time-oriented data, and little guidance on forecasting. There are some very good high level books on time series analysis. These books are mostly written for technical specialists who are taking a doctoral-level course or doing research in the field. They tend to be very theoretical and often focus on a few specific topics or techniques. We have written this book to fill the gap between these two extremes.

This book is intended for practitioners who make real-world forecasts. Our focus is on short-to medium-term forecasting where statistical methods are useful. Since many organizations can improve their effectiveness and business results by making better short-to medium-term forecasts, this book should be useful to a wide variety of professionals. The book can also be used as a textbook for an applied forecasting and time series analysis course at the advanced undergraduate or first-year graduate level. Students in this course could come from engineering, business, statistics, operations research, mathematics, computer science, and any area of application where making forecasts is important. Readers need a background in basic statistics (previous exposure to linear regression would be helpful but not essential), and some knowledge of matrix algebra, although matrices appear mostly in the chapter on regression, and if one is interested mainly in the results, the details involving matrix manipulation can be skipped. Integrals and derivatives appear in a few places in the book, but no detailed working knowledge of calculus is required.

Successful time series analysis and forecasting requires that the analyst interact with computer software. The techniques and algorithms are just not suitable to manual calculations. We have chosen to demonstrate the techniques presented using three packages, Minitab, JMP, and SAS. We have selected these packages because they are widely used in practice and because they have generally good capability for analyzing time series data and generating forecasts. However, the basic principles that underlie most of our presentation are not specific to any particular software package. Readers can use any software that they like or have available that has basic statistical forecasting capability. While the text examples do utilize Minitab, JMP, and SAS, and illustrate the features and capability of those packages, these features or similar ones are found in many other software packages.

There are three basic approaches to generating forecasts: regression-based methods, heuristic smoothing methods, and general time series models. Because all three of these basic approaches are useful, we give an introduction to all of them. Chapter 1 introduces the basic forecasting problem, defines terminology, and illustrates many of the common features of time series data. Chapter 2 contains many of the basic statistical tools used in analyzing time series data. Topics include plots, numerical summaries of time series data including the autocovariance and autocorrelation functions, transformations, differencing, and decomposing a time series into trend and seasonal components. We also introduce metrics for evaluating forecast errors, and methods for evaluating and tracking forecasting performance over time. Chapter 3 discusses regression analysis and its use in forecasting. We discuss both cross-section and time series regression data, least squares and maximum likelihood model fitting, model adequacy checking, prediction intervals, and weighted and generalized least squares. The first part of this chapter covers many of the topics typically seen in an introductory treatment of regression, either in a stand-alone course or as part of another applied statistics course. It should be a reasonable review for many readers. Chapter 4 presents exponential smoothing techniques, both for time series with polynomial components and for seasonal data. We discuss and illustrate methods for selecting the smoothing constant(s), forecasting, and constructing prediction intervals. The explicit time series modeling approach to forecasting that we have chosen to emphasize is the autoregressive integrated moving average (ARIMA) model approach. Chapter 5 introduces ARIMA models and illustrates how to identify and fit these models for both nonseasonal and seasonal time series. Forecasting and prediction interval construction are also discussed and illustrated. Chapter 6 extends this discussion into transfer function models and intervention modeling and analysis. Chapter 7 surveys several other useful topics from time series analysis and forecasting, including multivariate time series problems, ARCH and GARCH models, and combinations of forecasts. We also give some practical advice for using statistical approaches to forecasting and provide some information about realistic expectations. The last two chapters of the book are somewhat higher in level than the first five.

Each chapter has a set of exercises. Some of these exercises involve analyzing the data sets given in Appendix B. These data sets represent an interesting cross section of real time series data, typical of those encountered in practical forecasting problems. Most of these data sets are used in exercises in two or more chapters, an indication that there are usually several approaches to analyzing, modeling, and forecasting a time series. There are other good sources of data for practicing the techniques given in this book. Some of the ones that we have found very interesting and useful include the U.S. Department of Labor-Bureau of Labor Statistics (http://www.bls.gov/data/home.htm), the U.S. Department of Agriculture-National Agricultural Statistics Service, Quick Stats Agricultural Statistics Data (http://www.nass.usda.gov/Data_and_Statistics/Quick_Stats/index. asp), the U.S. Census Bureau (http://www.census.gov), and the U.S. Department of the Treasury (http://www.treas.gov/offices/domestic-finance/debt-management/interest-rate/). The time series data library created by Rob Hyndman at Monash University (http://www-personal.buseco.monash.edu.au/~hyndman/TSDL/index.htm) and the time series data library at the Mathematics Department of the University of York (http://www.york.ac.uk/depts/maths/data/ts) also contain many excellent data sets. Some of these sources provide links to other data. Data sets and other materials related to this book can be found at ftp://ftp.wiley.com/public/scitechmed/ timeseries.

We have placed a premium in the book on bridging the gap between theory and practice. We have not emphasized proofs or technical details and have tried to give intuitive explanations of the material whenever possible. The result is a book that can be used with a wide variety of audiences, with different interests and technical backgrounds, whose common interests are understanding how to analyze time-oriented data and constructing good short-term statistically based forecasts.

We express our appreciation to the individuals and organizations who have given their permission to use copyrighted material. These materials are noted in the text. Portions of the output contained in this book are printed with permission of Minitab Inc. All material remains the exclusive property and copyright of Minitab Inc. All rights reserved.

DOUGLAS C. MONTGOMERY CHERYL L. JENNINGS MURAT KULAHCI

CHAPTER 1

Introduction to Forecasting

It is difficult to make predictions, especially about the future.

NEILS BOHR, Danish physicist

1.1 THE NATURE AND USES OF FORECASTS

A forecast is a prediction of some future event or events. As suggested by Neils Bohr, making good predictions is not always easy. Famously “bad” forecasts include the following from the book Bad Predictions:

“The population is constant in size and will remain so right up to the end of mankind.” L’Encyclopedic, 1756.

“1930 will be a splendid employment year.” U.S. Department of Labor, New Year’s Forecast in 1929, just before the market crash on October 29.

“Computers are multiplying at a rapid rate. By the turn of the century there will be 220,000 in the U.S.” Wall Street Journal, 1966.

Forecasting is an important problem that spans many fields including business and industry, government, economics, environmental sciences, medicine, social science, politics, and finance. Forecasting problems are often classified as short-term, medium-term, and long-term. Short-term forecasting problems involve predicting events only a few time periods (days, weeks, months) into the future. Medium-term forecasts extend from one to two years into the future, and long-term forecasting problems can extend beyond that by many years. Short- and medium-term forecasts are required for activities that range from operations management to budgeting and selecting new research and development projects. Long-term forecasts impact issues such as strategic planning. Short- and medium-term forecasting is typically based on identifying, modeling, and extrapolating the patterns found in historical data. Because these historical data usually exhibit inertia and do not change dramatically very quickly, statistical methods are very useful for short- and medium-term forecasting. This book is about the use of these statistical methods.

FIGURE 1.1 Time series plot of the market yield on U.S. Treasury Securities at 10-year constant maturity. (Source: U.S. Treasury.)

Most forecasting problems involve the use of time series data. A time series is a time-oriented or chronological sequence of observations on a variable of interest. For example, Figure 1.1 shows the market yield on U.S. Treasury Securities at 10-year constant maturity from April 1953 through December 2006 (data in Appendix B, Table B.1). This graph is called a time series plot. The rate variable is collected at equally spaced time periods, as is typical in most time series and forecasting applications. Many business applications of forecasting utilize daily, weekly, monthly, quarterly, or annual data, but any reporting interval may be used. Furthermore, the data may be instantaneous, such as the viscosity of a chemical product at the point in time where it is measured; it may be cumulative, such as the total sales of a product during the month; or it may be a statistic that in some way reflects the activity of the variable during the time period, such as the daily closing price of a specific stock on the New York Stock Exchange.

The reason that forecasting is so important is that prediction of future events is a critical input into many types of planning and decision making processes, with application to areas such as the following:

1. Operations Management. Business organizations routinely use forecasts of product sales or demand for services in order to schedule production, control inventories, manage the supply chain, determine staffing requirements, and plan capacity. Forecasts may also be used to determine the mix of products or services to be offered and the locations at which products are to be produced.

2. Marketing. Forecasting is important in many marketing decisions. Forecasts of sales response to advertising expenditures, new promotions, or changes in pricing polices enable businesses to evaluate their effectiveness, determine whether goals are being met, and make adjustments.

3. Finance and Risk Management. Investors in financial assets are interested in forecasting the returns from their investments. These assets include but are not limited to stocks, bonds, and commodities; other investment decisions can be made relative to forecasts of interest rates, options, and currency exchange rates. Financial risk management requires forecasts of the volatility of asset returns so that the risks associated with investment portfolios can be evaluated and insured, and so that financial derivatives can be properly priced.

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