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Manual of Forecasting: Methods and Applications Book · January 2000 DOI: 10.13140/RG.2.1.2528.4880

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Preface In preparing the manuscript for the third edition of Forecasting: methods and applications, one of our primary goals has been to make the book as complete and thorough as possible in order that it might best meet its intended objectives. The same set of principles has guided us in preparing this instructors manual. Our intent has been not only to provide solutions to the exercises but to go beyond and suggest several other types of teaching materials and suggestions to help those who teach forecasting. We hope that you will find that this manual delivers on those objectives. The instructors manual is in four parts. To avoid confusion with the chapters in the textbook, we will refer to these as Parts A through D. When the word Chapter is used, it refers to that chapter in the text itself. Part A is aimed at providing different course outlines for a number of different settings in which the text has been used. These range from short executive seminars to subsegments of a required college course to full-length courses at the graduate level on the subject of forecasting. In Part B, we have provided some teaching suggestions as to how we would teach a course based on the book. When teaching, we always use a range of additional teaching materials as complements to the text. In Part C, we discuss the use of case studies and provide suggestions for projects and exam questions. There use will depend on the overall structure, teaching style and design selected for the course. Part D provides solutions to the end-of-chapter exercises. We have provided solutions that can be used in teaching the course rather than just for grading student work. We hope the graphs, tables and descriptions will be useful in preparing overhead transparencies or handouts for students. We have long found it useful to teach forecasting using a computer package for the computational aspects of the subject. We have chosen in this edition not to emphasize a particular package but to comment on the facilities available in a range of packages (see Appendix I of the text). It is important to have a package that the students can iii

iv

Preface

learn relatively quickly and which provides as many of the statistical facilities as possible. This will depend on the students’ background and the type of course being taught. In preparing the solutions, we have mainly used Minitab version 11 and SAS version 6.12. Be aware that other packages may give slightly different numerical results due to different algorithms being used. We would like to express special thanks to a number of instructors who have helped us with this manual and given us their feedback from use of the second edition of Forecasting: methods and applications. At Stanford University, the book has been used by Professor Fred Shepardson and Professor Peter Reiss. Teaching materials were also provided by two of our colleagues at the University of Virginia, Professor Jim Freeland and Professor Bob Landel and by Dr Gary Grunwald who provided some ideas for the exercises while teaching at the University of Melbourne. Spyros Makridakis Fontainebleau, France Steven Wheelwright Boston, Massachusetts Rob Hyndman Melbourne, Australia November 1997

Contents

v

A/Planning a forecasting course Since the late 1940s organizational forecasting has been directly affected by numerous developments in estimation and prediction. Today organizations of all sizes find it essential to make forecasts in order to reduce the uncertainty of their environment and take advantage of opportunities. We have been involved in the forecasting area for more than 25 years, and this book is the culmination of our efforts and experiences at teaching forecasting to both full-time students and practicing managers with forecasting responsibility. This book grew out of our perception that a book was needed that would cover a full range of forecasting methods, be accurate and complete in describing the essential characteristics and steps for applying those methods in practice, and yet not get bogged down in theoretical questions underlying the development of individual techniques. The purpose of the book is to fill a gap in the literature by presenting in terms that are easily understandable (but that accurately and rigorously describe the techniques) a wide range of forecasting methods useful to students and practitioners of management, economics, engineering, and other disciplines requiring effective forecasting. The material included in the book has been particularly effective in a wide range of teaching activities. These have included seminars for middle management, seminars for practicing forecasters, and classes taught to both graduates and undergraduates in business and in more specialized topic areas such as courses for statisticians, economists, and management scientists. Four objectives guided the structuring and development of the major sections of this book. It is useful to understand these objectives before describing the way in which various materials might be combined to develop an effective course or program on forecasting and planning. Part A of this manual reviews these objectives and describes how the materials in the text can be used to teach forecasting effectively. Alternative course outlines that indicate how various parts of the book might be used for different audiences and in programs of different length are also suggested. Parts B and C of this manual provide additional information on the use of supplementary material and the teaching of individual sessions in a forecasting course. The various uses of the book can probably best be seen by considering specific course 1

2

Part A. Planning a forecasting course

outlines built around the text. Several such outlines are discussed and presented in Section A/3.

A/1 Objectives The major objectives we pursued in structuring and developing the four parts in this book included the following: 1. Presenting essential aspects of a wide range of forecasting methods with sufficient detail and clarity that they could be applied easily. 2. Presenting alternative forecasting methods in such a form that a minimum of technical background would be required to understand each technique, yet including enough of the essential concepts and their theoretical basis that students could gain a thorough understanding of each technique if desired. 3. Providing information concerning the operating and performance characteristics associated with each of the major forecasting methodologies so that criteria for selecting the most appropriate methods can be developed and applied. 4. Examining the important factors and issues in the application of forecasting and planning and the effective use of forecasting resources in an ongoing organization. It should be emphasized that while the body of each chapter is geared to the reader with only a basic background in algebra, the aim has still been to present a complete description of each technique and those factors that are relevant in deciding where and how to use it.

A/2 Course materials In addition to having a class study individual chapters, an instructor has four other major sources of material to use as building blocks in developing a forecasting course. Exercises As indicated in the preceding section, many of the chapters in this book deal with specific techniques for forecasting. At the end of these methodology-oriented chapters are exercises that we have found particularly helpful in teaching forecasting. The purpose of these exercises is to give students the opportunity to test their

A/3 Sample course outlines

3

knowledge of the basic mechanics of individual methodologies and their understanding of the strengths and limitations of those approaches. Complete solutions for these exercises are provided in Part D of this manual. All data sets used in the exercises are available from the book’s web page at www.maths.monash.edu.au/˜hyndman/forecasting/ Case studies Case studies on forecasting situations can take a variety of forms including basic exercises with a few paragraphs describing the situation surrounding that exercise, providing organizational descriptions of a forecasting system, and even outlining the management perspective of a decision maker who must use forecasts prepared in accordance with a given methodology. Part C outlines procedures for using such cases and suggests how those cases might be used to supplement the exercises at the end of each chapter. Part C concludes with several additional exercises, project suggestions and examination questions that might be useful in a forecasting course. Computer programs A number of computer programs that implement many of the quantitative techniques described in the text are available and their facilities are summarized in Appendix I of the book. We have found that it is essential to use a computer package when teaching Forecasting to enable students to apply the techniques themselves without unnecessarily detailed calculations. In addition, use of such a package has proved a particularly appropriate means of making students aware of the strengths and weakness of individual techniques on the basis of their applicability and results in a range of situations. Additional readings At the end of each chapter is a set of references that we have found relevant to topics raised in the chapter. These references provide more depth on the most important issues and suggest follow-up material for students with a special interest in a given topic.

A/3 Sample course outlines We have used the materials included in Forecasting: Methods and Applications, third edition, for a variety of purposes. These have varied in terms of (1) number of class sessions in the course, (2) the audience at which the course is directed, and (3) the mix of lecture and case sessions. In each course, we find it useful to use real case studies. These are most successful when the subject matter is of relevance to those attending the course. For longer courses designed for users, it is important to include computer laboratory sessions giving those

4

Part A. Planning a forecasting course

attending real experience in forecasting real data. Often the concepts that are covered in lectures are only understood after a person has carried out the procedure and seen the results on a computer. Here we provide six course outlines that vary along these three dimensions. Chapter references are given for background reading. In longer courses, we would cover all (or almost all) of the material in each chapter. In shorter courses, we given only a brief introduction to the material listed.

Course A: 5-session (one-day) course for potential users of forecasting Course A: for potential users of forecasting Session 1: Introduction

(Chapter 1)

Evaluating the performance of a forecasting system, identifying issues in organizational forecasting, and providing an overview of forecasting methodology. Session 2: Time series approaches to forecasting

(Sections 4/2 – 4/3)

Exponential smoothing methods and ARIMA models. Session 3: Regression approaches to forecasting

(Sections 5/1 – 5/3; 6/1 – 6/5)

Correlation analysis, simple regression, and multiple regression. Session 4: Evaluating alternative forecasting methods

(Section 2/4, Chapter 11)

Introduction of a basic framework. Session 5: Managing the organization’s forecasting function

(Chapter 12)

Auditing the status of forecasting, identifying key problems, designing, and implementing effective action plans.

The first outline is for a one-day course that would serve as a primer for potential users of forecasting in a business organization. Given the audience, the teaching approach is a combination of lectures and case discussions that can draw into the class the experience of individual participants. It is assumed that those attending such a course have some basic mathematical abilities; if they also have some forecasting experience, it is assumed that it is in a fairly limited area or that it involves a fairly narrow set of methodologies. As outlined below, selected sections of half a dozen chapters in this book can be used to illustrate various methodologies and the ways in which forecasting management issues can be tackled. The references to the textbook are given in italics.

A/3 Sample course outlines

5

Course B: 12-session (two-day) course for practicing forecasters In the second course outline, 12 sessions spread over two days are used to introduce those with forecasting responsibility in a company to various methodologies and to an interactive forecasting computer package. This program has been used with companies who have have assigned forecasting responsibility to several people acting as business managers for various product lines or as product/market managers for various segments of the company’s business. The text chapters indicated serve as further reference on specific topics. The basic information on those topics would be covered during lecture class sessions and would be applied during the sessions that use the computer. Cases are used as the source of data and as illustrations of various management issues.

Course C: 11-session (two-day) course for managers who will use forecasting This is also a two-day program (11 sessions) aimed at managers who make use of forecasting rather than at forecasters. These may be managers who must do their own forecasting or managers who interface with a forecasting group, such as at a public utility or in a large consumer products firm. Again, this program assumes that an interactive forecasting package is available, and output from the package is used to illustrate methodologies. However, the participants do not actually use the package during the course. Much of the class time is spent describing the way in which alternative forecasting techniques can tackle representative management problems. The objectives of such a course would be to help managers identify situations in which a quantitative forecasting approach would be appropriate and how to describe and define such situations so that such a technique could be applied.

Course D: 9-session (three-day) course for potential users of forecasting This course gives a nine-session program that could be provided as an elective in a regular college or university curriculum or it might be compressed into a three-day program for managers or practicing forecasters. It would go into more depth on the topics covered in the outlines for one- or two-day courses and take a management orientation by concentrating on case applications and their discussion in the classroom. As indicated, this course outline includes specific time for group work after individual preparation of the cases.

6

Part A. Planning a forecasting course

Course B: for practicing forecasters Session 1: Introduction and defining the forecasting problem

(Chapters 1 and 2)

Major issues in forecasting, the concept of a forecasting strategy, a framework for classifying forecasting methodologies, and measuring forecasting error. Session 2: Exponential smoothing methods of forecasting

(Chapter 4)

Single exponential smoothing, Holt’s method, and Holt-Winters’ seasonal method. Session 3: Using an interactive forecasting package A laboratory session introducing a forecasting package. Inputting data, plotting data, using exponential smoothing methods for tackling a given case study. Session 4: Time series analysis

(Chapter 7)

Autoregressive and moving average models for time series analysis, autocorrelation analysis, and model selection. Session 5: Data preparation for forecasting Determining what to forecast, deciding how to forecast, and securing the required data. Use examples from past experience. Session 6: Optional laboratory session Application of smoothing methods and time series analysis to given case studies. Session 7: Simple regression

(Chapter 5)

Correlation analysis, simple linear regression, and statistical tests of significance. Session 8: Multiple regression

(Chapter 6)

Basics of time series multiple regression, causal factors in multiple regression, statistical characteristics of this method. Session 9: Laboratory session Applying simple and multiple regression to given case studies. Session 10: Selection of a forecasting methodology

(Section 2/4 and Chapter 11)

Techniques for analyzing the characteristics of a given data set, criteria for selecting a forecasting methodology, and comparing the results obtained from alternative techniques Session 11: Systematic improvement of forecasting

(Chapters 12)

Designing a forecasting strategy, measuring performance, and considering organizational aspects of the forecasting function

7

A/3 Sample course outlines

Course C: for managers who will use forecasting Session 1: Overview of forecasting for management

(Chapter 1)

A management perspective on forecasting, introduction to forecasting methodologies, accuracy as a performance criterion, additional criteria for selecting a forecasting method. Session 2: Seasonal indices and decomposition

(Chapter 3/1, 3/2 and 3/4)

Classical decomposition, computation of seasonal indices, deseasonalizing a data series. Session 3: Exponential smoothing

(Chapter 4)

Basic smoothing models, performance on trend and seasonal patterns. Session 4: Interactive forecasting package Philosophy and characteristics, overview of program structure and components, data creation and handling, inputs and outputs to the program, sample runs, limitations, and applications. Session 5: Autocorrelation analysis

(Section 2/3/3 and Section 7/1)

Time dependence, correlation error analysis. smoothing method. Session 6: Time series analysis

Apply to errors from exponential

(Sections 7/2 – 7/8)

Concepts and theory, performance, and practice. Session 7: Simple regression

(Chapter 5)

Linear time trend analysis, least squares estimation, performance, and practice. Session 8: Multiple regression

(Chapter 6)

Causal relationships, illustrative examples, performance, and practice. Session 9: Management use of forecasting

(Chapters 10 and 12)

Computer programs, business cycles, and judgmental inputs. Session 10: Implementing quantitative forecasting techniques

(Chapters 12)

Selecting the right data, specifying the forecasting project, determining support requirements, using available methodologies.

8

Part A. Planning a forecasting course

Course D: for potential users of forecasting Session 1: Introduction and overview Session 2: Role of forecasting in decision making, planning, and control (Chapter 1) Session 3: Time series decomposition

(Chapter 3)

Classical decomposition, computation of seasonal indices, deseasonalizing a data series. Value of decomposition for forecasting. Session 4: Exponential smoothing

(Chapter 4)

Group work on exercises for smoothing time series. Discussion of time series smoothing and evaluation of accuracy Session 5: Time series analysis

(Chapter 7)

Introduction to autoregressive and moving average models for time series analysis, autocorrelation analysis, and model selection. Some exercises on model selection and autocorrelation analysis of errors. Interpretation of forecasts and prediction intervals. Session 6: Simple regression

(Chapter 5)

Correlation, least squares estimation, simple linear regression, statistical tests of significance, forecasts. Session 7: Multiple regression

(Chapter 6)

Seasonal dummy variables, other explanatory variables, variable selection, estimation, forecasts, collinearity. Session 8: Regression analysis in practice

(Chapter 6)

Group discussion of a case study. Identification of appropriate variables, variable selection, interpretation of computer output, use of model for forecasting, forecasting explanatory variables. Session 9: The forecasting function in the firm

(Chapter 10 and 12)

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A/3 Sample course outlines

Course E: segment of production course on short-range forecasting Session 1: Introduction to short-range forecasting Session 2: Analysis of time series data

(Chapters 1, 2, and 12)

(Chapters 3 and 7)

Plots, seasonality, autocorrelation. Session 3: Exponential smoothing

(Chapter 4)

Simple models, calculation of forecasts, autocorrelation of errors. Session 4: ARIMA models

(Chapter 7)

Overview of models, model selection, interpretation of forecasts, and prediction intervals.

Course E: 4-session segment of production course on short-range forecasting This short course on forecasting is actually a four session segment of an MBA elective course on production and operations management that focuses on short-range forecasting as an input to production planning and control. Students are expected to use an interactive forecasting package in applying selected methods to a variety of production planning situations.

Course F: 18-session course for MBA elective on forecasting for management This course is designed as a full-quarter elective course for MBA students who may accept job assignments with forecasting responsibility immediately upon graduation. Thus, students electing this course would tend to have a fairly good background in mathematics and would be particularly interested in the knowledge needed to apply individual techniques. The majority of the class sessions would involve lectures dealing with various topics related to quantitative forecasting techniques and their application. However, at the end of each of several major sections of the course, one or more cases would be used to show the practical application of the concepts being discussed and some of the difficulties encountered in their implementation. A forecasting competition would be conducted during the course so students could forecast a variable, obtain results, and then prepare further forecasts. We have also used class visitors at the end of a long course of this type. These would be practicing forecasters from business or industry who would provide a fresh perspective on the practice of forecasting and would also discuss non-statistical problems they have encountered in forecasting.

10

Part A. Planning a forecasting course

Course F: for MBA elective on forecasting for management Session 1: Introduction to forecasting - Why forecast?

(Chapter 1)

Summarizing data and relationships: a review of some useful concepts Explanatory methods of forecasting. Forecasting and causality Session 2: Summarizing data and relationships

(Chapters 2 and 3)

A review of some simple quantitative concepts, time series, moving average smoothing. Session 3: Decomposition

(Chapter 3)

Trend analysis and seasonality, detrending, classical decomposition, overview of other decomposition methods. Session 4: Exponential smoothing methods

(Chapter 4)

Simple exponential smoothing, Holt’s method, Holt-Winters’ method. Session 5: Regression

(Chapter 5)

Regression analysis, correlation, least squares estimation, tests of significance. Session 6: Multiple regression

(Chapter 6)

Multiple regression models, significance tests, variable selection, Session 7: Multiple regression

(Chapter 6)

More on estimating regressions, diagnostics, transformations, prediction with regression models. Session 8: Multiple regression

(Chapter 6)

Lagged variables, spurious regressions. Session 9: Multiple regression

(Chapter 6)

Econometrics and economic models. Session 10: Time series analysis

(Chapter 7)

tests for autocorrelation, analysis of errors from forecasting methods, autoregressive models. Session 11: Time series analysis

(Chapter 7)

ARMA and ARIMA models Session 12: Time series analysis

(Chapter 7)

Box-Jenkins procedures, estimation, tests

11

A/3 Sample course outlines

Session 13: Time series analysis

(Chapter 7)

Examples and applications. Session 14: Advanced forecasting methods

(Chapter 8)

Overview of some more advanced methods. Session 15: Long-term forecasting

(Chapter 9)

Cycle analysis and indexes, cycles and the forecast problem, index construction, leading indicators. Session 16: Judgmental forecasting

(Chapter 10)

Subjective methods, estimation, tests, model selection criteria, estimation, and diagnostics, choosing a method to fit the problem. Session 17: Comparing forecasts

(Chapter 11)

Measures of accuracy. How do we compare forecasts? Canonical procedures, rules of thumb. Session 18: Conclusion Review and award presentation for the competition

A major project is also a worthwhile adjunct to such a course. One option would be to give students a case and ask them to prepare the complete analysis and forecasts in both written and oral form for the management of that company. Such a project allows students not only to test their knowledge of various techniques but also to handle the problems of deciding what to forecast and how, and determining what data would be most appropriate. A final variation of this forecasting project that we have used with some success is to identify a forecasting situation in an existing company and to have the manager in that situation come to class so that the students, acting as forecasters, can define the task and determine what information is needed. Students can work on the project in teams and make their final presentations to the manager involved. This type of project is perhaps the best possible test of students’ knowledge of the subject area and their ability to handle the practical aspects of forecasting in an organization. The final presentation could serve as the final exam for the students. Many other course outlines based on the material in this book are possible, of course. Here we wish to introduce the range of such outlines and suggest how they might be adapted to use the complementary materials described in Parts B and C of this manual.

B/Teaching suggestions Chapter 1: The forecasting perspective The fate of a course is often decided early. So motivating the subject is critical. This chapter needs help to make it a live opening for the course, and here are some suggestions. 1. Spend time making the three points on page 5 meaningful. (a) Scheduling existing resources (b) Acquiring additional resources (c) Determining what future resources are needed If possible, invite a couple of business practitioners into the first class to make the reality of these matters clear. 2. Consider Figure 1-1 and study it from the point of view of the group responsible for making the sales forecast for a company. Sales forecasting is one of the most routine and most fundamental business tasks. It directly impinges on budget policy at all levels of the firm. Make the point that there are two directions to look:

where does it come from? 

Sales Forecast

where does it go? -

3. Following the previous point, try to establish (i) the context within which a “forecast” is made, and (ii) the environment in which a “forecast” is received. There is more to forecasting than learning a mathematical method, getting data, running a computer program, and reporting on extrapolated future values. Sales of a product group take place within the context of sales of competing products (both internal to the company and externally). Sales depend on market demand. Market demand 12

13

Chapter 1: The forecasting perspective

depends on market conditions. And so on. It is a dynamic interlocking system out there. When the sales forecast has been made it is passed on in the hierarchy, to be examined, modified by expert judgment, repackaged, and passed on again. Forecasting is not a passive activity. The physical environment within which forecasts are created is dynamic, and the organizational consumer of forecasts shares these dynamics and adds its own human relations. 4. Table 1-1 offers opportunities for class participation. What forecasting scenarios are hot topics right now—in the local environment (what will acid rain do to our forests)? in the nation (will the national deficit ever decrease)? in any firm that you know about (will our ski businesses survive in the uncertain weather)? And so on. 5. Sections 1/2 and 1/3 begin to introduce the jargon of the field of forecasting, and it is important to establish some of the most commonly used terms. For example, a lot will be said about time series methods and explanatory methods. Distinctions are also made between quantitative methods and qualitative methods. And you can no doubt think up other dichotomous categories. The purpose of these categorizations is to aid in communication. For example, with the two mentioned above, we can identify four cells: time series quantitative time series non-quantitative

causal non-quantitative causal quantitative

As more and more dichotomous categories are invented we can generate many, many possible (methodology) cells—but some of them will not be possible. It might be enough to try to think of examples for the four cells given above. 6. Be careful with the distinction between explanatory models and time series models. There is some semantic confusion with this choice of words, because “explanatory models” can deal with “time series” data. So the context has to help out. When we say time series models or time series methods we usually mean to talk about one set of (time-dependent) data and we will try to develop a model (an equation) which can be though of as the “generating process” of these data. We specifically will not look at its relationship with other variables. When we causal models we are specifically looking for other variables (which may be time series) which offer explanations (or linkages) to the main variable (which may also be a time series). 7. It is important to see forecasting as a process involving several steps as outlined in Section 1/3. Many people focus on just Step 4: choosing and fitting models. To emphasize the basic steps in a forecasting task, it is helpful to take a particular example and lead a class discussion on what is involved in each of the five steps.

14

Part B. Teaching suggestions

Chapter 2: Basic forecasting tools Chapter 2 introduces some of the basic quantitative tools that will be used extensively later. Therefore it is important the material covered here is well understood. 1. The two examples in section 2/1 will help to fix the distinction between explanatory and time series methods. 2. The patterns discussed in Section 2/2/1 are important for later reference. In the many time series to be dealt with in the book, it will become second nature to ask (i) is there a trend? (ii) is there an annual pattern (seasonality)? (iii) is there a longer (than one year) cycle? It is important to emphasize that when we talk about “business cycles” we mean something different from “seasonality” which refers to the annual pattern. Cycles are almost always longer than seasonal patterns. 3. The plots in Section 2/2, particularly the time plot and the scatterplot, will be used extensively in subsequent chapters. These also help students understand the difference between explanatory and time series methods. 4. In dealing with the familiar descriptive statistics in Section 2/3, the following aspects could be stressed. (a) Cross-sectional versus time series data: Some students find it difficult to switch their thinking from elementary descriptive statistics of cross-sectional data (e.g., the weights of 100 mice) to time series data which may have trend, seasonality and longer term business cycles (e.g., housing starts). The mean of “housing starts” does not convey much information because of the trend, seasonality and cycle. Similarly, the variance of “housing starts” is hard to interpret. So in section 2/3, as each descriptive statistic is defined, it is useful to ask: is there any difference in meaning when we talk about cross-sectional data or time series data? (b) Absolute values: The sum of the deviations from the mean for any data set always equals zero. The sum of the “absolute” deviations from the mean does better. But it is an awkward statistic to deal with. (For those with a little calculus background, ask them to differentiate the MAD equation (2.3) with respect to the mean, and the point will be established.) It is easier to deal with the mean of the squared values. (Again, ask those with calculus background to differentiate the MS equation (2.4)).

Chapter 2: Basic forecasting tools

15

(c) Mean square versus variance: There is a tendency for current statistics texts to define variance using (n − 1) in the denominator without explanation. Then there is the inevitable question, “Why (n−1) and not n?” When we write down formulas for computing summary numbers (statistics) the formulas don’t know anything about assumptions, so from their point of view it makes no difference. Only when a statistical model has been defined (using random variables with their accompanying probability distributions) does it make any sense to talk about “unbiased” or “biased” estimates, and then a meaning can be given to the issue. To avoid some of the problems in this connection, we have chosen to talk about the mean square as the simple average of squares, and the variance will use (n − 1) in the denominator. This is a matter of convenience. Since this issue is tied up with d.f. (degrees of freedom), it is useful to make a point that d.f. are defined as (number of independent data points) minus (number of parameters estimated). For example, if there are n = 20 students in the class and each has a score on a test, then how many numbers must you sing out to a stranger before they know all the scores? Twenty. So there are 20 d.f. for the raw data. Now compute the mean. Find the deviation from the mean for all 20 students. How many of these deviations must be sung out before a stranger can know them all? Nineteen. After hearing 19 of them, the last one must be such that they sum to zero. One d.f. was lost in computing the mean. (d) Bivariate statistics: Covariance and correlation are important concepts to understand. Note that variance is a special case of covariance (if you convert Y to X). Understanding correlation is important in regression (Chapters 5 and 6) and in autocorrelation (Chapter 7). (e) Autocorrelation: To give students a feel for the meaning of autocorrelation, show some plots of a time series plotted against itself, lagged one period, then lagged two periods, etc. The physical act of making one or two of these scatterplots is worth the time it takes. Another worthwhile activity is to do Exercise 2.4 in class with students guessing which plots align with which ACFs. Discussion is usually easy to stimulate with students justifying their answers. (f) Accuracy measurements: Dwell on the interpretation of these summary numbers, so that the distinctions between absolute statistical measures and relative measures are clearly understood. For example, in dealing with MSE versus MAPE, you could consider the X and F data in Table 2-9. (i) What happens to the values of MSE and MAPE if the data were all multiplied by 2? (ii) Similarly consider what happens to the values of MSE and MAPE if 100 was subtracted from all numbers.

16

Part B. Teaching suggestions In case (i) the MSE is going to be 4 times larger but the MAPE won’t change. In case (ii) the MSE won’t change but the MAPE is considerably changed. What happens if 119 is subtracted from all data? (g) Theil’s U statistic: This is worth understanding. It becomes easy to use in practice because when U < 1 the forecasting method is better than the na¨ıve one (using last period’s observation as the forecast for the next period). When U > 1 it is worse then the na¨ıve method. Unfortunately, not many computer forecasting programs calculate Theil’s U . (h) ACF of forecast error: It is good practice to look at the ACF of the errors obtained from any forecasting method. (See Section 7/1) It is worth emphasizing this throughout Chapters 4–8 whenever a forecast is calculated.

5. Throughout the book, we emphasize the calculation of forecast intervals wherever possible because it provides a sense of how dependable a forecast is. Section 2/5 allows some simple forecast intervals to be computed using only the MSE. This helps students understand the MSE better and introduces the basic concept of a forecast interval. 6. The reason for giving a special section (2/6) to least squares is simply because it is almost all-pervasive in the model building world. It has been found to be extremely valuable in statistics (e.g., in the fitting of regression models to data). 7. Many forecasting tasks can be simplified by transforming or adjusting the data in some way. It is helpful to ask students how they would transform or adjust a few given data sets. This can lead to some interesting discussion and emphasizes the need to understand the data before doing any forecasting.

Chapter 3: Time series decomposition Decomposition methods are among the oldest of all the forecasting procedures. They are easy to understand, at least in principle, because most people dealing with time series data assume the presence of trend, the influence of a business cycle, seasonality (if we’re dealing with monthly or quarterly data, for instance), and the ever-present noise (the error term, the disturbance term, or the random shock). The following points should be noted in teaching this chapter. 1. The components implied by decomposition are invariably described as trend, cycle, seasonality and noise (or other words to describe this uncontrollable part). when we speak of trend it seems easy to understand, but in fact it is not all that clear. it

Chapter 4: Exponential smoothing methods

17

is often inextricably mixed up with the so-called cycle (which itself is not a mathematical cycle—such as a sine wave—but rather a irregularly shaped up and down movement associated with “general business conditions”) and the only way trend can be separated from cycle is by arbitrary definition of trend. Because of this complexity many decomposition methods (e.g., the Census II method) identify trend/cycle as one component. Seasonality is less ambiguous and it refers to systematic patterns that occur within the calendar year. Suggestion: Have students come up with a written definition of these four components. 2. The ratio-to-moving averages method is easy to compute (see Table 3-6) and it is good to see plots of the original data, the moving average and the ratio-to-moving average. When the ratio-to-moving average values are portrayed as in Table 3-6 they can be visualized in a seasonal plot (Section 2/2/2) which allows for the stability of the seasonal pattern to be assessed. Students should be clear on the meaning of each of the columns in Table 3-6. 3. The Wall Street Journal, Business Week, Fortune and other business magazines all make repeated reference to seasonally adjusted time series and it is important that all students know exactly what this means. The ratio-moving averages are in fact seasonal indices plus the random noise component. By averaging these seasonal indices for each month (or quarter as the case may be), the random component is reduced, and the resulting seasonal index is a measure of the impact of the season. By dividing the original data by this seasonal index we are left with seasonally adjusted data—which has in it, trend and cycle and noise. That is, the influence of the season has been removed. The Census II method talks about preliminary and final “seasonal adjustment factors” (same as “seasonal indices”), and preliminary and final seasonally adjusted series. 4. If software is available for one or more of these decomposition methods, it is interesting for students to compare the results. In particular, investigate what the methods produce when the series contains some unusual behavior such as a level shift or some outliers. The classical decomposition method is not designed to cope with such behaviors, but the Census II and STL methods both contain some robustness facilities.

Chapter 4: Exponential smoothing methods Exponential smoothing methods can be useful as an introduction to some of the ideas of time series forecasting, particularly the concept of forecasts being weighted averages of

18

Part B. Teaching suggestions

time-lagged observations. They are also useful forecasting methods in their own right. 1. These are time series methods as opposed to explanatory methods. 2. In dealing with time-dependent data the concept of a moving average is valuable because it is dynamic. It moves with time. 3. Whereas moving averages involve equal weights over a set of observations, the simple exponential smoothing (SES) method is fundamentally different in that it implies unequal (exponentially decaying) weights. Aside: You can engage the students in a discussion on how to weight past data in making a forecast. Should the latest data count more than earlier data? When is this true? (E.g., when older data was based on a different manufacturing process.) When might it not be true? (E.g., when current data occurs during a strike.) 4. In order to appreciate the fact that all methodologies have built-in limitations, it is useful to do what engineers typically do, namely, test the methods on some standard types of input series. This can be demonstrated by constructing a simple series of 20 observations containing a level shift part way through the series. Then apply both a moving average and SES to the data to see how each method accommodates the step. This can be a most enlightening experience for students and a valuable base for latter work. Similar test series might contain a single outlier, or a trend. 5. Following the previous point, you can ask if SES and moving averages keep pace with trend. And then discuss seasonality as a complicating factor. Can moving averages and SES take care of seasonal indices? 6. Discuss Pegel’s two-way classification with the students to emphasize the difference between linear and multiplicative trend and seasonality. The flexibility of Pegels’ classification has yet to be fully appreciated, and it is worth discussing some of the cells other than those corresponding to SES (A-1), Holt’s method (B-1) and HoltWinters’ method (B-2 and B-3). For example, consider cell C-3 which will often outperform Holt-Winters’ method. 7. A very important point to establish for exponential smoothing methods is the fact that an initialization process (for getting a method going) has to be defined. Since the initialization procedure has an influence on all subsequent smoothed values it has to be handled with care. Therefore, in deciding how well an SES model fits, for example, it is wise to define a “test period” which excludes that early part of the series which is still “settling down during the initializing phase”. Contrast this with regression models where we can define “errors of fit” for all data points at once. For this reason, we have given explicit statements about each strategy and some general

Chapter 5: Simple regression

19

comments on alternative initialization strategies in Section 4/5/1. Note that these are not the only ways of going about it. Students may be able to come up with there own suggestions. 8. We have designed this chapter to be complete in the sense that the equations are all given and fully worked examples are provided. Table 4-11 is something that you might work toward. It gives in one place a comparison of how all the methods do on one set of seasonal data. If your students have come to the course with their own data sets they should be encouraged to work toward a table similar to this. 9. Also note that it is appropriate to discuss the extensive forecasting experiments described in Chapter 11. 10. A good exercise in discussing ARRSES would be to ask students to generate a time series using a slowly changing α value, and then do an ARRSES analysis to see if the method comes close to what was simulated.

Chapter 5: Simple regression Many students will have already have had a first course in statistics and will have done some simple regression—mostly in the context of cross-sectional data. Chapters 5 and 6 of this text should accomplish the following: (a) Consolidate understanding of simple and multiple regression for crosssectional data. (b) Discuss the importance and limitations of the correlation coefficient. (c) Discuss the use of regression in a forecasting (time series) context. (d) Deal with the practical application of simple regression, multiple regression and econometric models. The following suggestions will assist in teaching this material. 1. Discuss the data setup for simple regression, multiple regression and econometric models. Mention that in econometric modelling the dependent and independent variables become mixed up—in the sense that Y variables appear on the right hand side of econometric equations as well as on the left hand side. 2. Review the details of simple regression of Y on X, make sure everyone knows the fundamental statistics (mean, variance, covariance, correlation, regression coefficients),

20

Part B. Teaching suggestions and then deal with the definition and role of the overall F test, the t-tests for individual coefficients and the sampling fluctuation of the coefficients.

3. The correlation coefficient is a very widely used statistic and therefore should be understood well. Mention that it is a measure of linear association, that it is therefore unaffected by any linear transformation, that its sampling fluctuation is large for small sample sizes (so beware of those regressions based on 10 observations!), and that it can be severely affected by skewness (or outliers). 4. Emphasize the difference between regression for cross-sectional data and for time series data. Cross-sectional regression can be useful in a forecasting context (e.g., the automobile data in chapter 2) but time-indexed data and time series regression pose special problems. The error terms in cross-sectional regression are usually assumed independent, but in time series regression this independence is often suspect, and in some cases (e.g., in dynamic regression models) the errors are carefully defined not to be independent. The autocorrelation coefficient is simple to define and to compute, but its sampling distribution is more difficult to handle than the sampling distribution of the correlation coefficient. 5. Many textbooks talk about regression as a forecasting tool but very few actually do forecasting with regression. For example, if we regress Y t on Xt−1 , Yt = 3 + 5Xt−1 + (error), and we want to forecast Yt+1 , then the equation allows us to do that: Yˆt+1 = 3 + 5Xt . We already know Xt , and so can obtain Yt+1 . However, if we regress Yt on Xt : Yt = 3 + 5Xt + (error), then in order to forecast Yt+1 we will need to know Xt+1 , and we do not know this. So we will have to forecast Xt+1 before we can forecast Yt+1 . 6. Discuss the meaning of equations (5.3) and (5.4) for slope and intercept. The slope is actually (covariance of X and Y ) divided by (variance of X) and the intercept is the mean of Y minus the slope times the mean of X. 7. There are no assumptions involved in calculating a correlation coefficient. Equation (5.10) for r is merely a formula. When it comes to regressing Y on X and some statistical regression model is defined, then the correlation between X and Y , when squared, has another useful interpretation. It is “the proportion of variance explained by the linear relationship between X and Y ”. What this means is that, knowing the X values we will be able to recover a certain proportion of the variance of Y , and this proportion is r 2 .

Chapter 6: Multiple regression

21

8. The r value is a measure of linear association so point out the message in Figure 5-7 when a strong nonlinear association cannot be picked up by r. Note also that for small samples the correlation coefficient is “notoriously unstable” (a phrase Kendall and Stuart use in the Advanced Theory of Statistics, Vol. 1). Finally, emphasise how skewness can have a profound effect on r. The King Kong example (see Figure 5-8) is a useful illustration of this effect and the answers to exercise 5.6 present further evidence. 9. It is a good idea to contrast equations (5.13) and (5.14) and ask the question: “Where are the random variables in each equation?” In (5.13) there is only one random variable—ε. In (5.14) there are three random variables—a, b and e. The values of a and b are estimates for the unknown parameters, α and β in (5.13). This is why we can define a standard error for the slope and a standard error for the intercept— standard errors which are needed to define t tests for the slope and intercept. 10. Emphasise that the F statistic involves a numerator degree of freedom and a denominator degree of freedom, and make sure that students know how to read the F tables (Appendix III, Table C). A pragmatic point is that the F test should be done first when appraising a regression analysis, and afterward the individual t tests can be examined. In the case of simple regression there is no difference, because the t test is a special case of the F test, but in multiple regression (chapter 6) this is important. 11. In simple regression there is an intimate connection between the slope and the intercept. Since the least squares regression line always goes through the mean of X and the mean of Y , it stands to reason that if the slope is changed the intercept is changed and vice versa. If the mean of X and Y are both positive then an increase in the slope will cause a decrease in the intercept, and vice versa. Equations (5.17) and (5.18) should be studied. Note in (5.17) that the second term under the square root sign will be small if the denominator (representing the “spread of the Xs”) is large relative to the numerator (which is the mean of X). In (5.18) the standard error of the slope depends on how spread out the Xs are. If they are well spread out, the standard error is small.

Chapter 6: Multiple regression 1. As for many other chapters, it will be helpful here to have a readily available regression package for students to work on, so that they can check various things in the chapter and can run their own data through various regression models.

22

Part B. Teaching suggestions

2. If at all possible, students should run their own data through the various analyses for maximum understanding. We sometimes adapt the exercises at the end of the chapter to use the data sets of interests to our students. 3. In multiple regression for cross-sectional data it is important to point out that the significance of individual coefficients is contingent upon the other regressors present in the regression. We emphasise this and say: “the coefficient for X 3 is significant in the presence of the other regressors”. 4. In real world regression problems a considerable amount of time is spent selecting independent variables and coming up with a reasonable model specification. To illustrate this we have used a mutual savings bank data set and have done a detailed analysis of most of the stages that led to a model which was actually used by a large metropolitan bank. The example is sometimes a little complicated, but we feel it is worth the effort to get into it in detail. 5. In the notes on chapter 5 we pointed out that regression is often spoken of as a forecasting methodology, but seldom actually used explicitly in a forecasting context. In this chapter we carry the bank study through to its conclusion by forecasting with a final regression model. We explore the difficulties of having to forecast the independent variables before we can forecast the dependent variable. 6. As in chapter 5, it is worthwhile to understand the cross-sectional regression model thoroughly, and then consider where the time series regression applications violate certain assumptions. Since this text is not addressed to formal statisticians, it is enough to discuss the implications of correlated errors, improper specification (e.g., linear when it might be curvilinear, or two regressors when it should be four regressors), multicollinearity, etc., and refer interested students to texts such as Draper and Smith (1981) for more details. 7. Table 6-1 and Table 6-2: Take time to get to know these data sets because they will be used a lot during the chapter. Have students graph the data sets and keep them handy for class discussion. 8. The Durbin-Watson statistic is described in equation (6.9) and Table 6-7. Students don’t have too much difficulty learning how this statistic is computed. However, learning to use the D-W tables (Appendix III, Table F) is not so straightforward. Please spend time going through a couple of examples in using the tables. 9. Selecting variables for inclusion in regression is a meaty subject and we give only an introduction to the major ideas. In the context of the bank example we go through some of the procedures without explaining all the details (for example, we talk about

Chapter 7: The Box-Jenkins methodology for ARIMA models

23

using principal components to get a short list of variables). However, any serious multiple regression analysis will need to consider variable selection carefully. 10. Section 6/4 (Multicollinearity) gives some information that is not often given about multicollinearity. We hear too often that “multicollinearity is present” when the highest correlation among any pair of regressors is only .7, say. And we hear too often that “multicollinearity is not a problem” when there are no large correlations (i.e., not larger than say 0.5). Both of these statements are incorrect. Table 6-12 shows quite clearly that even when the correlations among regressors never get bigger than 0.333 we can have perfect multicollinearity. 11. Standard error formula (6.13) is a multivariate equivalents of (5.19). It is a little harder to interpret because it is written in matrix notation, but it should be part of a regression package so that confidence intervals can be determined. 12. Table 6-14 should be studied very carefully to ensure students understand how these forecasts are obtained. It takes a little while to get the time intervals straight, but it’s a real issue. 13. In discussing econometric models (Section 6/6) we have only given an introduction to how econometric models are related to the multiple regression models which are the subject of this chapter. Our aims are to give students an appreciation for econometric models, their breadth and depth, and the need for specialized skills to develop and use them effectively. The topic of econometric modelling is itself an extensive field and we have not chosen to cover it in this book. An instructor may choose to include other materials on econometric methods (such as Johnston, 1984; Judge et al., 1988; or Pindyck and Rubenfeld, 1991) to complement the materials in this chapter. A useful introductory perspective is provided by Aykac and Borges “Econometric methods for managerial applications” in the Handbook of Forecasting, Makridakis and Wheelwright (editors), (New York: Wiley and Sons, 1982).

Chapter 7: The Box-Jenkins methodology for ARIMA models 1. We have chosen to present this material in a different order from that found in most other textbooks. Students always find this material a little difficult at first, and we have found the order given in the textbook the most successful approach in leading students through ARIMA modeling. 2. Section 7/1 allows students to firmly grasp the idea of white noise and the use of the ACF and PACF before considering ARIMA models. The white noise tests can

24

Part B. Teaching suggestions be applied to the residuals from regression models or exponential smoothing models. Introducing residual analysis in the context of these earlier forecasting methods emphasises that these ideas are not only applicable to ARIMA models, but to any forecasting methodology. It also allows students to become familiar with some of the tools used in ARIMA modelling before having to learn about ARIMA models themselves.

3. Next we introduce the ideas of stationarity and differencing in Section 7/2. We find it better to introduce these ideas before ARMA models (rather than after as is often done) because it allows ARMA models to be applied to a much wider range of time series from the start. A common approach is to consider only stationary series at first and non-stationary series later, but this gives students the initial impression that ARMA models are not very widely applicable. We find that the approach given in the book leads students to be more positive about these very useful models. 4. You can anticipate that we will be wanting to use the backward shift operator notation later (particularly in chapter 8) and should make a decision whether to introduce it at this time or not. Students do not seem to find this difficult to digest. 5. Students usually find autoregressive models relatively easy to understand as an extension of multiple regression. This is the way we normally introduce them. Next consider the connection between exponential smoothing and AR processes. An exponential smoothing process also involves weighted past values and so is a special case of an AR process. 6. Moving average models are usually more difficult to understand at first. Some instructors try to connect them with previous forecasting methods such as exponential smoothing. We have not found that students find this particularly helpful. Instead we just say they are like a multiple regression, but with past errors as the “explanatory variables”. Get the point across that linear functions of past values of the error series are called MA processes, and linear functions of past values of the observations are called AR processes. 7. Note the potential confusion between moving average models, moving average smoothing and moving average forecasting. Students find this unfortunate duplication of terminology difficult and it needs to be explained very carefully. 8. If you have the facilities it is recommended that you have students generate time series using some of the simpler models so that they really know what is implied. Students will learn more about the models if they have to generate data using a spreadsheet package, than if they use a package with built-in data generation facilities. Generating data with known properties and then studying the shape of the theoretical ACF and PACF gives a valuable insight into the BJ models.

Chapter 7: The Box-Jenkins methodology for ARIMA models

25

Make a point about learning in one direction and analyzing real data in the opposite direction. In schematic form, this is: (a) generate data with known properties (b) study the theoretical ACF and PACF (c) store the simulated data and then analyze it. See if the empirical ACF and PACF match the properties that we started with. In real world data, it is the other way around: (a) analyze the observed data (b) study the empirical ACF and PACF (c) try to identify a theoretical underlying model that could have given rise to the observed data 9. Note that the constant term c in these models is not the same as the mean of the time series if there is an AR component. For example, if the mean of Y t is µ, then Yt − µ = φ(Yt−1 − µ) + et so that c = µ(1 − φ). In general, the constant term will be c = µ(1 − φ1 − φ2 − · · · − φp ). For an MA model, the constant is equal to the mean of the series. 10. It will become increasingly important for students to be able to write out the equation for any ARIMA model, so practice at this time is important. 11. The ACF and PACF are used repeatedly later Sections of this chapter, so they should be learned thoroughly. Give students some sample ACFs and PACFs and have them guess the type of series they come from. 12. Before proceeding to Section 7/5, it would be wise to have students very comfortable with the ideas of stationarity, ACFs, PACFs, ARIMA models and how they are all related. They should be able to write the equations for simple ARIMA models. 13. In trying to generate time series that are from an ARIMA(p, d, q) process, students will need to be aware of the restrictions on the values that the AR and MA coefficients can take on.

26

Part B. Teaching suggestions

14. Students should not be fooled into believing that it will be easy identifying ARIMA models for real data series. As soon as the model becomes a mixed model—even the very simplest ARIMA(1, 0, 1)—the shape of the ACF and PACF can become confusing. It is good to lean the properties of the elementary models, and it is good to remember that they will seldom make themselves known unequivocally in real data series. 15. Estimation (Section 7/4) is often taught in two parts: preliminary estimation and final estimation (using some iterative process such as Marquardt’s algorithm). Computationally, this is the way it must be done, but from the point of view of a forecaster using a computer package, the computational details are not relevant. Therefore, we have focussed on the results which are obtained from a computer package as these are of most relevance to practising forecasters. 16. The use of the AIC in Section 7/6 is not common in introductory forecasting books, but we have found it extremely useful in practice and many computer packages are not giving it as part of the standard output. 17. The conversion of an ARIMA equation into a form suitable for forecasting (Section 7/8/1) takes a little bit of algebraic multiplication and rearrangement of terms. However, since all forecasting packages will provide forecasts from an ARIMA model automatically, students will probably never need to do the calculation themselves. The purpose of including the details in this section is to show how the equations give forecasts, something which may not be immediately obvious to students, particularly when there is an MA component. 18. The material in Section 7/8/3 is very poorly understood, even by some experienced forecasters. The effect of differencing on the forecasts is worth understanding. Too often differencing is carried out without thought for its implications later on. 19. Students learn most from working through an analysis of a time series from start to finish. Exercises 7.8 and 7.9 are useful for this purpose. See also Section C/2/7 which can be used as a student project in longer forecasting courses as it enables each student to choose a different set of data to analyze.

Chapter 8: Advanced forecasting models 1. There is a lot of material in this chapter, and the instructor may wish to select only a few topics to cover. In a shorter course, we suggest omitting Section 8/5 (Multivariate autoregressive models) and Section 8/6 (State space models). Please note that Sections 8/1, 8/2 and 8/3 are sequential. Therefore it is important to

Chapter 8: Advanced forecasting models

27

cover Section 8/1 (Regression with ARIMA errors) well before going on to Sections 8/2 (Dynamic regression models) and 8/3 (Intervention analysis). 2. The approach we have adopted for Sections 8/1 and 8/2 is very different from that found in most books. We have followed the Pankratz approach (and terminology) to modelling rather than the more traditional Box-Jenkins’ approach. We have taught using both approaches many times and have found students find the Pankratz approach very much easier to follow and use. In our own consulting work, we have also found it a much simpler methodology when fitting dynamic regression (transfer function) models. 3. It is essential when considering Sections 8/2 and 8/3 that students are comfortable with the backshift operator notation. 4. Unfortunately, there are not many software packages which allow the range of models covered in this chapter to be fitted. We have taught the material to a range of students using click-and-point interface available in the SAS Forecasting system. It provides particularly good facilities for dynamic regression and intervention models. We have had most success in teaching this material through case studies with the students spending most of the class time doing the analysis on PCs. 5. After studying Sections 8/1 and 8/2, have the students find examples in the real world where one input variable influences another variable dynamically over future time. One of our students came up with three series that seemed to go round robin: monthly gas prices, domestic autos produced and autos sold. 6. For intervention analysis, a good class project is to ask students to read an article involving the application of intervention analysis, then prepare their own report or oral presentation on what was done. We have done this with the Ledolter and Chan (1996) article with good results. 7. For Sections 8/4 through 8/7, we only provide a brief introduction to the ideas involved, plus some applications. Our aim here is to provide students with enough information to know when these models might be applicable and in what circumstances they might be useful. If students wish to use these models, they will need to learn much more about them than is described in our book. 8. An interesting activity is to have students hunt for illustrations in the literature which make use of one or more of the methods covered in this chapter. Each student can give a brief presentation based on one application and lead a discussion on whether the model was appropriate to the problem.

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Part B. Teaching suggestions

Chapter 9: Forecasting the long term 1. The first thing which it is important to bring across to the students is long-term mega-trends. A good way for doing so is to present Figure 9-1 and ask if the data presents a trend (most students will say “No”), then Figure 9-2 asking the same question. At this point several students would say that 14 years is enough to establish a long-term trend. One can then show Figure 9-3, which indicates that the 14 years of Figure 9-2, shown as a shaded region of Figure 9-3, are merely a small part of Figure 9-3. Finally, one can show Figure 9-4 and discuss why mega-trends can only be established by going back to the beginning of the Industrial Revolution (that is around 1800). Another figure which can be used to help illustrate this starting date as well as the persistence of mega-trends is Figure 9-6 which shows wheat prices in constant £ and goes back to the middle of the 13th century. 2. Once long-term mega-trends have been identified they can be extrapolated unless we believe that they will change due to some other revolution similar to the industrial one. If that is the case then we have to make our predictions not by extrapolation but by using analogies or by making various scenarios about the implications of large changes like those of the forthcoming Information Revolution. 3. When forecasting for the long term, deviations around the long-term mega-trends are of critical importance as cycles can last for many years or even decades. Moreover, since cycles are mostly random walks we have to go beyond pure quantitative models to predict them. This is a point worth making and can be illustrated by generating random numbers, cumulating their effects, and showing the result on a graph. Such graphs show that predicting turning points is impossible quantitatively since they present random walks. 4. Chapter 9 has a lot of figures that usually generate a great deal of interest from students. The way to present them is by discussing the implications if they are extrapolated in the long run, ending up with the question of what will happen when our buying power increases (at a double rate since real prices drop exponentially and real income increases exponentially) and we get a situation of over-abundance, while at the same time huge inequalities between rich and poor nations, and rich and poor citizens in single nations. 5. The discussion about such implications, as well as those that would come up by talking about various analogies and scenarios between the Industrial and forthcoming Information Revolution, generate great interest and strong opinions which provide the basis for a lively debate.

Chapter 10: Judgmental forecasting and adjustments

29

Chapter 10: Judgmental forecasting and adjustments 1. One way of introducing the topic of judgmental forecasting is to give Figure 10-4 to the class and then ask them to make forecasts after consulting the figure. Tell one third of the people that the product shown in the figure is mature, the second third that it is old, and the final third that it is new. The results of their forecasts can be summarised and presented. They usually are similar to those shown in Figure 10-5 which indicates how pre-conceived ideas are being used and how they can bias the forecasting process (after all it can be indicated that the great majority of new products fail after a couple of years). 2. Large errors in judgmental forecasts can also be illustrated by comparing the performance of professional investment managers to those of randomly selected stocks. The consistent under-performance of expert managers is remarkable and can be used as the basis to discuss what is wrong as well as illustrating how one can improve investment returns without having to pay any fees to professional experts by simply selecting bond, stocks and other investments randomly. In addition one can discuss why people prefer “experts” to manage their investments (obviously, they feel more secure by doing so, or alternatively they think that they reduce their uncertainty) while clearly such a choice results in smaller returns and extra fees. 3. Another interesting topic in Chapter 10 is the use of decision rules instead of intuitive, global judgment when the judgmental inputs can be quantified. Again there is a lot of material for interesting discussion starting with the finding that decision models in the form of multiple regression equations can predict more accurately the performance (their average GPA) of candidates for universities than admissions officers. This and similar types of decision rules can therefore be discussed as ways of improving future oriented decision-making. 4. The last part of this chapter deals with ways of debiasing decision making so that the advantages of both quantitative models and judgment can be exploited while avoiding their disadvantages. 5. The following is a list of judgmental exercises (there are two versions: one to be given to half the class and the second to the other half). These exercises provide an excellent way to show the students their biases as their answers from the two versions vary considerably.

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Part B. Teaching suggestions

Judgmental exercises 1. What is the percentage of countries in the UN that are African? To make your estimate, I would suggest that you start with a value of 65% (this percentage was found in the computer by generating a random number between 0 and 100). First decide whether this value is too high or too low—then move upward or downward from that value to what you feel is the true value. Your final estimate as to the true percentage of African nations in the UN is . 2. A psychological test was administered to a group of 100 people. The group consisted of 30 engineers and 70 lawyers. The following descriptions were obtained for Peter Jones: “Peter Jones is of high intelligence and exhibits a strong drive for competence. He has a need for order and clarity and for neat and tidy systems in which every detail finds its appropriate place. His writing is enlivened by somewhat corny puns and by flashes of imagination. He seems to have little feel and little sympathy for other people and does not enjoy interacting with others. Self-centred, he nonetheless has a deep moral sense.” If you had to place a bet on whether a participant in the test named Peter Jones was an engineer or a lawyer, what would you say? . .

Peter Jones is an engineer Peter Jones is a lawyer Please put a cross on the appropriate line.

3. You are the chief executive officer of a company faced with a difficult choice. Because of worsening economic conditions, 12,000 people will need to be fired to reduce the payroll costs and avoid serious financial problems. Two alternative programs to combat the firings have been proposed to you. The estimates of the consequences of the programs are as follows: • If program A is adopted, 4,000 jobs will be saved.

• If program B is adopted, there is a two-thirds probability that no jobs will be saved and a one-third probability that 12,000 jobs will be saved. Which of the two research and development projects would you select? A Please tick the appropriate box.



B



31

Chapter 10: Judgmental forecasting and adjustments

4. The figure below shows the sales of “Electrack”, a video game produced by Jeutronics, a medium sized French toy company. Provide optimistic, most likely and pessimistic forecasts for the year 2001.

100000 50000

Sales 4,433 60,298 67,884 89,512

0

Year 1993 1994 1995 1996

150000

200000

250000

Figure 1: Actual sales of ’Electrack’

1993

1995

1997

1999

2001

Your Forecasts of “Electrack” Sales in 2001: Pessimistic Most Likely Optimistic

. . .

5. You are in a store about to buy a new watch which will cost 350FF. As you wait for the sales clerk, a friend comes by and tells you that an identical watch is available in another store two blocks away for 200FF. You know that the service and reliability of the other store are just as good as this one. Will you travel two blocks to save 150FF? 6. FINISHED FILES ARE THE RESULT OF YEARS OF SCIENTIFIC STUDY COMBINED WITH THE EXPERIENCE OF YEARS Please indicate the number of Fs which appear in the above sentence.

.

How confident are you of your above answer? Indicate your confidence on a scale of 0 to 100 with 0 indicating no confidence and 100 indicating full confidence. . How many times did you read the sentence “FINISHED . . . OF YEARS”?

.

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Part B. Teaching suggestions

Judgmental exercises 1. What is the percentage of countries in the UN that are African? To make your estimate, I would suggest that you start with a value of 10% (this percentage was found in the computer by generating a random number between 0 and 100). First decide whether this value is too high or too low—then move upward or downward from that value to what you feel is the true value. Your final estimate as to the true percentage of African nations in the UN is . 2. A psychological test was administered to a group of 100 people. The group consisted of 30 engineers and 70 lawyers. If you had to place a bet on whether a participant in the test named Peter Jones was an engineer or a lawyer, what would you say? Peter Jones is an engineer Peter Jones is a lawyer

. .

Please put a cross on the appropriate line. 3. You are the chief executive officer of a company faced with a difficult choice. Because of worsening economic conditions, 12,000 people will need to be fired to reduce the payroll costs and avoid serious financial problems. Two alternative programs to combat the firings have been proposed to you. The estimates of the consequences of the programs are as follows: • If program A is adopted, 8,000 people will be fired.

• If program B is adopted, there is a one-third probablity that no nobody will be fired and a two-thirds probability that 12,000 people will be fired. Which of the two research and development projects would you select? A



B



Please tick the appropriate box. 4. The figure below shows the sales of “Electrack”, a video game produced by Jeutronics, a medium sized French toy company. Figure 2 shows the most likely predictions from a widely-used computerized mathematical model for new products. After having looked at Figures 1 and 2, provide an optimistic, most likely and pessimistic forecast for the year 2001.

33

Chapter 10: Judgmental forecasting and adjustments Year 1993 1994 1995 1996

Sales 4,433 60,298 67,884 89,512

200000

150000

Figure 2: Actual and predicted sales of ’Electrack’

250000

Figure 1: Actual sales of ’Electrack’

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50000

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50000

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0

0

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1993

1995

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1993

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Your Forecasts of “Electrack” Sales in 2001: Pessimistic Most Likely Optimistic

. . .

5. You are in a store about to buy a new video camera that costs 4000FF. As you wait for the sales clerk, a friend comes by and tells you that an identical camera is available in another store two blocks away for 3850FF. You know that the service and reliability of the other store are just as good as this one. Will you travel two blocks to save 150FF? 6. FINISHED FILES ARE THE RESULT OF YEARS OF SCIENTIFIC STUDY COMBINED WITH THE EXPERIENCE OF YEARS (Please do not read the above sentence again) Please indicate the number of Fs which appear in the above sentence.

.

How confident are you of your above answer? Indicate your confidence on a scale of 0 to 100 with 0 indicating no confidence and 100 indicating full confidence. .

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Part B. Teaching suggestions

Chapter 11: The use of forecasting methods in practice 1. The material of Chapter 11 relates to that of Chapter 10. The major question which arises at the beginning of the chapter is the choice between judgmental and statistical forecasting methods. As the quote by Sanders and Manrodt explains “Like past investigations (surveys) we found that judgmental methods are the dominant forecasting procedure used in practice.”

This means that there is a great deal of potential improvement as managers now realize the potential for higher forecasting accuracy (and therefore reduced costs) and they are continuously pushed, at the same time, to operate more efficiently and effectively in order to reduce their operational cost. Thus the time is right to persuade them of the considerable benefits they can obtain by using the knowledge and experience we have accumulated which clearly indicates the benefits from statistical forecasting and how it can be best integrated (this is also the major topic for the next chapter) with judgmental predictions. 2. The findings in Section 11/2 are very useful and relevant in putting forecasting on a practical grounding. In this section, the major empirical evidence is being summarized and various tables and figures are available for backing up the findings. The major question then for forecasting users is to select the most appropriate method for their specific situation. This topic is discussed in Section 11/3. It is worthwhile for the instructor to present each one of these factors and discuss them in class. Obviously a critical factor is the last one (the number and frequency of forecasts). Such factor signifies the necessity for simple methods that can be completely automated when the number of forecasts required is very large and they are needed frequently. 3. In the absence of clear factors, when guidelines are not obvious or in case of doubt as to what method to select, the best alternative is to combine three or four simple methods and use their average as a way of predicting the future. As is well known, through many empirical studies, such a simple average of the combined forecasting methods is both more accurate than the individual methods being combined while at the same time variance of the forecasting errors of combining is smaller than that of the individual methods involved. Combining can, therefore, be presented as a practical alternative which improves forecasting accuracy and reduces the chances of errors (in particular large ones).

Chapter 12: Implementing forecasting: its uses, advantages, and limitations

35

Chapter 12: Implementing forecasting: its uses, advantages, and limitations 1. In Chapter 12 it is important to emphasize what can and cannot be predicted or in other words present and discuss the limits of predictability. This can be done in terms of short-, medium- and long-term predictions as the horizon of our forecasts presents different challenges and problems as far as predictability is concerned. Critical in such a discussion is the medium-term which must predict the ups and downs of business cycles. Such predictions are extremely difficult and present a major challenge for businesses when they attempt to make budget estimates. The same is true for the longer run (18 months to 5 years) when predictions for the five years Business Plan are made and when longer term cyclical deviations around the long term trend must be dealt with. The topic of predictability, or the lack of it, can be related to the introductory chapters of the book and to our experience from the forecasting practice (including the findings from the surveys among forecasting users presented in the previous chapter). 2. The second topic of Chapter 12 deals with the organisational aspects of forecasting and the need to deal with the various forecasting problems that are encountered in organisations which are using forecasting methods and the possible solutions to such problems. There is enough information in the corresponding part of Chapter 12 to describe these problems and discuss suggestions for solving them satisfactorily. 3. The third section of Chapter 12 (Extrapolative predictions versus creative insights) discusses the role and value of forecasting beyond its operational applications. As the title implies, its greatest value is when the forecasts are creative in nature, which by definition means that they cannot be based on simply extrapolating historical information. On the contrary it may be necessary to go against conventional wisdom in order to come up with creative insights about future changes or what the future might hold. 4. In the last part of this chapter the instructor can discuss and possibly develop his/her own ideas about how forecasting is going to evolve in the future. Central to such a conception will have to be the creation of a learning process which will result in organisational learning (rather than the experience of each individual person concerning forecasting resting with such person and disappearing when he or she changes jobs or company). Creating learning about forecasting is more practical and cost efficient these days through the use of groupware (or intranets) which allow the people in organisations working on forecasting not only to exchange ideas, information and inside knowledge, but also to record the forecasting process they have been using as well as their successes and failures so that they can be reviewed in

36

Part B. Teaching suggestions the future by themselves or others in ways that can enhance learning. In other words ways must be developed which can help organisations to improve their forecasting process by knowing and avoiding past mistakes while using practices that they have been found to be successful in the past.

C/Additional materials for teaching forecasting This chapter suggests additional materials that might be used to complement the contents of Forecasting: Methods and Applications, 3rd ed., in a teaching situation. Since cases can be a valuable addition to a forecasting course and yet often represent a very different style of teaching from straight lectures and problem sets, Section C/1 outlines ways in which cases can be effectively integrated into the teaching of forecasting. Section C/2 provides some suggestions for special project assignments. These provide a context for forecasting but are shorter than field-based cases. Lastly, Section C/3 consists of exam questions.

C/1 Using cases in teaching forecasting Many different types of cases can be used to meet very different purposes in teaching. Briefly, these can be grouped into three categories. The first would be case exercises in which the case is simply an expanded problem (that is, what was traditionally described as a “story” problem) providing data and their context for the students to use in applying a specific tool or technique being covered in a forecasting course. The second type would be the management process case in which the case describes the forecasting process in an organization and allows students to consider the process itself as opposed to specific techniques applied to specific data sets. The final category would be a mix of these two but generally oriented toward applying a technique to specific data and then looking at the management decision-making implications of the resulting forecasts. This third category is closely tied to the implementation of what the forecaster recommends to management. A case course is taught using a problem-centered, participant-involved method of instruction. For most class sessions in such a course, a case is assigned to be read and prepared for discussion and analysis in the classroom. Each case describes a specific management problem and seeks to describe selected aspects of an everyday situation that either the forecasting specialist or the management user of forecasting might encounter. 37

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Part C. Additional materials for teaching forecasting

C/1/1 The nature of a case In spite of the realism that an instructor seeks to build into all cases, they cannot be completely true-to-life management situations for the following reasons. 1. The information comes to the student in a neatly presented form. By contrast, managers and forecasters must gather facts in ongoing situations from memos, conversations, statistical reports, and the public press in a much less organized fashion. 2. A case is designed to fit a particular unit of class time and to focus on a certain category of problem, such as a forecasting technique, forecasting system, or the management use of a given forecast. Consequently, it may omit elements of the real situation—people or organizational issues for example—in order to focus attention on what the instructor would like the class to see. 3. A case is a “snapshot” taken at a given point in time. In reality, business problems form a continuum requiring some action today, further consideration and action tomorrow, and so on. It is very seldom that a manager can wrap up his or her problems, put them away and go on to the next “case” as is done in a course. 4. While students studying cases are required to make decisions, they do not have the responsibility for implementing those decisions and do not have to bear the burden of ineffective implementation. This can be a particularly important shortcoming when training forecasting specialists, who will have to interface with managers, who in turn must make decisions based on their forecasts.

C/1/2 The educational purposes of cases Cases can help forecasters and managers sharpen their analytical skills by exposing them to facts and figures which must be evaluated and used to produce both quantitative and qualitative evidence to support recommendations and decisions. In case discussions students are typically challenged by instructors and peers to defend their arguments and analyses. This can have the cumulative effect on the students of helping to develop a problem-solving methodology and heightened ability to think, reason, and apply specific techniques in a rigorous fashion. Case studies cut across a range of organizational situations and provide exposure to a far greater number of situations than would be likely on a single job involving normal day-to-day routine. Thus, cases permit building knowledge across a range of subjects and situations by dealing selectively and intensively with problems in each field. Students come to recognize that the problems they face as a manager or forecaster are not unique

C/1 Using cases in teaching forecasting

39

to one organization or even a system of organizations. This helps them to develop a more professional sense of their tasks and the way in which they can be handled most effectively. Cases and the related class discussions can provide the focal point around which the student’s past experience, expertise, observations, and rules of thumb can be brought together in a framework for effectively tackling new situations. What each class member brings to identifying the central problems in a case, analyzing them and proposing solutions to them, is as important as the content of the case itself. The lessons of experience can be tested as students present and defend their analyses against those of participants with different experiences and attitudes. This is one place where common problems, interdependencies, differences of perception, and organizational needs can be highlighted and resolved in a systematic fashion. An important benefit of using cases is that they help students learn how to ask the right questions. It has often been said that 90% of the task of a good manager is to ask useful questions. Answers can be relatively easy to find once the appropriate questions are asked. Even when assignment questions are used with individual cases, students should be pressed to ask themselves, “What are the real problems that the individual forecaster or manager must resolve in this situation?” One final benefit that an instructor often seeks to achieve through the use of cases is to transfer to the students the excitement and challenge that can come from pursuing management and forecasting careers. In the cases they prepare, students often see some problems they are glad they do not have to face in real life, and others that they recognize from first-hand experience. They should come to recognize that being a manager or working as a forecaster for a manager can be a great challenge intellectually, politically, and socially.

C/1/3 How students should prepare a case There is no single form of case preparation that works best for everyone. However, some general guidelines can be offered that might well be adapted to the way each individual student does his or her work. These guidelines would include the following. 1. Go through the case almost as fast as you can turn the pages, asking what the case is about and what types of information are being provided for analysis. 2. Read the case very carefully, underlining key facts and perhaps writing those and key issues in the margin. The students should try to put themselves in the position of the manager or forecaster in the case and develop a sense of involvement in that person’s problem.

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Part C. Additional materials for teaching forecasting

3. After a thorough reading of the case it’s often useful to prepare a list of the key problems. The case can then be gone through once more, picking up those considerations and data that are relevant for resolving each of those problem areas. 4. Perform the analysis that will enable the student to conclude what recommendations are most consistent with the situation and the facts provided. 5. Develop a set of recommendations that can be supported by case data and the student’s analysis, and indicate how those recommendations should be implemented in this situation. These five steps can best be applied by the student working individually. The next step which can aid in case preparation is to have students meet in discussion groups to present their arguments to others as well as to listen to the arguments of their peers. This testing of the student’s analysis and recommendations is an important preparatory step for class discussion. The purpose of the group discussion is not to develop a consensus or a group plan of action, but rather to help each member refine, adjust, and amplify their own thinking. It is not necessary or even desirable that the discussion group members agree. In class the instructor will usually let the class direct the discussion toward those topics where most of the individuals have concentrated their attention. However, the faculty member is also likely to prod the class to explore fully those avenues that are most relevant based on the faulty member’s experience and based on the purposes for which the case was included in that course. Often the faculty member will summarize the discussion and draw out the useful lessons and observations toward the end of the class discussion. However, this might also be done by asking a student to provide that type of summary. It should be emphasized that learning through the case method results from rigorous discussion and controversy. Each member of the class and the instructor must assume responsibility for preparing a case and for contributing ideas to the class discussion.

C/1/4 Use of cases in course design Just as there are many types of cases, there are many purposes for which they may be used in course design. Perhaps the simplest is to select exercise cases that can be used to present data for the students to use with various forecasting techniques. Many of the cases that will be suggested later in this chapter are of this type and, thus, can be used simply as exercises or work problems for students. A second way in which cases are often used by instructors who do not teach largely by the case method is as a way to get at implementation and management decision-making issues. In such a use of cases, an instructor might choose to cover a topic such as regression

C/1 Using cases in teaching forecasting

41

analysis using a more traditional lecture method with exercises and problems and then conclude that section of the course with one or two classes built around management process-oriented cases. These can be viewed as a way for students not only to apply the techniques related to regression that they have learned but to look at the implications of those applications for managers and to consider how they might be effectively “sold” to management and implemented. If this were to be the only use of cases the instructor might simply choose to end each of four or five major sections of the course with one or two cases, resulting in a course that is approximately 80% exercises and lecture/problems, and 20% case applications. A third way in which cases often have been used effectively in teaching forecasting and planning is to teach the basic techniques and their applications using exercises and problems and then at the every end of the course to have a major section on implementation. In that section, cases requiring the use of different techniques and illustrating the range of management issues related to implementation could be addressed. If this approach is followed a class of thirty sessions might have only the last four or five built around case studies of the management process and implementation type. Still a fourth approach to utilizing cases in this subject area would be to build the entire course around cases. While this is certainly feasible given the amount of material readily available, this can often be a most challenging task if the students are not used to case courses from other parts of their curriculum. The authors’ experience would suggest that it is best to use one of the foregoing forms of case use initially, before building an entire course around cases. When students are not particularly well versed in the case method, it has often been found effective to assign study groups to meet in preparation for the classes in which cases will be used and then to have two or three individuals briefly (5 minutes each) present their recommendations and analysis at the start of class in order to get the discussion going. That provides a complete set of thoughts and ideas on the case situation that the rest of the class discussion can build on. It also ensures that students are well prepared for that class since they know they might be called on to make such a starting presentation

C/1/5 Obtaining cases Prepared cases with teaching notes are available from Harvard Business School Publishing, 60 Harvard Way, Box 230-5A, Boston, MA 02163. They can also be obtained through the internet at www.hbsp.harvard.edu These can be reprinted by the instructor and used to complement the exercises in the text itself.

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Part C. Additional materials for teaching forecasting

C/2 Assignments C/2/1 The Phrygian thread factory (Prepared by Professor Fred Shepardson, Stanford University. Used by permission.) The Phrygian Thread Factory was founded in 1947 by Ikos Matzakis shortly after emigrating from Greece. The enterprise had begun on a small scale, supplying thread for the local garment industry. In these days, Matzakis would buy cotton fiber from relatives in Greece, import it to the United States, and dye it and spin it produce a rather wide range of end products. Since those humble beginnings Phrygian had grown to become a not inconsequential thread supplier for the Northeastern United States. In addition to supplying the garment industry and various distributors and retailers of sewing threads, Phrygian was now supplying large industrial users. Major customers included the auto industry (for upholstery and seat belts) and the telecommunications industry (for wrapping and insulating cables) Similarly, Phrygian no longer restricted itself to cotton thread. The bulk of its output was now nylon, although significant amounts of rayon, cotton, and silk were produced as well. Phrygian’s product line was virtually unlimited, for it was standard operating procedure to do custom dye jobs to match customer color specifications. However, color notwithstanding, there still remained nearly a hundred distinct items in the product line. One of Phrygian’s most important products was NC-216. This was a bonded nylon thread customarily used by the auto industry in sewing seams in upholstery. To make it Phrygian began with raw nylon fiber of weight 210 denier. Two strands were spun together with a right-hand twist to form a thread. Then three threads were twisted together, again with a right-hand twist, to form the final thread. Once this was ready, it was loosely wound into large spools and sent to the dyehouse. The dyehouse staff would dye the thread into batches of up to ninety pounds. From the dye vats, the thread would go directly to large walk-in ovens to accelerate drying. After 24–48 hours in the ovens the thread was moved to large drying rooms to finish drying. After one to five days in the drying room, the thread was ready to be sent upstairs for bonding. In the bonding room the thread was passed slowly through a hot liquid plastic solution and then through heaters and on to winders. Once this process was completed, random samples were taken and tested, primarily for breaking strength. The thread was finally sent down to the spooling room to be put on customer-specified spools (usually one pound spools). Once finished, the completed order was sent down to shipping to be packaged and shipped

C/2 Assignments

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Other products followed the same general flow, although some required additional procedures (such as skeining before dying) and others required fewer (for example, no bonding). In the office suite life was characterized by a constant effort to track down and expedite orders. Orders were phoned in by salesmen in the field. In the case of custom color requirements, color samples followed by mail. For most orders, salesmen, wanted to know a projected delivery date. The production manager, Roy, and his assistant, Fred would characteristically supply delivery dates off the top of their heads. In this process they relied on their intelligence guided by experience. For all products they were aware of the normal production time. They also knew that these lead times were quite flexible. With constant monitoring, a product could be shipped in a much shorter time than its expected processing time. However, with no monitoring a product often took much longer than the normal lead time. For very important orders, Roy and Fred would promise an early date and then ride the department managers closely to make sure the date was met. Other aspects of production control were done in the same sort of ad hoc manner. Workforce levels, overtime, and extra shifts were decided on pretty much a day-to-day basis. Bernie, the new plant manager, had decided things had to change. This decision had been made in response to the latest catastrophe. Phrygian had just received a large rush order from Non-Specific Motors for three thousand pounds of NC-216, a thousand pounds in each of three colors. Bernie was at first jubilant when he heard of it. But his jubilation was short-lived when he learned that there was not enough 210 denier nylon to meet such a large order. There was already additional nylon backordered but it was not scheduled to arrive in time to be of use for the Non-Specific order. While Roy and Fred scrambled orders, robbing Peter to pay Paul, and combed the countryside for additional supplies of 210 nylon, Bernie sat in his office and plotted strategy. Bernie decided the first requirement was a good forecasting system so they would not be caught off guard like this again. By going over orders for the last three years, he noticed that each year Non-Specific had placed a large rush order for NC-216 at about this same time. He felt sure such information could be used in planning operations. He decided to call Roy and Fred in and have them set up a forecasting system. The next day Bernie made his pitch. “Look, you guys, things have been going pretty well. Phrygian’s profits are up, even our market share is up. But now we’re getting bigger and I think we’ve just about hit our breaking point. Last year you hired Fred, Roy. That took some of the load off you, but already it’s getting ahead of you again. We are getting more customer complaints about orders being shipped late. And now we’ve gotten caught short on 210 for the Non-Specific order.” “Look, Bernie, we’re getting burdened on this I know. But it’s the first time this has

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Part C. Additional materials for teaching forecasting 1972 January February March April May June July August September October November December a b c

260 550 930

1973 1340 1500 1570 1360 1350 1400 1610 2280 2730 3210 3350 3620

1974 3690 3520 3330 3120 2880 2670 2790 3540 3920 4310 4200 4070

1975 4110 3870 3550 3420 3250 2910 3080 3890 4310 4860 4660 4520

1976 4500 4290 4010 3830 3570 3250 3520 4280 4830 5310 5180 5030

1977 2600 5830 5400 4210 3900 3640 4010 4830 5270 5960 5830 5510

1988 5330 5290 4960 4730 4370 4020 4020b 4830 4880 5540 5430 5210

1979 5140 4900 4400 4090 4600 c 4540 4930 5920 6480 7170 7080 6930

1980 6820 6540 6030 5770 5510 5000 5430 6520 7180

As a matter of policy, no special discounts or sales campaigns have been held for NC-216. Phrygian dropped NC-336, from their regular product line, retaining it as an option at a price premium. Phrygian introduced NC-236, identical to NC-216, except with a right hand spin and a left-hand twist. Monthly order for NC in poundsa

happened. Give us a break.” “Roy, I know it’s the first time and I’m not really blaming you. But I want to make sure it doesn’t happen a second time. I want you two to develop a forecasting system so you’ll have a better idea of what to expect.” “You mean some sort of automated technique for the new computer you got?” “That’s right, Fred. Since sales orders are now being entered into a data file for the billing system anyway, there must be some way of accessing that information and using it.” After the meeting Roy and Fred sat in Roy’s office kicking around ideas. “Look, Roy, here’s the orders for NC-216 since late 1972. That’s when we introduced NC-216, to meet the demand caused by federally mandated seat belts (see Table 3-1). Suppose we just use this one item for discussion purposes. Now what do you propose to do?” “Well, Fred, I’m not sure. In the past I always used to talk with the salesmen periodically to get a feel for what they thought was coming. Then I’d use that with my intuition

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C/2 Assignments 1979 January February March April May June July August September October November December ∗

7018 7353 7009 6795

1980 5983 6767 6355 6380 5326 4960 5565 6646 7980 8006 7736 7578

1981 6796∗ 7481 6910 6305 6116 5698 6105 7350

For January 1981 through September 1981, inflation factor was calculated using data from October 1980. Fred’s forecasts for October 1979 through September 1981

to make decisions on ordering raw materials, setting up vacation and maintenance schedules.” “Well, your intuition didn’t intuit the big Non-Specific order.” “Actually, I’d thought of it. That’s why we have had that big order already in on 210. It’s just I though we wouldn’t need it for almost another month.” “Hey, Roy, maybe we could just use last year’s figure for a month’s demand, plus an inflation factor to get a forecast for the same month this year.” “That might make sense, Fred. But look here with the NC-216. Notice in January of 1977 the low figure. That’s probably because of our wildcat strike that entire month. If we’d used that figure to predict January 1978 we would have really been caught short.” “Well, perhaps this forecasting system should have a manual component as well. A place for our intuition to pick up on facts like that.” “We’re busy enough already, Fred. With almost a hundred products, not considering color differences, we’d be buried under the reams of data and the damned computer output. Of course we should be able to take advantage of the fact that most of our products fall into one of three or four demand patterns.” “How’s this then, Roy? Suppose we are trying to forecast January 1981. We need

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Part C. Additional materials for teaching forecasting

the forecast about three months in advance; that means the first week of October, 1980. Suppose we take the average January demand for the five years preceding and inflate it by a certain percentage. Since we will know October 1980’s real demand by then, we will let the inflation percentage be the percentage difference between October 1980’s real demand and the five preceding years’ average of October demands. Following that procedure, we should be able to develop forecasts for even the next twelve months.” (See Table 3-2.) “That sounds really good, Fred. But let’s test it by going back in the data and forecasting our last twelve months’ demand. Then we can compare it with reality (see Table 3-2). In any case, just to protect ourselves, I think we should hire a consultant and see what ideas he has.” “Sounds good to me. Boy, this could cut Phrygian Thread’s Gordian Knot.”

Assignment You are to act as a consultant to Phrygian Thread. Prepare a careful analysis of the demand pattern for NC-216. Then present a forecasting system for Fred and Roy to consider for their product line. Evaluate your model on whatever criteria you feel to be appropriate. Make explicit any assumptions you make.

C/2 Assignments

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C/2/2 Nike stock price predictions (Prepared by Professor Peter Reiss, Stanford University. Used by permission.) M.A. Verage is a retired stockbroker who on occasion still tries her hand at picking stocks. Last summer she invested her entire life’s savings in Nike, a company that sells not only running shoes, but also a wide range of athletic apparel. Having made a bundle on her original investment, M.A. is now concerned that the athletic fad will fade. (She privately fears that this is already happening.) She is also concerned that the current speculative bubble of optimism on Wall Street will burst, and this too will send Nike stock prices plummeting. Her dilemma is this: since last August, the value of her stock has ranged from one and a half to two times what she paid for it. If she sells now, she can realize a sure return on her original investment. On the other hand, Nike’s stock price has historically been quite strong—even in recessions—and it may return to its 1983 high of $24.00. Not wanting to make a foolish decision either way, she decides to call in an expert who can predict what will happen to the price of Nike stock over the next 15 trading days. For a rather modest fee, you have a suddenly become an expert on stock price behavior. Having not had a course in finance and not having access to any information about Nike’s prospects in the athletic apparel market, you must resort to forecasting Nike’s stock price using only information on past stock prices.

Assignment 1. In a four-page (or less) document, you or your group must present your forecast of how Nike’s stock price will behave. If you present more than one set of forecasts, you must state which one is your most preferred (and why). 2. You may assume that she has taken this subject and is familiar with the material covered in the text. 3. You should spend at least three-quarters of your discussion describing the data (e.g., plots, transformations, statistics . . .) and why you have elected to use your forecasting techniques. Running every possible method on the data would be terribly time consuming and burdensome. Please limit your efforts by first looking at the data and then deciding what to do. You will be graded not so much on how you do at forecasting the price and the sophistication of your techniques, as you will on the thoughtfulness and clarity of your discussion. Remember, the idea here is not so much to give you a once-and-for all grade, as it is for the experiment with techniques used in class. If you get to the point where you are disgusted with moving

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Part C. Additional materials for teaching forecasting averages, exponential smoothing, etc, you have probably tried too many techniques and written too much.

4. Your paper is limited to four double-spaced pages of text. You can attach as many plots or copies of output as you want. 5. The data on Nike’s stock price are below. Please use only these data. The data are arrayed by date and price. February data are given first, followed by March. Date 01 09 16 23 02 09 16 23

Price 19.875 20.125 22.625 22.625 17.125 16.000 16.125 16.250

Date 02 10 17 24 03 10 17 24

Price 19.375 20.250 23.125 19.125 16.625 16.375 16.000 15.875

Date 03 11 18 25 04 11 18

Price 19.500 21.500 23.125 17.000 16.125 16.250 15.625

Date 04 14 21 28 07 14 21

Price 20.125 22.750 23.125 17.875 16.000 16.125 15.875

Date 07 15 22 01 08 15 22

Price 20.000 23.000 22.625 17.625 15.625 16.125 16.000

C/2 Assignments

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C/2/3 Final Paper Option (Prepared by Professor Peter Reiss, Stanford University. Used by permission.)

Paper requirements All papers must be typed, double-spaced and have neat corrections. Your paper can be anywhere from seven to sixteen pages in length. I suggest you aim for seven to ten pages, but if you require more pages to make your arguments clear, you may use it. Footnotes, references and exhibits are not counted in the above page limits. In writing your papers, remember that, although content is paramount, style and clear prose are also important. Any sources that contribute significant or little known facts must be referenced with footnotes. You are also not permitted to turn in any paper submitted for another course (or any modified course paper). All data sources must be clearly detailed.

Topics You pretty much have freedom to choose your own topic as long as it is related to a forecasting problem. This forecasting problem can be an event change study, a time series study, or a characteristics-based study. You must use real data. This data can be gathered from public or private sources. You must reserve 5 to 10 per cent of your data for out of sample predictions. You may not use this data when fitting your model. Once you have settled on perhaps several models, you are then to forecast the out of sample data, as well as periods (or phenomena) beyond your data. You are then to write up your results, evaluating how you did in forecasting your reserved data and how you expect to do with your future forecasts. You must not alter your models once you have simulated them over the reserved data. I will not penalize you for bad out of sample predictions as long as you intelligently evaluate why they occurred and you indicate how you might have gone about fixing your model once these new data were revealed (you can actually revise your model if you find it a compelling exercise—but you must report your initial models and results first). As far as content, you should in the beginning of the paper lay out the issue or topic you are discussing. This should include a statement of the forecasting problem or topic, any analytical or numerical frameworks you wish to use, and a brief statement of the forecasting methods you have chosen to explore. You then should present your analysis and facts back to your analysis . Finally, you should give a brief summary of your models,

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Part C. Additional materials for teaching forecasting

and their out of sample predictions, and an evaluation of the reliability of your model. Suggestions for actions based on your forecasts should also be made at this point. Sample topics might include: 1. A model that predicts the sales of a company’s product line. 2. A model of movements in macroeconomic variables. 3. A model of a firm’s choice of strategies (e.g., capacity utilization, price, output, quality, advertising, etc.). 4. A model of seasonality, cycles and trend in macroeconomic or company data. 5. Inventory modeling and production scheduling problems. 6. Judgmental forecasting

C/2 Assignments

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C/2/4 Demand for blood tests (Prepared by Professor Fred Shepardson, Stanford University. Adapted with permission.) This assignment is based on the article “Box-Jenkins vs multiple regression: some adventures in forecasting the demand for blood tests” by Everette S. Gardner, Interfaces, Vol. 9, August 1979. This paper is a report on consulting activities performed for the Clinical Coagulation Laboratory at the North Carolina Memorial Hospital. You are to put yourself in the position of the Laboratory Director. Consider the papers as the consultants’ final report to you on their study of the problem of forecasting the Laboratory’s demand for blood tests. While you are responsible for this project, your boss has taken a keen interest in it as well and she is eager to have the Laboratory start using an analytically based forecasting method. She has also received a copy of the consultants’ report and you can assume she has read it. She is now awaiting your report on the project and how you will proceed. Your presentation should include a brief analysis of the consultants’ work and your own recommendation for implementing the proposed forecasting procedure. What is your presentation?

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Part C. Additional materials for teaching forecasting

C/2/5 Winning Wines As the operations manager for Winning Wines, Sandra McDougal has become quite concerned about managing her product release. The firm produces a large range of wines primarily for local consumption. Lately Winning Wines has found a valuable new export market in South East Asia. Currently, of the 39 distinct products Winning Wines produces, 12 are for this new market. Ms McDougal is worried about managing the release of her products gradually so that customer demand is satisfied without affecting the price by oversupply. Consequently she has decided to introduce a formal forecasting procedure at Winning Wines. In order to begin her analysis of the demand pattern for the company’s products, Ms McDougal has selected a product for which the company has a detailed history—a popular sparkling white wine. From marketing she has obtained extensive monthly sales data for this wine. Graphs of the data with ACF and PACF plots are attached. What forecasting techniques should Ms McDougal be considering at this point? Defend your choice. What advice can you give to help her in developing a forecasting system for demand for Winning Wines entire product line?

53

C/2 Assignments

Sparkling wine sales

600

o o

o

o

500

o

o

liters

400

o o o

o

300

o

o

o o

o

o o

o

o

o

200

o o o

o

o

o o o

o

o o

o

o

o

o

o

o

o

o o

o

o

o

o o

o

o

o o

o o o

100

o

3

4

o o

5

6

0.2

-0.4

-0.2

0.0

0.0

ACF

PACF

0.4

0.4

0.6

0.8

2

o

o o

o o o o

o

o

1

o

o o

0

10

20

30

0

10

Sparkling wine sales (liters) for Winning Wines.

20

30

54

Part C. Additional materials for teaching forecasting

C/2/6 Decomposition and exponential smoothing assignment Select one time series of real data. The series can be selected from among those available in the Time Series Data Library (www.maths.monash.edu.au/~hyndman/tseries/) or can be published data or data you have collected. The data series must be seasonal and comprise at least 30 observations. 1. Make a time plot of your data and describe the main features of the series. 2. Transform your series if necessary. Explain which transformation was used and why. If no transformation was used, explain why not. 3. Decompose the transformed series using an additive model. Produce a decomposition plot and a seasonal sub-series plot for the decomposition. 4. Forecast the next two years of your series using Holt–Winters’ additive method. Give the parameters of the method and report the MSE, MAPE and MAD of the one-step forecasts from your method. If you transformed the series, give the forecasts on the original scale. 5. Find a prediction interval for the next observation using the MSE. Check the assumption of normality. 6. Add your forecasts and prediction interval to the graph of the data. 7. Explain why the MSE cannot be used to obtain prediction intervals for longer-term forecasts.

C/2 Assignments

55

C/2/7 ARIMA Assignment Select one time series of real data. The series can be selected from among those available in the Time Series Data Library (www.maths.monash.edu.au/~hyndman/tseries/) or can be published data or data you have collected. The data series must comprise at least 30 observations. You should produce forecasts of the series using an ARIMA model. Write a brief report (about 4 pages) of your analysis including • transformations • model selection • estimation • diagnostics • forecasts and prediction intervals Explain carefully what you have done and why you have done it. You should also compare your results with those obtained using an exponential smoothing method. Which method do you think gives the better forecasts? You should write as if your report is to a client who is interested in forecasts of your data. You may assume that your client is familiar with the material covered in the text. You will be graded not so much on the sophistication of your techniques, as you will on the thoughtfulness and clarity of your discussion and the communication of your results. You will also be required to give a presentation of your analysis in class.

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Part C. Additional materials for teaching forecasting

C/3 Exams C/3/1 Sample exam questions for a time series and forecasting subject Question 1 The graphs in Figure 1 concern the production of sulphuric acid in Australia between March 1956 and March 1992. 1.1 Describe the series in a few sentences. Does transforming or differencing seem appropriate? 1.2 Given that Australian economic policy was radically different between 1972 and 1975 due to the Whitlam Labor Government and that there was a severe recession in 1991–1992, explain in a couple of sentences some of the unusual features of the time series plot. What other information might be helpful in modelling this series? 1.3 Your client has asked you to provide forecasts of this series for the next two years. She has no specific idea of the expected behavior of the forecasts and does not require forecast intervals. Consider each of the methods listed below. Say, in a few words each, if and why you think each of the methods listed might be appropriate or not for this situation. If you find more than one method that might be appropriate, discuss in about two sentences the relative merits of the appropriate methods. Assume methods a)–e) will be applied to the data as given, without any preceding actions taken. a) b) c) d) e) f)

Single exponential forecasting Holt’s method Holt–Winter’s method AR(1) with −1 < φ < 1 ARIMA(0,1,1) with −1 < θ < 1 with a constant ARIMA(0, 1, 4) applied to the data differenced at lag 4.

57

C/3 Exams

Quarterly production of sulphuric acid in Australia o o

600

o

o o o

o o

o o

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400

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o o o o o o o o o o o o

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200

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o

1960

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0.2

Partial ACF

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ACF

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Time

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Lag

Figure 1: Graphs relating to production of sulphuric acid in Australia.

58

0

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Part C. Additional materials for teaching forecasting

1500

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0.4 0.0

0.2

Partial ACF

0.4 0.2

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ACF

0.6

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Year

0

5

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Lag

20

25

5

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Lag

Figure 2: Graphs relating to the Beveridge wheat price index.

25

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C/3 Exams

Question 2 The graphs in Figure 2 concern the Beveridge wheat price index from 1500– 1869. 2.1 Describe the series in a few sentences. Explain why taking logarithms of the series is appropriate. Does differencing also seem appropriate? Explain why or why not. 2.2 Suppose you wish to forecast the series for the next 10 years. Consider each of the methods listed below. Say, in a few words each, if and why you think each of the methods listed might be appropriate or not for this situation. If you find more than one method that might be appropriate, discuss in about two sentences the relative merits of the appropriate methods. Assume methods (a)–(f) will be applied to the logged data without any other actions taken. (a) (b) (c) (d) (e) (f) (g)

Single exponential forecasting Holt’s method Holt–Winter’s method AR(1) with −1 < φ < 1 ARIMA(p,1,q) with no constant ARIMA(p,1,q) with a constant ARMA(p,q) applied to the logged data differenced at lag 12.

Question 3 The graph in Figure 3 is of the number of housing starts in the US each month for nine years. 3.1 Consider forecasting the time series using the various methods listed in the previous question. Say, in a few words each, if and why you think each of the methods might be appropriate or not for the client in this situation. If you find more than one method that might be appropriate, discuss in about two sentences the relative merits of the appropriate methods. Assume methods (a)–(f) will be applied to the data as given without any preceding actions taken. 3.2 Describe what the ACF would probably look like for this series and describe any actions you would take before trying to fit a stationary ARMA model. 3.3 Discuss in about 4–5 sentences (but without giving any equations) what actions you would take after you have obtained the parameter estimates from your ARMA model but before you produce any forecasts.

60

150 50

100

thousands

200

Part C. Additional materials for teaching forecasting

1966

1967

1968

1969

1970

1971

1972

1973

1974

1975

Figure 3: U.S. monthly housing starts, January 1966–December 1974.

Question 4 The graphs in Figure 4 concern the total building and construction activity in Australia each quarter. The units represent the value of work done in millions of dollars at 1984/1985 prices. Data are available from July 1976 to September 1994. However, the graphs are based on a restricted set of data. The first quarter on the graph is July–September 1976; the last quarter on the graph is July–September 1991. Let Yt denote the raw series shown in the time plot and let X t denote the series after differencing at lags 1 and 4. The ACF and PACF graphs are for the X t series. 4.1 Describe the series in a few sentences. Does transforming seem appropriate? If so, what transformation would you try? What features of the series suggest differencing is appropriate? 4.2 You are developing a forecasting model for the Housing Industry Association and you wish to test the model by forecasting the data from December 1991 to September 1994. Consider each of the methods listed below. Comment, in a few words each, on whether the methods listed might be appropriate for these data. If more than one method might be appropriate, discuss in about two sentences the relative merits of the appropriate methods. Assume methods a)–e) will be applied to the data, Yt , without any preceding actions taken.

61

C/3 Exams

Building and construction activity in Australia o o

5000

o

o

4500

o o

o

o o

o

o o

4000

o o o

o

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o

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3500

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o

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3000

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o

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o o

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o

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o

o

o

o

o

o o

o

2500

o o o

1976

1978

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1982

1984

1986

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1990

1992

-0.2

Partial ACF

0.4 -0.4

-0.4

-0.2

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ACF

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ACF and PACF for differenced data

0

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10 Lag

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5

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20

Lag

Figure 4: Graphs relating to quarterly totals of building and construction activity in Australia, First quarter: Jul–Sep 1976; last quarter: Jul–Sep 1991.

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Part C. Additional materials for teaching forecasting a) b) c) d) e) f) g)

Single exponential forecasting Holt’s method Holt–Winter’s method AR(1) with −1 < φ < 1 ARIMA(0, 4, 1) AR(8) applied to Xt ARIMA(1,0,0) applied to Xt

4.3 Explain why it is better when evaluating forecast performance to fit the model using the data up to September 1991 rather than using the complete data set up to September 1994.

63

C/3 Exams

C/3/2 Two hour exam for a time series and forecasting subject All questions of this exam involve the series plotted below.

5

10

15

20

25

Monthly retail turnover recreational goods (Tasmania)

1982

1984

1986

1988

1990

1992

1994

1996

Figure 1: Time plot of monthly retail turnover ($ million) of recreational goods in Tasmania between April 1982 and March 1996.

The last 15 months of data are given below: Jan Feb Mar Apr May Jun 1995 13.7 14.7 14.8 13.0 14.0 13.4 1996 16.9 16.3 14.7

Jul 13.6

Aug 14.9

Sep 13.5

Oct 14.7

Nov 15.7

Dec 21.9

Question 1 1.1 Describe the series plotted above in a few sentences. Comment on trend, seasonality, cycles and changes in variance and discuss the causes for these. 1.2 Explain why it is easier to analyze the logarithms of the data rather than the raw data. 1.3 Your client has asked you to provide forecasts of this series for the next two years. Consider each of the methods listed below. Assume the methods will be applied to the logged data. Say, in a few words each, if and why you think each of the methods listed might be appropriate or not for this situation. If

64

Part C. Additional materials for teaching forecasting you find more than one method that might be appropriate, discuss in about two sentences the relative merits of the appropriate methods. a) b) c) d) e)

Single exponential forecasting Holt’s method Holt–Winter’s method AR(1) with −1 < φ < 1 ARIMA(0,1,1) with −1 < θ < 1 and with the mean removed after differencing at lag 1 f) ARMA(p,q) model fitted to the series after differencing at lag 12. g) Seasonal means method.

Question 2 Figure 2 shows the results of a STL decomposition applied to the logarithm of the data shown in Figure 1. The seasonal component is assumed to be constant from year to year. Figure 3 shows the seasonal pattern. 2.1 Say which quantities are plotted in each graph of Figures 2 and 3. 2.2 Explain how seasonally adjusted data can be obtained using the quantities plotted in Figure 2. 2.3 If you were using a classical decomposition, what sort of moving average smoother would be appropriate for estimating the trend of the series? Express the smoother as a weighted moving average smoother and explain how the weights ensure there is no seasonal contamination of the trend estimate. 2.4 Explain why there is a problem with computing a moving average smoother near the ends of the series. Explain why a loess smoother does not have this problem. 2.5 What sort of decomposition would have been necessary if we had used the raw data instead of the logged data?

65

-0.10

0.0

0.10

-0.1

0.1

0.3

1.6

2.0

2.4

2.8

1.5

2.0

2.5

3.0

C/3 Exams

1982

1984

1986

1988

1990

1992

1994

1996

time

Figure 2: STL decomposition of the logarithm of the data shown in Figure 1.

66

Part C. Additional materials for teaching forecasting

-0.1

0.0

0.1

0.2

0.3

Seasonal pattern

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

3.0

Figure 3: Seasonal pattern (or indices) based on the STL decomposition in Figure 2.

1.5

2.0

2.5

Forecasts

1982

1984

1986

1988

1990

1992

1994

1996

1998

Figure 4: Forecasts of the logarithms of the data shown in Figure 1, computed using HoltWinters’ method with parameters a = 0.47, b = 0.41 and c = 0.0.

67

C/3 Exams

Question 3 Holt-Winters’ method was used to forecast the logged data. The forecasts are shown in Figure 4. The MSE for the one-step forecasts is 0.0045. The first few forecasts are: Apr 96 2.69

May 96 2.74

Jun 96 2.70

Jul 96 2.74

Aug 96 2.79

3.1 Give the forecasts for April–August 1996 on the original scale. 3.2 Compute a 95% prediction interval for the first forecast (on the original scale). 3.3 The smoothing parameters (α, β and γ) were chosen to minimize the one-step MSE. Using the Holt-Winters’ equations, explain what γ = 0 implies about the data. Discuss how this feature of the data is also seen in the seasonal decomposition in Figure 2.

Question 4 Let the series plotted in Figure 1 be denoted by {X t }, let Yt = log(Xt ) and let Wt = Yt − Yt−12 . Then the following model was fitted: Wt = 0.52Wt−1 + 0.38Wt−2 + Zt where {Zt } is white noise with variance 0.0063. 4.1 What sort of ARIMA model is Wt (i.e. what are p, d and q)? 4.2 Is the model for Wt stationary? 4.3 Write down the model for Yt . Is the model for Yt stationary? 4.4 Compute the forecasts for Xt for April 1996 and May 1996. 4.5 Compare the forecast performance of this model with the method used in Question 3, referring to the one-step MSE for both models. 4.6 The Ljung-Box statistic for h = 24 is 50.3. Complete the Ljung-Box test and comment on the adequacy of the model.

End of Exam

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Part C. Additional materials for teaching forecasting

C/3/3 Final Exam: Elective MBA Course (Prepared by Professor Peter Reiss, Stanford University. Adapted by permission.) Part I: True or False (Worth 100 points. Suggested time allotment: 1 hour) The following statements are either always true or they are false. Answer each statement “True” or “False” and give a one or two sentence justification that supports your conclusion. You will receive zero for an incorrect answer, one for the right answer, and 1 to 3 more points for your justification. 1. A forecast error cost function based upon absolute deviations (MAD) penalizes you more for large errors than does mean squared error (MSE). 2. The larger R2 , the better the model. 3. The smaller the standard error of a multiple regression model, the more accurate are the predictions of the future. 4. The first four randomness tests we discussed in class will (with an extremely high probability) perform equally well in detecting any trend or seasonality in a time series. 5. Given that we now know how to use Box-Jenkins procedures, there is absolutely no reason to ever use a simple moving average of past values of the time series. 6. The mean of a time series (that has any variance) can never be an optimal forecasting rule. ˆ t = (Xt−1 + Xt−2 )/2) is always sta7. A single moving average (for example, X tionary. 8. Single moving averages as a smoothing and forecasting tool work well on data that have a strong linear trend. 9. Single Exponential Smoothing (SES) puts most of its averaging weight on values that are mid-way between the start and end of the observations being averaged. 10. Holt’s method is always better than single exponential smoothing because it removes more trend. 11. In SES, if the value of α that minimizes MSE is equal to one, this implies X t ˆ t = Xt−1 . is best represented as a moving average of length one (i.e. X 12. If you definitely have seasonality, trend and randomness in your time series data, Winters’ method should be used before any other smoothing method. 13. In trying to pick the best exponential smoothing method, it is always best to compare MSE. The method with the smallest MSE is always preferred.

69

C/3 Exams

14. If a variable appears to be insignificant in a regression, it should be dropped. 15. A high F statistic for your regression indicates that you have a good forecasting model. 16. A multiplicative decomposition method will usually be preferred to an additive decomposition method when the variability of the data increases through time. 17. The adjusted R2 is a much better measure of goodness-of-fit and forecasting accuracy because it penalizes you for data mining. 18. If the residuals from a regression show evidence of first-order serial correlation, then we can obtain better forecasts of the dependent variable by incorporating that serial dependence in our forecasting rule. 19. The F sub-block test in regression analysis is not useful if the linear regression slope coefficients change through time. 20. Econometric system models are always better forecasting tools than linear regression models because simultaneous equations models use more information about the process generating the data. 21. Correlation always implies causality. 22. Autocorrelation coefficients are not very useful for detecting the presence of an autoregressive process in time series data. 23. Differencing techniques can always make a time series stationary. Besides, we can never over-difference a time series. 24. Box-Jenkins forecasting methods do not work well in the presence of quadratic time trends in the data. 25. If there are patterns in your out-of-sample forecast errors, but your withinsample forecast errors appear random, it was just by chance that you obtained this nonrandomness out of sample. Part II: Thought Questions (Worth 70 points. Suggested time allotment:

3 4

hour.)

Please answer only one of the following questions. Long answers are not required, so please pay attention to your time. The situations described are hypothetical and the names have been changed to protect the offenders. 1. Recently you received a forecasting report that used a multiple regression model to construct the forecasts. The author of the report chose his model because “of the several hundred tried, it had by far the highest R-squared, adjusted R-squared and F statistic. It also had the lowest sum of squared residuals and a Durbin-Watson of 1.40 for n = 100 and k = 5.” The author also claims that on a reserved sample of the data the model has “very little bias and the MSE is only slightly worse than within sample.” Given only this information,

70

Part C. Additional materials for teaching forecasting (a) Comment on the author’s judgment in choosing his forecasting model. (b) Suggest a number of diagnostics or additional steps you would like to see the author take. 2. Recently the Bank of the GSB issued its forecasts of macroeconomic aggregate variables (e.g. GNP, the money supply, interest rates, etc.) for 1998. In their report they say that they have a ten equation model of the U.S. economy. Discuss: (a) What types of data (or models) they would have to have in order to predict the 1998 values. (b) What sorts of questions you as a consumer of these forecasts would ask in order to be more confident that their forecast of a 20 percent inflation rate was reasonable.

Part III: Forecasting analysis (Worth 130 points. hours).

Suggested time allotment: 1 14

(a) Look at the following retail sales data (see the attached pages) and write a short description of the data. (b) A colleague has proposed that “Box-Jenkins, time trend and seasonal regression, and decomposition appear to work best as forecasting tools.” Looking at the attached output that contains data and diagnostics on each of these best methods, evaluate the above statement. Comment on the strengths and weakness of the three approaches.

71

C/3 Exams

Quarterly Retail turnover

1700

o

o o

o o

o

1600

o

o o

1500

o

o

o o

o o

o

1400

o o

o

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o

1300

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o

o

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o

o

o

o

o

o

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2

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PACF

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14

2

Graphs of the retail turnover data.

4

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1500 50

-20

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Remainder

10

20

-50 0

Seasonal

150

1400

Trend-cycle

1500

1300

Data

1700

Part C. Additional materials for teaching forecasting

2

4

6

8

time

Decomposition plot of the retail turnover data.

Part of computer output for Seasonal decomposition: Seasonal indices -51.80256 -51.52647

189.64777 -86.31869

10

73

C/3 Exams

Part of computer output for Regression: Regression Coefficients:

(Intercept) t Q1 Q2 Q3

Value Std. Error 1271.2339 9.1093 5.0954 0.2961 36.8263 9.5463 35.6839 9.5417 275.2755 9.5463

t value 139.5535 17.2103 3.8577 3.7398 28.8360

Pr(>|t|) 0.0000 0.0000 0.0005 0.0007 0.0000

Residual standard error: 20.77 on 34 degrees of freedom Multiple R-Squared: 0.9771 F statistic: 363.2 on 4 and 34 degrees of freedom, the p-value is 0

Part of computer output for ARIMA modelling: Period(s) of Differencing = 1,4. Number of observations = 34 NOTE: The first 5 observations were eliminated by differencing.

Parameter MA1,1

Estimate 0.74282

Approx. Std Error 0.20763

T Ratio 3.58

Lag 4

Variance Estimate = 37259.2677 Std Error Estimate = 193.026599 AIC = 458.540303 SBC = 460.066663 Number of Residuals= 34 To Lag 6 12 18 24

Chi Square 2.05 5.92 13.49 20.71

Autocorrelations DF 5 11 17 23

Prob 0.842 0.878 0.703 0.599

-0.007 -0.005 -0.164 0.106 -0.083 -0.074 -0.100 0.055 0.191 -0.161 -0.004 -0.054 -0.118 -0.067 0.112 0.233 0.165 0.015 -0.118 -0.031 0.091 -0.225 0.019 -0.014

Model for variable TURNOVER No mean term in this model. Period(s) of Differencing = 1,4. Moving Average Factors Factor 1: 1 - 0.74282 B**(4)

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Part C. Additional materials for teaching forecasting

Residuals from regression o

40

o o o

o

o o

o

20 o

o o o

o

o

o

o

o o

o

0

o o

o o

o

o

o

o

-20

o

o

o

o

o

o

o

o

-40

o

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0.2

PACF

ACF

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0.2 0.0

-0.2

-0.2

-0.4 2

4

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14

2

4

6

Regression residuals for the retail turnover data.

8

10

12

14

75

C/3 Exams

Residuals from ARIMA model o

40 o

o

o o

20

o

o o o o o

o

o

o o

o

0

o

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o

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-20

o o

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-40

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0.2 PACF

ACF

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-0.2

0.0

-0.2

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4

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14

2

4

6

ARIMA residuals for the retail turnover data.

8

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12

14

D/Solutions to exercises Chapter 1: The forecasting perspective 1.1 Look for pragmatic applications in the real world. Note that there are no fixed answers in this problem. (a) Dow theory: There is an element of belief that past patterns will continue into the future. So first, look for the patterns (support and resistance levels) and then project them ahead for the market and individual stocks. This is a quantitative time series method. (b) Random walk theory: This is quantitative, and involves a time series rather than an explanatory approach. However, the forecasts are very simple because of the lack of any meaningful information. The best prediction of tomorrow’s closing price is today’s closing price. In other words, if we look at first differences of closing prices (i.e., today’s closing price minus yesterday’s closing price) there will be no pattern to discover. (c) Prices and earnings: Here instead of dealing with only one time series (i.e., the stock price series) we look at the relation between stock price and earnings per share to see if there is a relationship—maybe with a lag, maybe not. Therefore this is an explanatory approach to forecasting and would typically involve regression analysis. 1.2 Step 1: Problem definition This would involve understanding the nature of the individual product lines to be forecast. For example, are they high-demand products or specialty biscuits produced for individual clients? It is also important to learn who requires the forecasts and how they will be used. Are the forecasts to be used in scheduling production, or in inventory management, or for budgetary planning? Will the forecasts be studied by senior management, or by the production manager, or someone else? Have there been stock shortages so that demand has gone unsatisfied in the recent past? If so, would it be better to try to forecast demand rather than sales so that we can try to prevent this 76

Chapter 1: The forecasting perspective

77

happening again in the future? The forecaster will also need to learn whether the company requires one-off forecasts or whether the company is planning on introducing a new forecasting system. If the latter, are they intending it to be managed by their own employees and, if so, what software facilities do they have available and what forecasting expertise do they have in-house? Step 2: Gathering information It will be necessary to collect historical data on each of the product lines we wish to forecast. The company may be interested in forecasting each of the product lines for individual selling points. If so, it is important to check that there are sufficient data to allow reasonable forecasts to be obtained. For each variable the company wishes to forecast, at least a few years of data will be needed. There may be other variables which impact the biscuit sales, such as economic fluctuations, advertising campaigns, introduction of new product lines by a competitor, advertising campaigns of competitors, production difficulties. This information is best obtained by key personnel within the company. It will be necessary to conduct a range of discussions with relevant people to try to build an understanding of the market forces. If there are any relevant explanatory variables, these will need to be collected. Step 3: Preliminary (exploratory) analysis Each series of interest should be graphed and its features studied. Try to identify consistent patterns such as trend and seasonality. Check for outliers. Can they be explained? Do any of the explanatory variables appear to be strongly related to biscuit sales? Step 4: Choosing and fitting models A range of models will be fitted. These models will be chosen on the basis of the analysis in Step 3. Step 5: Using and evaluating a forecasting model Forecasts of each product line will be made using the best forecasting model identified in Step 4. These forecasts will be compared with expert in-house opinion and monitored over the period for which forecasts have been made. There will be work to be done in explaining how the forecasting models work to company personnel. There may even be substantial resistance to the introduction of a mathematical approach to forecasting. Some people may feel threatened. A period of education will probably be necessary. A review of the forecasting models should be planned.

78

Part D. Solutions to exercises

Chapter 2: Basic forecasting tools 2.1 (a) One simple answer: choose the mean temperature in June 1994 as the forecast for June 1995. That is, 17.2 ◦ C. (b) The time plot below shows clear seasonality with average temperature higher in summer.

20 18

Celsius

16 14 12 10 8 6 1994 Jan

1994 Feb

1994 May

1994 Jul

1994 Sep

1994 Nov

1995 Jan

1995 Mar

1995 May

Month

Exercise 2.1(b): Time plot of average monthly temperature in Paris (January 1994–May 1995).

2.2 (a) Rapidly increasing trend, little or no seasonality. (b) Seasonal pattern of period 24 (low when asleep); occasional peaks due to strenuous activity. (c) Seasonal pattern of period 7 with peaks at weekends; possibly also peaks during holiday periods such as Easter or Christmas. (d) Strong seasonality with a weekly pattern (low on weekends) and a yearly pattern. Peaks in either summer (air-conditioning) or winter (heating) or both depending on climate. Probably increasing trend with variation increasing with trend. 2.3 (a) Smooth series with several large jumps or direction changes; very large range of values; logs help stabilize variance. (b) Downward trend (or early level shift); cycles of about 15 days; outlier at day 8; no transformation necessary. (c) Cycles of about 9–10 years; large range and little variation at low points indicating transformation will help; logs help stabilize variance.

79

Chapter 2: Basic forecasting tools

(d) No clear trend; seasonality of period 12; high in July; no transformation necessary. (e) Initial trend; level shift end of 1982; seasonal period 4 (high in Q2 and Q3, low in Q1); no transformation necessary. 2.4 1-B, 2-A, 3-D, 4-C. The easiest approach to this question is to first identify D. Because it has a peak at lag 12, the time series must have a pattern of period 12. Therefore it is likely to be monthly. The slow decay in plot D shows the series has trend. The only series with both trend and seasonality of period 12 is Series 3. Next consider plot C which has a peak at lag 10. Obviously this cannot reflect a seasonal pattern since the only series remaining which is seasonal is series 2 and that has period 12. Series 4 is strongly cyclic with period approximately 10 and series 1 has no seasonal or strong cyclic patterns. Therefore C must correspond to series 4. Plot A shows a peak at lag 12 indicating seasonality of period 12. Therefore, it must correspond with series 2. That leaves plot B aligned with series 1. 2.5 (a) Mean Median MAD MSE St.dev.

X 52.99 52.60 3.11 15.94 4.14

Y 43.70 44.42 2.47 8.02 2.94

(b) Mean and median give a measure of center; MAD, MSE and St.dev. are measures of spread. (c) r = −0.660. See plot on next page.

(d) It is inappropriate to compute autocorrelations since there is no time component to these data. The data are from 14 different runners. (Autocorrelation would be appropriate if they were data from the same runner at 14 different times.) 2.6 (a) See plot on following page. (b) and (c) Notation:

Error 1 = Error 2 =

(actual demand) − (method 1 forecast) (actual demand) − (method 2 forecast)

80

46 44 42 40

Y: maximal aerobic capacity

48

Part D. Solutions to exercises

48

50

52

54

56

58

60

X: running times

200 180 160

Actual Forecast Method 1 Forecast Method 2

140

Demand

220

240

Exercise 2.5(c): Plot of running times versus maximal aerobic capacity.

5

10

15

Month

Exercise 2.6(a): Time plots of data and forecasts.

20

81

Chapter 2: Basic forecasting tools Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Actual 139 137 174 142 141 162 180 164 171 206 193 207 218 229 225 204 227 223 242 239

Analysis of errors (periods 1–20)

Method 1 157 145 140 162 149 144 156 172 167 169 193 193 202 213 223 224 211 221 222 235

Error 1 −18 −8 34 −20 −8 18 24 −8 4 37 0 14 16 16 2 −20 16 2 20 4

ME MAE MSE MPE MAPE Theil’s U

6.25 14.45 307.25 2.55 7.87 0.94

Method 2 170 162 157 173 164 158 166 179 177 180 199 202 211 221 232 235 225 232 233 243

Error 2 −31 −25 17 −31 −23 4 14 −15 −6 26 −6 5 7 8 −7 −31 2 −9 9 −4 −4.80 14.29 294.00 −3.61 8.24 0.85

On MAE and MSE, Method 2 is better than Method 1. On MAPE, Method 1 is better than Method 2. Note that this is different from the conclusion drawn in Section 4/2/3 where these two methods are compared. The difference is that we have used a different time period over which to compare the results. Holt’s method (Method 2) performs quite poorly at the start of the series. In Chapter 4, this period is excluded from the analysis of errors. 2.7 (a) Changes: −0.25, −0.26, 0.13, . . . , −0.09, −0.77. There are 78 observations in the DOWJONES.DAT file. Therefore there are 77 changes. (b) Average change: 0.1336. So the next 20 changes are each forecast to be 0.1336. (c) The last value of the series is 121.23. So the next 20 are forecast to be: ˆ 79 = 121.23 + 0.1336 = 121.36 X ˆ 80 = 121.36 + 0.1336 = 121.50 X ˆ 81 = 121.50 + 0.1336 = 121.63 X

etc.

82

Part D. Solutions to exercises ˆ 79+h = 121.23 + h(0.1336). In general, X

120 115 110

Dow Jones index

(d) See the plot below.

0

20

40

60

80

100

day

Exercise 2.7(d): Plot of Dow Jones index (DOWJONES.DAT)

(e) The average change is c = Xn + hc. Therefore,

1 n−1

Pn

t=2 (Xt

ˆ n+h = − Xt−1 ) and the forecasts are X n

ˆ n+h = Xn + h X = Xn +

1 X (Xt − Xt−1 ) n − 1 t=2

h (Xn − X1 ). n−1

This is a straight line with slope equal to (X n − X1 )/(n − 1). When h = 0, ˆ n+h = Xn and when h = −(n − 1), X ˆ n+h = X1 . Therefore, the line is drawn X between the first and last observations. 2.8 (a) See the plot on the next page. The variation when the production is low is much less than the variation in the series when the production is high. This indicates a transformation is required. (b) See the plot on the next page. (c) See the table on page 84.

83

Chapter 2: Basic forecasting tools

10 12 2

4

6

8

Forecast

1960

1970

1980

1990

1950

1960

1970

1980

1990

6

8

1950

4

Logarithms of vehicles

0

Vehicles (thousands)

•

Exercise 2.8 (a) and (b): Time plots of Japanese automobile production and the logarithms of Japanese automobile production.

84

Part D. Solutions to exercises

Year 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990

Data 11 20 29 32 38 39 50 70 69 111 182 188 263 482 814 991 1284 1702 1876 2286 3146 4086 4675 5289 5811 6294 7083 6552 6942 7842 8514 9269 9636 11043 11180 10732 11112 11465 12271 12260 12249 12700 13026

Log 2.40 3.00 3.37 3.47 3.64 3.66 3.91 4.25 4.23 4.71 5.20 5.24 5.57 6.18 6.70 6.90 7.16 7.44 7.54 7.73 8.05 8.32 8.45 8.57 8.67 8.75 8.87 8.79 8.85 8.97 9.05 9.13 9.17 9.31 9.32 9.28 9.32 9.35 9.41 9.41 9.41 9.45 9.47

Forecast

Error

Error2

|Error/Log|

2.40 3.00 3.37 3.47 3.64 3.66 3.91 4.25 4.23 4.71 5.20 5.24 5.57 6.18 6.70 6.90 7.16 7.44 7.54 7.73 8.05 8.32 8.45 8.57 8.67 8.75 8.87 8.79 8.85 8.97 9.05 9.13 9.17 9.31 9.32 9.28 9.32 9.35 9.41 9.41 9.41 9.45 9.47

0.598 0.372 0.098 0.172 0.026 0.249 0.337 −0.014 0.475 0.495 0.032 0.336 0.606 0.524 0.197 0.259 0.282 0.097 0.198 0.319 0.261 0.135 0.123 0.094 0.080 0.118 −0.078 0.058 0.122 0.082 0.085 0.039 0.136 0.012 −0.041 0.035 0.031 0.068 −0.001 −0.001 0.036 0.025

0.357 0.138 0.010 0.030 0.001 0.062 0.113 0.000 0.226 0.245 0.001 0.113 0.367 0.275 0.039 0.067 0.079 0.009 0.039 0.102 0.068 0.018 0.015 0.009 0.006 0.014 0.006 0.003 0.015 0.007 0.007 0.002 0.019 0.000 0.002 0.001 0.001 0.005 0.000 0.000 0.001 0.001

0.1996 0.1103 0.0284 0.0472 0.0071 0.0635 0.0792 0.0034 0.1009 0.0950 0.0062 0.0602 0.0981 0.0782 0.0285 0.0362 0.0379 0.0129 0.0256 0.0396 0.0314 0.0159 0.0144 0.0109 0.0091 0.0133 0.0089 0.0065 0.0136 0.0091 0.0093 0.0042 0.0146 0.0013 0.0044 0.0037 0.0033 0.0072 0.0001 0.0001 0.0038 0.0027

Exercise 2.8 (c) and (d).

Chapter 2: Basic forecasting tools

85

(d) MSE=0.059 (average of column headed Error 2 ) MAPE=3.21% (average of values in last column multiplied by 100). (e) See graph. Forecast is e9.47 = 13026. (f ) There are a large number of possible methods. One method, which is discussed in Chapter 5, is to consider only data after 1970 and use a straight line fitted through the original data (i.e. without taking logarithms). (g) The data for 1974 is lower than would be expected. If this information could be included in the forecasts, the MSE and MAPE would both be smaller because the forecast error in 1974 would be smaller.

86

Part D. Solutions to exercises

Chapter 3: Time series decomposition 3.1

3-MA 55.50 70.33 94.67 115.67 129.33 142.33 155.33 168.33 185.00 204.00 226.33 255.33 291.67 339.67 364.00

5-MA 70.33 81.50 91.60 111.40 128.40 142.60 155.60 170.00 187.40 206.00 230.00 261.40 298.80 316.50 339.67

3 × 3-MA 62.92 73.50 93.56 113.22 129.11 142.33 155.33 169.56 185.78 205.11 228.56 257.78 295.56 331.78 351.83

7-MA 81.50 91.60 99.83 107.57 126.00 141.86 156.71 172.86 189.29 210.71 236.86 268.29 283.00 298.80 316.50

5 × 5-MA 81.14 88.71 96.65 111.10 125.92 141.60 156.80 172.32 189.80 210.96 236.72 262.54 289.27 304.09 318.32

400

Y 42 69 100 115 132 141 154 171 180 204 228 247 291 337 391

100

200

300

3-MA 3x3 MA 5-MA 5x5 MA 7-MA

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Exercise 3.1: Smoothers fitted to the shipments data.

15

16

17

87

Chapter 3: Time series decomposition

The graph on the previous page shows the five smoothers. Because moving average smoothers are “flat” at the ends, the best smoother in this case is the one with the smallest number of terms, namely the 3-MA. 3.2

Tˆt =

=

1 3

1

5 (Yt−3 + Yt−2 + Yt−1 + Yt + Yt+1 ) + 15 (Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2 )  + 15 (Yt−1 + Yt + Yt+1 + Yt+2 + Yt+3 ) 1 2 1 1 1 15 Yt−3 + 15 Yt−2 + 5 Yt−1 + 5 Yt + 5 Yt+1

+

2 15 Yt+2

+

1 15 Yt+3 .

3.3 (a) The 4 MA is designed to eliminate seasonal variation because each quarter receives equal weight. The 2 MA is designed to center the estimated trend at the data points. The combination 2 × 4 MA also gives equal weight to each quarter. (b) Tˆt = 1 Yt−2 + 1 Yt−1 + 1 Yt + 1 Yt+1 + 1 Yt+2 . 8

4

4

4

8

3.4 (a) Use 2×4 MA to get trend. If the end-points are ignored, we obtain the following results. Data: Trend: Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 Q1 99 120 139 160 Q1 110.250 129.250 150.125 Q2 88 108 127 148 Q2 114.875 134.500 154.750 Q3 93 111 131 150 Q3 100.375 119.635 138.875 Q4 111 130 152 170 Q4 105.500 124.375 145.125 Data – trend: Y1 Y2 Q1 9.750 Q2 –6.875 Q3 –7.375 –8.625 Q4 5.500 5.625

Y3 9.750 –7.500 –8.875 6.875

Y4 9.875 –6.500

Ave 9.792 –7.042 –8.292 6.000

(b) Hence, the seasonal indices are: Sˆ1 = 9.8, Sˆ2 = −7.0, Sˆ3 = −8.3 and Sˆ4 = 6.0. The seasonal component consists of replications of these indices. (c) End points ignored. Other approaches are possible. 3.5 (a) See the top plot on the next page. There is clear trend which appears close to linear, and strong seasonality with a peak in August–October and a trough in January–March. (b) Calculations are given at the bottom of the next page. The decomposition plot is shown at the top of the next page.

88

Part D. Solutions to exercises

1200 110

104

70 80 90

96

100

remainder

seasonal

1000

1200

trend-cycle

800

data

1600

Plastic sales

2

Year J F Data 1 742 697 2 741 700 3 896 793 4 951 861 5 1030 1032 2×12 MA Trend 1 2 1000.5 1011.2 3 1117.4 1121.5 4 1208.7 1221.3 5 1374.8 1382.2 Ratios 1 2 74.1 69.2 3 80.2 70.7 4 78.7 70.5 5 74.9 74.7 Seasonal indices Ave 77.0 71.3

3

4

5

M

A

M

J

J

A

S

O

N

D

776 774 885 938 1126

898 932 1055 1109 1285

1030 1099 1204 1274 1468

1107 1223 1326 1422 1637

1165 1290 1303 1486 1611

1216 1349 1436 1555 1608

1208 1341 1473 1604 1528

1131 1296 1453 1600 1420

971 1066 1170 1403 1119

783 901 1023 1209 1013

1022.3 1130.7 1231.7 1381.2

1034.7 1142.7 1243.3 1370.6

1045.5 1153.6 1259.1 1351.2

75.7 78.3 76.2 81.5

90.1 92.3 89.2 93.8

105.1 104.4 101.2 108.6

116.0 114.0 111.4 123.0

77.9

91.3

104.8

116.1

977.0 977.0 977.1 978.4 982.7 990.4 1054.4 1065.8 1076.1 1084.6 1094.4 1103.9 1112.5 1163.0 1170.4 1175.5 1180.5 1185.0 1190.2 1197.1 1276.6 1287.6 1298.0 1313.0 1328.2 1343.6 1360.6 1331.2 119.2 121.0 111.3 115.4

124.5 125.4 122.2 119.8

123.6 123.6 124.8 122.2

115.6 118.4 122.6 120.5

98.8 96.6 98.3 104.4

79.1 81.0 85.5 88.9

116.8

122.9

123.6

119.3

99.5

83.6

Exercise 3.5(a) and (b): Multiplicative classical decomposition of plastic sales data.

89

Chapter 3: Time series decomposition

(c) The trend does appear almost linear except for a slight drop at the end. The seasonal pattern is as expected. Note that it does not make much difference whether these data are analyzed using a multiplicative decomposition or an additive decomposition. 3.6

Period t 61 62 63 64 65 66 67 68 69 70 71 72

Trend Tt 1433.96 1442.81 1451.66 1460.51 1469.36 1478.21 1487.06 1495.91 1504.76 1513.61 1522.46 1531.31

Seasonal St 76.96 71.27 77.91 91.34 104.83 116.09 116.76 122.94 123.55 119.28 99.53 83.59

Forecast Yˆt = Tt St /100 1103.6 1028.3 1131.0 1334.0 1540.3 1716.1 1736.3 1839.1 1859.1 1805.4 1515.3 1280.0

3.7 (a) See the top of the figure on the previous page. (b) The calculations are given below. Year Q1 Q2 Data 1 362 385 2 382 409 3 473 513 4 544 582 5 628 707 6 627 725 4×2 MA 1 2 399.3 413.3 3 478.3 499.6 4 557.9 580.6 5 654.8 670.6 6 689.4 708.1 Ratios 1 2 95.7 99.0 3 98.9 102.7 4 97.5 100.2 5 95.9 105.4 6 91.0 102.4 Seasonal indices Ave 95.8 101.9

Q3

Q4

432 498 582 681 773 854

341 387 474 557 592 661

382.5 430.4 519.4 601.5 674.9

388.0 454.8 536.9 627.6 677.0

112.9 115.7 112.1 113.2 114.5

87.9 85.1 88.3 88.7 87.4

113.7

87.5

90

600 105 90 95

96

98

100

remainder

seasonal

400

500

600

trend-cycle

700

400

data

Part D. Solutions to exercises

2

3

4

5

6

Exercise 3.7: Decomposition plot for exports from French company.

(c) Multiplicative decomposition seems appropriate here because the variance is increasing with the level of the series. The most interesting feature of the decomposition is that the trend has levelled off in the last year or so. Any forecast method should take this change in the trend into account. 3.8 (a) The top plot shows the original data followed by trend-cycle, seasonal and irregular components. The bottom plot shows the seasonal sub-series. (b) The trend-cycle is almost linear and the small seasonal component is very small compared to the trend-cycle. The seasonal pattern is difficult to see in time plot of original data. Values are high in March, September and December and low in January and August. For the last six years, the December peak and March peak have been almost constant. Before that, the December peak was growing and the March peak was dropping. There are several possible outliers in 1991.

91

Chapter 3: Time series decomposition

(c) The recession is seen by several negative outliers in the irregular component. This is also apparent in the data time plot. Note: the recession could be made part of the trend-cycle component by reducing the span of the loess smoother. 3.9 (a) and (b) Calculations are given below. Note that the seasonal indices are computed by averaging the de-trended values within each half-year. Data 1.09 1.07 1.10 1.06 1.08 1.03 1.04 1.01 1.03 0.96

2×2 MA Trend 1.0825 1.0825 1.0750 1.0625 1.0450 1.0300 1.0225 1.0075

Detrended Data -0.0125 0.0175 -0.0150 0.0175 -0.0150 0.0100 -0.0125 0.0225

Seasonal Component 0.017 -0.014 0.017 -0.014 0.017 -0.014 0.017 -0.014 0.017 -0.014

Seasonal Adjusted Data 1.073 1.084 1.083 1.074 1.063 1.044 1.023 1.024 1.013 0.974

(c) With more data, we could take moving averages of the detrended values for each half-year rather than a simple average. This would result in a seasonal component which changed over time.

92

Part D. Solutions to exercises

Chapter 4: Exponential smoothing methods 4.1 Period Data MA(3) SES(α = 0.7) ˆ t Yt Et Et Yt Yˆt 1974 1 1 5.4 2 2 5.3 5.40 -0.10 3 3 5.3 5.33 -0.03 4 4 5.6 5.33 0.27 5.31 0.29 1975 1 5 6.9 5.40 1.50 5.51 1.39 2 6 7.2 5.93 1.27 6.48 0.72 3 7 7.2 6.57 0.63 6.99 0.21 4 8 7.10 7.14 Accuracy statistics from period 4 through 7 ME 0.92 0.65 MAE 0.92 0.65 MAPE 13.22 9.56 MSE 1.08 0.64 Theil’s U 1.40 1.14 Theil’s U statistic suggests that the na¨ıve (or last value) method is better than either of these. If SES is used with an optimal value of α chosen, then α = 1 is selected. This is equivalent to the na¨ıve method. Note different packages may give slightly different results for SES depending on how they initialize the method. Some packages will also allow α > 1. 4.2 (a) Forecasts for May 1992 Method Forecast MSE

MA(3) 24.0 1484.3

MA(5) 48.6 1031.2

MA(7) 55.6 757.5

α = 0.3 41.9 1211.80

α = 0.5 33.1 1193.98

MA(9) 51.7 860.8

MA(11) 53.1 1313.8

(b) Forecasts for May 1992 Method Forecast MSE

α = 0.1 45.5 1421.35

α = 0.7 29.1 1225.40

α = 0.9 28.7 1298.49

(c) Of these forecasting methods, the best MA(k) method has k = 7 and the best SES method has α = 0.5. However, it should be noted that the MSE values for the MA methods are taken over different periods. For example, the MSE for the MA(7) method is computed only over 9 observations because it is not possible to compute an MA(7) forecast for the first seven observations. So the MSE

93

Chapter 4: Exponential smoothing methods

values are not strictly comparable for the MA forecasts. It would be better to use a holdout sample but there are too few data. 4.3 Optimizing α for SES over the period 3 through 10: α MAPE MSE

0.1 65.60 79.34

0.2 53.46 47.24

0.3 44.43 29.95

0.4 37.60 20.10

0.5 32.32 14.17

0.6 28.16 10.41

0.7 24.82 7.91

0.8 22.08 6.17

0.9 19.80 4.92

1.0 17.86 4.00

The optimal value is α = 1. With Holt’s method, any combination of α and β will give MAPE=0. This is so because the differences between successive values of (4.13) are always going to be zero with this errorless series. Using α = 1 for SES and α = 0.5 and β = 0.5 for Holt’s method gives the following results. Data Yt 2 4 6 8 10 12 14 16 18 20

SES ˆ Yt Et

Holt’s ˆ Yt Et

2 4 6 8 10 12 14 16 18

4 6 8 10 12 14 16 18 20

2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0

(a) Clearly Holt’s method is better as it allows for the trend in the data. (b) For SES, α = 1. Because of the trend, the forecasts will always lag behind the actual values so that the forecast errors will always be at least 2. Choosing α = 1 makes the forecast errors as small as possible for SES. (c) See above. 4.4 (a) (b) and (c) See the table on the following page. (d) There’s not much to choose between these methods. They are both bad! Look at Theil’s U values for instance. The last value method over the same period (13–28) gives MSE=6.0, MAPE=2.05 and Theil’s U=1.0.

94

Part D. Solutions to exercises

Period Data Forecast Errors t Yt MA(12) Et 1 108 2 108 3 110 4 106 5 108 6 108 7 105 8 100 9 97 10 95 11 95 12 92 13 95 102.67 -7.67 14 95 101.58 -6.58 15 98 100.50 -2.50 16 97 99.50 -2.50 17 101 98.75 2.25 18 104 98.17 5.83 19 101 97.83 3.17 20 99 97.50 1.50 21 95 97.42 -2.42 22 95 97.25 -2.25 23 96 97.25 -1.25 24 96 97.33 -1.33 25 97 97.67 -0.67 26 98 97.83 0.17 27 94 98.08 -4.08 28 92 97.75 -5.75 29 97.33 Accuracy criteria: periods 13–28 ME -1.51 MAE 3.12 MSE 14.40 MAPE 3.23 Theil’s U 1.58

Forecast MA(6)

Errors Et

108.00 107.50 106.17 104.00 102.17 100.00 97.33 95.67 94.83 95.00 95.33 96.33 98.33 99.33 100.00 99.50 99.17 98.33 97.00 96.33 96.17 96.00 95.50

-3.00 -7.50 -9.17 -9.00 -7.17 -8.00 -2.33 -0.67 3.17 2.00 5.67 7.67 2.67 -0.33 -5.00 -4.50 -3.17 -2.33 0.00 1.67 -2.17 -4.00

Calculations for Exercise 4.4

-0.10 2.96 12.64 3.03 1.45

95

Chapter 4: Exponential smoothing methods 4.5 (a) (b) and (c)

Smoothing parameters Forecast Day Forecast Day Forecast Day Forecast Day MAE MSE MAPE Theil’s U

31 32 33 34

Paperbacks SES Holt α = 0.213 α = 0.335 β = 0.453 210.15 224.24 210.15 231.79 210.15 239.33 210.15 246.88 29.6 33.9 1252.2 1701.7 17.1 18.4 0.68 0.92

Hardcovers SES Holt α = 0.347 β = 0.437 β = 0.157 240.38 250.73 240.38 254.63 240.38 258.53 240.38 262.43 27.3 28.6 1060.6 1273.0 13.5 14.3 0.81 0.92

For both series, SES forecasting is performing better than Holt’s method. (d) SES forecasts are “flat” and Holt’s forecasts show a linear trend. Both series show an upward linear trend and we would expect the forecasts to reflect that trend. Perhaps an out-of-sample analysis would give a better indication of the merits of the two methods. (e) The autocorrelation functions of the forecast errors in each case are plotted on the next page. In each case, there is no noticeable pattern. Only a few spikes are just outside the critical bounds which is expected. 4.6 Here is a complete run for one set of values (β = 0.1 and α 1 = 0.1). Note that in this program we have chosen to make the first three values of α be equal to the starting value. This is not crucial, but it does make a difference. t 1 2 3 4 5 6 7 8 9 10 11 12

Yt 200.0 135.0 195.0 197.5 310.0 175.0 155.0 130.0 220.0 277.5 235.0

Ft 200.00 200.00 193.50 193.65 197.31 289.74 241.00 224.08 185.43 204.62 234.77 234.81

Et 0.00 -65.00 1.50 3.85 112.69 -114.74 -86.00 -94.08 34.57 72.88 0.23

At 0.00 -6.50 -5.70 -4.74 7.00 -5.18 -13.26 -21.34 -15.75 -6.89 -6.17

Mt 0.00 6.50 6.00 5.79 16.48 26.30 32.27 38.45 38.06 41.55 37.41

αt 0.100 0.100 0.100 0.950 0.820 0.425 0.197 0.411 0.555 0.414 0.166 0.165

96

Part D. Solutions to exercises Holt paperbacks

0.2 -0.2 -0.4

-0.4

6

8

10

12

14

2

4

6

8

10

Lag

Lag

SES hardbacks

Holt hardbacks

12

14

12

14

0.2 0.0 -0.2 -0.4

-0.4

-0.2

0.0

ACF

0.2

0.4

4

0.4

2

ACF

0.0

ACF

0.0 -0.2

ACF

0.2

0.4

0.4

SES paperbacks

2

4

6

8

10

12

14

2

4

Lag

6

8

10

Lag

Exercise 4.5 (e): Autocorrelation functions of forecast errors.

For other combinations of values for β and starting values for α, here is what the final α value is: α = 0.1 α = 0.2 α = 0.3

β = 0.1 0.165 0.058 0.140

β = 0.3 0.327 0.454 0.618

β = 0.5 0.732 0.783 0.797

β = 0.7 0.143 0.133 0.133

The time series is not very long and therefore the results are somewhat fickle. In any event, it is clear that the β value and the starting values for α have a profound effect on the final value of α. 4.7 Holt-Winters’ method is best because the data are seasonal. The variation increases with the level, so we use Holt-Winters’ multiplicative method. The optimal smoothing parameters (giving smallest MSE) are a = 0.479, b = 0.00 and c = 1.00. These give the following forecasts (read left to right):

97

Chapter 4: Exponential smoothing methods 309.1 327.5 345.9 364.2

312.1 330.5 348.9 367.3

315.2 333.6 352.0 370.4

318.3 336.7 355.0 373.4

321.3 339.7 358.1 376.5

324.4 342.8 361.2 379.6

4.8 First choose any values for the three parameters. Here we have used α = β = γ = 0.1. Different values will choose different initial values. Our program uses the method described in the textbook and gave the following results: ME -240.5

MAE 240.5

MSE 62469.9

MAPE 37.5

r1 0.70

Theil’s U 2.6

Now compare with the optimal values: α = 0.917, β = 0.234 and γ = 0.000. Using the same initialization, we obtain the results in Table 4-11, namely ME -9.46

MAE 24.00

MSE 824.75

MAPE 3.75

r1 0.17

Theil’s U 0.29

98

Part D. Solutions to exercises

Chapter 5: Simple regression 5.1 (a) −0.7, almost 1, 0.2

(b) False. The correlation is negative. So below-average values of one are associated with above-average values of the other variable. (c) Wages have been increasing over time due to inflation. At the same time, population has been increasing and consequently, new houses need to be built. So, because they are both increasing with time, they are positively correlated.

(d) There are many factors affecting unemployment and it is simplistic to draw a causal connection with inflation on the basis of correlation. As in the previous question, both vary with time and the correlation could be induced by their time trends. Or they could both be related to some third variable such as business confidence or government spending. (e) The older people in the survey had much less opportunity for education than the younger people. This negative correlation is caused by the increase in education levels over time. ¯ = 5, Y¯ = 25, P(Xi − X) ¯ 2 = 20, P(Xi − X)(Y ¯ i − Y¯ ) = 78. So b = 78/20 = 3.9 5.2 (a) X and a = 25 − 3.9(5) = 5.5. Hence, the regression line is Yˆ = 5.5 + 3.9X. p P P ˆ (b) (Yi − Y¯ )2 = 304.20, (Yi − Yˆi )2 = 25.80, σe = 25.80/(5 − 2) = 2.933. So F =

304.20/(2 − 1) = 35.4. 25.80/(5 − 2)

This has (2−1) = 1 df for the numerator and (5−2) = 3 df for the denominator. From Table C in Appendix III, the P -value is slightly smaller than 0.010. (Using a computer, it is 0.0095.) Standard errors: q s.e.(a) = (2.93) 15 + 25 20 = 3.53 q 1 s.e.(b) = (2.93) 20 = 0.656.

On 3 df, t∗ = 3.18 for a 95% confidence interval. Hence 95% intervals are α: β:

5.500 ± 3.18(3.53) = [−5.7, 16.7] 3.900 ± 3.18(0.656) = [1.8, 6.0]

99

Chapter 5: Simple regression

Output from Minitab for Exercise 5.2: Regression Analysis The regression equation is Y = 5.50 + 3.90X Predictor Constant X

Coef 5.500 3.9000

S = 2.933

StDev 3.531 0.6557

R-Sq = 92.2%

T 1.56 5.95

P 0.217 0.010

R-Sq(adj) = 89.6%

Analysis of Variance Source Regression Error Total

DF 1 3 4

SS 304.20 25.80 330.00

MS 304.20 8.60

F 35.37

P 0.010

(c) R2 = 0.922, rXY = rY Yˆ = 0.960. (d) The line through the middle of the graph is the line of best fit. The 95% prediction interval shown is the interval which would contain the Y value with probability 0.95 if the X value was 17. The 80% prediction interval shown is the interval which would contain the Y value with probability 0.80 if the X value was 26. The dotted line at the boundary of the light shaded region gives the ends of all the 95% prediction intervals. The dotted line at the boundary of the dark shaded region gives the ends of all the 80% prediction intervals. 5.3 (a) See the plot on the next page and the Minitab output on page 101. The straight line is Yˆ = 0.46 + 0.22X. (b) See the plot on the next page. The residuals may show a slight curvature (Λ shaped). However, the curvature is not strong and the fitted model appears reasonable. (c) R2 = 90.2%. Therefore, 90.2% of the variation in melanoma rates is explained by the linear regression. (d) From the Minitab output: Prediction: 9.286. Prediction interval: (6.749, 11.823)

100

6 2

4

melanoma

8

10

Part D. Solutions to exercises

10

20

30

40

ozone

0.5 0.0 -1.0

-0.5

Residuals

1.0

1.5

Exercise 5.3(a): Scatterplot of melanoma rate against ozone depletion.

10

20

30

40

ozone

Exercise 5.3(b): Scatterplot of residuals from the linear regression.

101

Chapter 5: Simple regression

Output using Minitab for Exercise 5.3: MTB > Regress ’Melanoma’ 1 ’Ozone’; SUBC> predict 40. The regression equation is Melanoma = 0.460 + 0.221 Ozone Predictor Constant Ozone

Coef 0.4598 0.22065

S = 0.9947

StDev 0.6258 0.02426

R-Sq = 90.2%

T 0.73 9.09

P 0.481 0.000

R-Sq(adj) = 89.1%

Analysis of Variance Source Regression Error Total Fit 9.286

DF 1 9 10

StDev Fit 0.517

SS 81.822 8.905 90.727

(

MS 81.822 0.989

95.0% CI 8.116, 10.456)

F 82.70

(

P 0.000

95.0% PI 6.749, 11.823)

Note that it is the prediction interval (PI) we want here. Minitab also gives the confidence interval (CI) for the line at this point, something we have not covered in the book. (e) This analysis has assumed that the susceptibility to melanoma among people living in the various locations is constant. This is unlikely to be true due to the diversity of racial mix and climate over the locations. Apart from ozone depletion, melanoma will be affected by skin type, climate, culture (e.g. is sun-baking encouraged?), diet, etc. 5.4 (a) See plot on the next page and computer output on page 103. (b) Coefficients: a = 4.184, b = 0.9431. Only b is significant, showing the relationship is significant. (We could refit the model without the intercept term.) (c) If X = 80, Yˆ = 4.184 + 0.9431(80) = 79.63. Standard error of forecast is 1.88 (from computer output).

102

30

40

50

60

Production rating

70

80

90

Part D. Solutions to exercises

20

40

60

80

Manual dexterity

100

Exercise 5.4(a): Scatterplot of production rating against manual dexterity test scores.

80

•

•

•

•

60

• • • •

40

• • •

•

•

•

•

• •

•

20

Production rating

• •

20

40

60

80

Manual dexterity

Exercise 5.4(e): 95% prediction intervals for production rating.

100

103

Chapter 5: Simple regression

Output using Minitab for Exercise 5.4 MTB > Regress ’Y’ 1 ’X’; SUBC> Predict ’newX’. The regression equation is Y = 4.18 + 0.943 X Predictor Constant X

Coef 4.184 0.94306

S = 5.126

StDev 3.476 0.05961

R-Sq = 93.3%

T 1.20 15.82

P 0.244 0.000

R-Sq(adj) = 92.9%

Analysis of Variance Source Regression Error Total Fit 23.05 41.91 60.77 79.63

DF 1 18 19

StDev Fit 2.38 1.46 1.18 1.88

SS 6576.8 473.0 7049.8

( ( ( (

95.0% CI 18.04, 38.85, 58.28, 75.68,

MS 6576.8 26.3

28.05) 44.97) 63.26) 83.58)

F 250.29

( ( ( (

95.0% PI 11.17, 30.71, 49.71, 68.16,

P 0.000

34.92) 53.10) 71.82) 91.10)

(d) For confidence and prediction intervals, use Table B with 18 df. 95% CI for β is 0.94306 ± 2.10(0.05961) = [0.82, 1.07].

(e) See output. Again it is the prediction interval (PI) we want here, not the confidence interval (CI). The prediction intervals are shown in the plot on the previous page.

5.5 (a) See the plot on the following page. The straight line regression model is Yˆ = 20.2−0.145X where Y = electricity consumption and X = temperature. There is a negative relationship because heating is used for lower temperatures, but there is no need to use heating for the higher temperatures. The temperatures are not sufficiently high to warrant the use of air conditioning. Hence, the electricity consumption is higher when the temperature is lower.

104

19 18 17 16

Electricity consumption (Mwh)

Part D. Solutions to exercises

10

15

20

25

30

Temperature

Exercise 5.5(a): Electricity consumption (Mwh) plotted against temperature (degrees Celsius).

-1

0

Residuals

1

2

Possible outlier

10

15

20

25

30

Temperature

Exercise 5.5(c): Residual plot for the straight line regression of electricity consumption against temperature.

105

Chapter 5: Simple regression (b) r = −0.791

(c) See the plot on the previous page. Apart from the possible outlier, the model appears to be adequate. There are no highly influential observations. (d) If X = 10, Yˆ = 20.2 − 0.145(10) = 18.75. If X = 35, Yˆ = 20.2 − 0.145(35) = 15.12. The first of these predictions seems reasonable. The second is unlikely. Note that X = 35 is outside the range of the data making prediction dangerous. For temperatures above about 20 ◦ C, it is unlikely electricity consumption would continue to fall because no heating would be used. Instead, at high temperatures (such as X = 35◦ C), electricity consumption is likely to increase again due to the use of air-conditioning. 5.6 (a) When H = 130 and W = 45, r = 0.553. (b) When H = 40 and W = 150, r = −0.001.

(c) The following table shows the influence of outliers at various positions. H 129 128 122 112 99 83 65 44 22 0

W 0 22 44 64 83 99 112 122 128 129

r -0.393 0.032 0.527 0.773 0.846 0.810 0.627 0.151 -0.365 -0.624

The point about all this is that an outlier (and skewness in general) can seriously affect the correlation coefficient. It is a good idea to look at the scatterplot before computing any correlation. 5.7 (a) See the plot on the next page. The winning time has been decreasing with year. There is an outlier in 1896. (b) The fitted line is Yˆ = 196−0.0768X where X denotes the year of the Olympics. Therefore the winning time has been decreasing an average 0.0768 seconds per year. (c) The residuals are plotted on the next page. The residuals show random scatter about 0 with only one usual point (the outlier in 1896). But note that the last five residuals are positive. This suggests that the straight line is “levelling out”—the winning time is decreasing at a slower rate now than it was earlier.

106

50 48 44

46

winning.time

52

54

Part D. Solutions to exercises

1900

1920

1940

1960

1980

2000

year

1 0 -1

fit$resid

2

3

Exercise 5.7(a): Scatterplot of winning times against year.

1900

1920

1940

1960

1980

year

Exercise 5.7(c): Residual plot for linear regression model of winning times.

2000

107

Chapter 5: Simple regression (d) The predicted winning time in the 2000 Olympics is Yˆ = 196 − 0.0768(2000) = 42.50 seconds.

This would smash the world record. But given the previous five results (with positive residuals), it would seem more likely that the actual winning time would be higher. A prediction interval is 42.50 ± 2.0796(1.1762) = 42.50 ± 2.45 = [40.05, 44.95]. 5.8 (a) There is strong seasonality with peaks in November and December and a trough in January. The surfing festival shows as a smaller peak in March from 1988. The variation in the series is increasing with the level and there is a strong positive trend due to sales growth. (b) Logarithms are necessary to stabilize the variance so it does not increase with the level of the series. (c) See the plot on the next page and the computer output on page 109. The fitted line is Yˆ = −526.57 + 0.2706X where X is the year and Y is the logged annual sales. (d)

Yˆ = −526.57 + 0.2706(1994) = 12.98 Yˆ = −526.57 + 0.2706(1995) = 13.25 Yˆ = −526.57 + 0.2706(1996) = 13.52

X = 1994 : X = 1995 : X = 1996 :

Prediction intervals (from computer output): X = 1994 :

[12.57, 13.40]

X = 1995 :

[12.80, 13.71]

X = 1996 :

[13.03, 14.02]

(e) We transform the forecasts and intervals with the exponential function: Total annual sales for 1994

exp(12.98) = $434, 443

Total annual sales for 1995

exp(13.25) = $569, 439

Total annual sales for 1996

exp(13.52) = $746, 383

Prediction intervals: X = 1994 :

[e12.57 , e13.40 ] = [286673, 658385]

X = 1995 :

[e12.80 , e13.71 ] = [361994, 895764]

X = 1996 :

[e13.03 , e14.02 ] = [455060, 1224208]

108

60000 40000 20000 0

1988

1990

1992

1994

11.5

12.0

12.5

Exercise 5.8(a): Time plot of sales figures.

Log Total annual sales

Sales

80000

Part D. Solutions to exercises

1987

1988

1989

1990

1991

1992

1993

Exercise 5.8(c): Regression line fitted to the logged sales data.

109

Chapter 5: Simple regression

Output using Minitab for Exercise 5.8: MTB > regress ’Log Sales’ 1 ’Year’; SUBC> predict ’new years’; The regression equation is Log Sales = - 527 + 0.271 Year Predictor Constant Year

Coef -526.57 0.27059

S = 0.1235

StDev 46.44 0.02334

R-Sq = 96.4%

T -11.34 11.60

P 0.000 0.000

R-Sq(adj) = 95.7%

Analysis of Variance Source Regression Error Total

DF 1 5 6

SS 2.0501 0.0762 2.1263

Fit StDev Fit 12.9818 0.1044 13.2524 0.1257 13.5230 0.1476 X denotes a row with

( ( ( X

MS 2.0501 0.0152

F 134.45

95.0% CI 12.7135, 13.2502) ( 12.9293, 13.5755) ( 13.1435, 13.9025) ( values away from the

P 0.000

95.0% PI 12.5661, 13.3975) 12.7994, 13.7054) X 13.0282, 14.0178) X center

These prediction intervals are very wide because we are only using annual totals in making these predictions. A more accurate method would be to fit a model to the monthly data allowing for the seasonal patterns. This is discussed in Chapter 7. (f ) One way would be to calculate the proportion of sales for each month compared to the total sales for that year. Averaging these proportions will give a rough guide as to how to split the annual totals into 12 monthly totals.

110

10 8 4

6

Percentage mortality

12

14

Part D. Solutions to exercises

0

20

40

60

80

100

Percentage Type A Birds

Exercise 5.9(a): Scatterplot of percentage mortality against percentage of Type A birds.

5.9 (a) The plot is shown above. The fitted line is Yˆ = 4.38 + 0.0154X where X = percentage of type A birds and Y = percentage mortality. (b) From the computer output: Predictor Constant % Type A

Coef 4.3817 0.015432

StDev 0.6848 0.007672

T 6.40 2.01

P 0.000 0.046

So the t-test is significant (since P < 0.05). A 95% confidence interval for the slope is 0.01543 ± 1.976(0.007672) = 0.01543 ± 0.01516 = [0.003, 0.031]. This suggests that the Type A birds have a higher mortality than the Type B birds, the opposite to what the farmers claim. (c) For a farmer using all Type A birds, X = 100. So Yˆ = 4.38 + 0.0154(100) = 5.92%. For a farmer using all Type B birds, X = 0. So Yˆ = 4.38%. Prediction intervals for these are [2.363, 9.487] and [0.587, 8.177] respectively. (d) R2 = 2.6. So only 2.6% of the variation in mortality is due to bird type.

111

140

Chapter 5: Simple regression

100 80 40

60

consumption

120

Model 1 Model 2

40

60

80

100

price

Exercise 5.10(b): Scatterplot of gas consumption against price.

(e) This information suggests that heat may be a lurking variable. If Type A birds are being used more in summer and the mortality is higher in summer, than the increased mortality of Type A birds may be due to the summer rather than the bird type. A proper randomized experiment would need to be done to properly assess whether bird type is having an effect here. 5.10 (a) Cross sectional data. There is no time component. (b) See the plot above. (c) When the price is higher, the consumption may be lower due to the pressure of increased cost. Therefore, we would expect b 1 < b2 < 0. (d) Model 1: First take logarithms of Y i , then use simple linear regression to obtain a = 5.10,

b = −0.0153,

σe2 = 0.0735.

Model 2: Split data into two groups. Fit each group separately using simple linear regression to obtain a1 = 221,

b1 = −2.91

and

a2 = 84.8,

Using the equation given in the question, we obtain σe2 = 2913.7/16 = 182.06. The fitted lines are shown on the graph above.

b2 = −0.447.

112

Part D. Solutions to exercises (e) Model 1: R2 = rY2 Yˆ = 0.721. 1

Model 2: R2 = rY2 Yˆ = 0.859. The second model is better with higher R 2 value. 2 The residual plots are given on the following page. Again, the second model is much better showing random scatter about zero. The first model show pattern in the residuals.

(f ) The graph on page 114 shows a local linear regression through the data. The fitted curve resembles the fitted lines for model 2. This suggests that model 2 is a reasonable model for the data. However, our approach has also meant the two lines do not join at X = 60. A better model would force them to join. This means the parameters must be restricted which makes the estimation much harder. (g) and (h) Using model 2, forecasts are obtained by  220.9 − 2.906X when X ≤ 60 ˆ Y = 84.8 − 0.447X when X > 60. and standard errors are obtained from (5.19): r √ 1 (X − 63)2 s.e.(Yˆ ) = 182.06 1 + + . 20 10672.11 The 95% PI are obtained using Yˆ ± t∗ (s.e.) where t∗ = 2.12 (from Table B with 16 df). Hence, we obtain the following values. X 40 60 80 100 120

Yˆ 104.67 46.55 49.03 40.09 31.15

s.e. 14.15 13.83 14.00 14.65 15.70

[ 95% PI ] [74.7 , 134.7] [17.2 , 75.9] [19.3 , 78.7] [ 9.0 , 71.1] [ -2.1 , 64.4]

For example, at a price of 80c, the gas consumption will lie between 19.3 and 78.7 for 95% of towns.

113

20 0 -20

Residuals model 1

40

Chapter 5: Simple regression

40

60

80

100

80

100

10 0 -10 -20

Residuals model 2

20

Price

40

60 Price

Exercise 5.10(e): Residual plots for the two models.

114

40

60

80

Consumption

100

120

140

Part D. Solutions to exercises

40

60

80

100

Price

80 60 40 20 0

Consumption

100

120

140

Exercise 5.10(f ): Local linear regression through the gas consumption data. The fitted line suggests that model 2 is more appropriate.

40

60

80

100

Price

Exercise 5.10(h): 95% prediction intervals for gas consumption.

120

115

Chapter 6: Multiple regression

Chapter 6: Multiple regression 6.1 (a) df for numerator = k and for denominator = n − k − 1 where n = number of observations and k = number of explanatory variables. Here, k = 16 so that n − 16 − 1 = 30. Hence, n = 30 + 16 + 1 = 47. (b)

¯ 2 = 1 − (1 − R2 ) n−1 = 1 − (1 − 0.943) R n−k−1

47 − 1 = 0.913. 48 − 16 − 1

(c) F = 31.04 on (17,30) df. From Table C in Appendix III, the P -value is much smaller than 0.01. So the regression is highly significant. (d) The coefficients should be compared with a t 30 distribution. From Table B in Appendix III, any value greater than 2.04 in absolute value will be significant at the 5% level. So the constant and variables 4, 8, 12, 13, 14, 15 and 17 are significant in the presence of other explanatory variables. Note that the significance level of 5% is arbitrary. There is no reason why some other significance level (e.g. 2%) could not be used. (e) The next stage would be to reduce the number of variables in the model by removing some of the least significant variables and re-fitting the model. 6.2 (a) The fitted model is Cˆ = 273.93−5.68P +0.034P 2 . For this model, R2 = 0.8315. [Recall: in exercise 5.6, model 1 had R 2 = 0.721 and model 2 had R2 = 0.859.] ¯ 2 values for each model are: So the R Model 1 Model 2 Model 3

¯ 2 = 1 − (1 − 0.721) n−1 = 1 − (1 − 0.721) 46 = 0.715. R n−k−1 45 46 2 n−1 ¯ = 1 − (1 − 0.859) R n−k−1 = 1 − (1 − 0.859) 43 = 0.849. ¯ 2 = 1 − (1 − 0.832) n−1 = 1 − (1 − 0.832) 46 = 0.824. R n−k−1 44

These values show that model 2 is the best model, followed by model 3. The t values for the coefficients are: Model 1 Model 2 Model 3

α : t = 10.22 α1 : t = 10.33 β0 : t = 8.83

β : t = −5.47 β1 : t = −6.61 β1 : t = −5.62

α2 : t = 4.11 β2 : t = 4.57

β2 : t = −1.99

Of these, only β2 from model 2 is not significantly different from zero. This suggests that a better model would be to allow the second part of model 2 to be a constant rather than a linear function. (b) From the computer output the following 95% prediction intervals are obtained.

116

Part D. Solutions to exercises

Output using Minitab for Exercise 6.2: MTB > regress ’C’ 2 ’P’ ’Psq’; SUBC> predict ’newP’ ’newPsq’. The regression equation is C = 274 - 5.68 P + 0.0339 Psq Predictor Constant P Psq

Coef 273.93 -5.676 0.033904

S = 14.37

StDev 31.03 1.009 0.007412

R-Sq = 83.2%

Analysis of Variance Source DF SS Regression 2 17327.0 Error 17 3511.0 Total 19 20838.0 Source P Psq

DF 1 1

T 8.83 -5.62 4.57

P 0.000 0.000 0.000

R-Sq(adj) = 81.2%

MS 8663.5 206.5

F 41.95

P 0.000

Seq SS 13005.7 4321.3

Fit StDev Fit 173.97 14.29 101.14 4.77 55.43 4.91 36.85 4.95 45.38 7.14 81.04 18.49 X denotes a row with XX denotes a row with

95.0% CI 95.0% PI ( 143.82, 204.13) ( 131.21, 216.74) XX ( 91.08, 111.21) ( 69.19, 133.10) ( 45.07, 65.80) ( 23.38, 87.48) ( 26.40, 47.29) ( 4.77, 68.92) ( 30.31, 60.46) ( 11.52, 79.25) ( 42.02, 120.06) ( 31.62, 130.46) XX X values away from the center very extreme X values

117

100 0

50

consumption

150

200

Chapter 6: Multiple regression

20

40

60

80

100

120

price

Exercise 6.2: Quadratic regression of gas consumption against price. 95% prediction intervals shown.

P 20 40 60 80 100 120

Cˆ 173.97 101.14 55.43 36.85 45.38 81.04

[ [ [ [ [ [ [

95% PI 131.21 , 216.74 69.19 , 133.10 23.38 , 87.48 4.77 , 68.92 11.52 , 79.25 31.62 , 130.46

] ] ] ] ] ] ]

It is clear from the plot that it is dangerous predicting outside the observed price range. In this case, the predictions at P = 20 and P = 120 are almost certainly wrong. Predicting outside the range of the explanatory variable is always dangerous, but much more so when a quadratic (or higher-order polynomial) is used. (c) rP P 2 = 0.990. If we were to use P , P 2 and P 3 , the correlations among these explanatory variables would be very high and we would have a serious multicollinearity problem on our hands. The coefficients estimates would be unstable (i.e. have large standard errors). Multicollinearity will often be a problem with polynomial regression.

118

Part D. Solutions to exercises

6.3 (a) From Table 6-15, we obtain the following values Period 54 55 56 57 58 59

Actual 4.646 1.060 -0.758 4.702 1.878 6.620

Forecast 1.863 1.221 0.114 2.779 1.959 5.789

Analysis of errors: periods 54 through 59. ME 0.74

MAE 1.11

MSE 2.15

MPE 34.82

MAPE 41.32

ACF1 -0.35

Theil’s U 0.34

Strictly speaking, we should not compute relative measures when the data cross the zero line (i.e., when there are positive and negative values) because relative measures will “blow up” if divided by zero. (b) and (c) Optimizing the coefficients for Holt’s method will give better forecasts. Another approach is to use a simple MA forecast. An MA(2) forecast actually works better than Holt’s method for both series. Other approaches are also possible. Calculate accuracy statistics for your forecasts and compare them with the forecasts in Table 6-14. 6.4 (a) The fitted equation is Yˆ = 73.40 + 1.52X1 + 0.38X2 − 0.27X3 . 95% confidence intervals for the parameters are calculated using a t 6 distribution. So the multiplier is 2.45: 73.40 ± 2.45(14.687) = [37.46, 109.3] 1.52 ± 2.45(0.1295) = [1.20, 1.84]

0.38 ± 2.45(0.1941) = [−0.09, 0.85]

−0.27 ± 2.45(0.1841) = [−0.72, 0.18]

(b) F = 123.3 on (3,6) df. P = 0.000. This means that the probability of results like this, if the three explanatory variables were not relevant, is very small. (c) The residual plots on page 120 show the model is satisfactory. There is no pattern in any of the residual plots. (d) R2 = 0.984. Therefore 98.4% of the variation in Y is explained by the regression relationship.

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Chapter 6: Multiple regression

Output using Minitab for Exercise 6.4: MTB > Regress ’Y’ 3 ’X1’ ’X2’ ’X3’; SUBC> Predict 10 40 30; SUBC> Confidence 90. The regression equation is Y = 73.4 + 1.52 X1 + 0.381 X2 - 0.268 X3 Predictor Constant X1 X2 X3

Coef 73.40 1.5162 0.3815 -0.2685

S = 2.326

StDev 14.69 0.1295 0.1941 0.1841

R-Sq = 98.4%

Analysis of Variance Source DF SS Regression 3 2001.54 Error 6 32.46 Total 9 2034.00 Source X1 X2 X3 Fit 95.762

DF 1 1 1 StDev Fit 1.632

T 5.00 11.71 1.97 -1.46

P 0.002 0.000 0.097 0.195

R-Sq(adj) = 97.6%

MS 667.18 5.41

F 123.32

P 0.000

Seq SS 1118.36 871.67 11.51

(

90.0% CI 92.590, 98.934)

(

90.0% PI 90.239, 101.285)

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Part D. Solutions to exercises

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Exercise 6.4(c): Residual plots for the cement data.

(e) The signs of the coefficients indicate the direction of the effect of each variable. X1 increases heat and has the greatest effect (the largest coefficient). The other variables are not significant, so they may not have any effect. If they do, the coefficients suggest that X2 might increase heat and X3 might decrease heat. (f ) For X1 = 10, X2 = 40 and X3 = 30, Yˆ = 73.40+1.52(10)+0.38(40)−0.27(30) = 95.76. 90% Prediction interval: [90.24,101.29] 6.5 The data for this exercise were taken from McGee and Carleton (1970) “Piecewise regression”, Journal of the American Statistical Association, 65, 1109–1124. It might be worthwhile to get this paper to compare what conventional regression can accomplish when there are special features in the data. In this case, the relationship

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Chapter 6: Multiple regression

between the Boston dollar volume and the NYSE-AME dollar volume underwent a series of changes over the time period of interest. In this paper, the solution was as follows: from from from from

Jan ’67 through Oct ’67 Nov ’67 through Jul ’68 Aug ’68 through Nov ’68 Dec ’68 through Nov ’69

Yˆ Yˆ Yˆ Yˆ

= = = =

8.748 + 0.0061X −20.905 + 0.0114X −79.043 + 0.0205X 11.075 + 0.0067X

Notice the slope coefficients in these four equations. They are small (because Boston’s dollar volume is small relative to the big board volumes) but they get increasingly stronger (from6 1 to 114 to 205) in successive periods of commission splitting. Then in Dec ’68, the SEC said “no more commission splitting” and it hurt the Boston dollar volume. The slope went back to 67, which is almost where it started. (a) The fitted equation is Yˆ = −66.2 + 0.014X. The following output was obtained from a computer package. Value Std. Error t value Pr(>|t|) (Intercept) -66.2193 39.6809 -1.6688 0.1046 X 0.0138 0.0029 4.7856 0.0000 F statistic: 22.9 on 1 and 33 degrees of freedom the p-value is 0.00003465 R-sq = 0.4097 Rbar-sq = 0.3918 D-W = 0.694 Clearly, the regression is significant, although the intercept is not significant. (b) Output from computer package: Value Std. Error t value Pr(>|t|) (Intercept) -67.2116 40.2550 -1.6696 0.1047 X 0.0135 0.0030 4.5025 0.0001 time 0.2737 0.6518 0.4199 0.6773 F statistic: 11.25 on 2 and 32 degrees of freedom the p-value is 0.0001992 R-sq = 0.4129 Rbar-sq = 0.3762 D-W = 0.6814 Here, the regression is significant, but time is not significant. In fact, comparing these two models shows that adding time to the regression equation is actually ¯ 2 values. And for both analyses, the D-W worse than not adding it. See the R

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Exercise 6.5(c): Connected scatterplot for the Boston and American stock exchanges.

statistic shows that there is a lot of pattern left in the residuals. A piecewise regression approach does far better with this data set. (c) See the plot above. 6.6 (a) and (b) Here are the seasonality indices based on the regression equations (6.10) and (6.12). They represent the intercept term in the regression for each of the 12 first differences. Mar-Feb Apr-Mar May-Apr Jun-May Jul-Jun Aug-Jul Sep-Aug Oct-Sep Nov-Oct Dec-Nov Jan-Dec Feb-Jan

Using (6.10) -2.6 -6.7 -3.5 -5.3 -3.6 -5.2 -5.9 -6.9 -4.1 -4.7 -0.8 -2.2

Using (6.12) -6.2 -10.6 -7.4 -9.2 -7.4 -9.2 -9.7 -10.7 -7.9 -8.5 -4.6 -6.2

These two sets of seasonal indices are not quite the same. In the first equa-

Chapter 6: Multiple regression

123

tion (6.10), all eleven dummy variables for seasonality were allowed to be in the regression. In the second equation (6.12), the best subsets regression procedure did not allow the first seasonal dummy into the final equation. The absolute values are not so important because, in the presence of different sets of explanatory variables, we expect the intercept terms to be different. (c) The seasonal indices should be the same regardless of which month is used as a base. 6.7 (a) Yt = 78.7 + 0.534xt + et (b) DW = 0.57. dL = 1.04 at 1% level. Therefore there is significant positive autocorrelation.

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Part D. Solutions to exercises

Chapter 7: The Box-Jenkins methodology for ARIMA models 7.1 (a) In general, the approximate standard error of the sample autocorrelations is √ 1/ n. So the larger the value of n, the smaller the standard error. Therefore, the ACF has more variation for small values of n than for large values of n. All three series show the autocorrelations mostly falling with the 95% bands. The few that lie just outside the bands are not of concern since we would expect about 5% of spikes to cross the bands. There is no reason to think these series are anything but white noise. √ (b) The lines shown are 95% critical values. These are calculated as ±1.96/ n. So they are closer to zero when n is larger. The autocorrelations vary randomly, but they mostly stay within the bounds. 7.2 The time plot shows the series as a non-stationary level. It wanders up and down over time in a similar way to a random walk. The ACF decays very slowly which also indicates non-stationarity in the level. Finally, the PACF has a very large value at lag 1, indicating the data should be differenced. 7.3 The five models are AR(1) MA(1) ARMA(1,1) AR(2) MA(2)

Yt Yt Yt Yt Yt

= 0.6Yt−1 + et . = et + 0.6et−1 . = 0.6Yt−1 + et + 0.6et−1 . = −0.8Yt−1 + 0.3Yt−2 + et . = et + 0.8et−1 − 0.3et−2 .

In each case, we assume Yt = 0 and et = 0 for t ≤ 0. The generated data are shown on the following two pages. There is a lot of similarity in the shapes of the series because they are based on exactly the same errors. 7.4 (a) The ACF is slow to die out and the time plot shows the series wandering in a non-stationary way. So we take first differences. The ACF of the first differences show one significant spike at lag 1 indicating an MA(1) is appropriate. So the model for the raw data is ARIMA(0,1,1). (b) There is not consistent trend in the raw data and the differenced data have mean close to zero. Therefore, there is no need to include a constant term. (c) (1 − B)Yt = (1 − θ1 B)et .

(d) See the output on page 127. There may be slight differences with different software packages and even different versions of the same package. The LjungBox statistics are not significant and the ACF and PACF of residuals show no significant differences from white noise.

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Chapter 7: The Box-Jenkins methodology for ARIMA models

t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

AR(1) 0.010 1.386 1.362 2.397 2.758 2.695 1.947 0.968 2.481 2.209 1.055 -0.797 -1.628 -1.047 1.062 0.917 0.560 1.276 -1.334 -0.711 0.484 2.050 2.070 0.112 0.987 2.262 0.327 -1.514 0.272 -0.427

MA(1) 0.010 1.386 1.358 1.898 2.268 1.832 0.954 -0.002 1.780 1.860 0.162 -1.592 -2.008 -0.760 1.648 1.294 0.178 0.946 -1.536 -1.170 0.964 2.306 1.896 -0.626 0.242 2.222 -0.028 -2.328 0.154 0.118

ARMA(1,1) 0.010 1.392 2.193 3.214 4.196 4.350 3.564 2.136 3.062 3.697 2.380 -0.164 -2.106 -2.024 0.434 1.554 1.111 1.612 -0.569 -1.511 0.057 2.340 3.300 1.354 1.054 2.855 1.685 -1.317 -0.636 -0.264

AR(2) 0.010 1.372 -0.565 2.443 -0.804 2.416 -1.844 2.000 -0.253 1.523 -1.564 0.278 -1.842 1.487 -0.052 0.768 -0.620 1.666 -3.619 3.485 -2.964 5.176 -4.190 3.775 -3.357 5.488 -6.428 5.079 -4.811 4.783

Generated data for Exercise 7.3

MA(2) 0.010 1.388 1.631 1.590 2.425 1.622 0.766 -0.248 1.641 2.300 -0.264 -1.862 -2.213 -0.561 1.979 1.653 -0.273 0.864 -1.351 -1.872 1.612 2.461 1.975 -0.986 -0.236 2.745 0.030 -3.035 0.121 0.867

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Exercise 7.3: Simulated ARMA series.

Chapter 7: The Box-Jenkins methodology for ARIMA models

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Output using Minitab for Exercise 7.4: MTB > ARIMA 0 1 1 ’Strikes’; SUBC> NoConstant; SUBC> Forecast 3. Final Estimates of Parameters Type Coef StDev MA 1 0.3174 0.1886

T 1.68

Differencing: 1 regular difference Number of observations: Original series 30, after differencing 29 Residuals: SS = 9256634 (backforecasts excluded) MS = 330594 DF = 28 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 Chi-Square 8.1(DF=11) 34.1(DF=23) Forecasts from period 30 Period 31 32 33

Forecast 4164.87 4164.87 4164.87

95 Percent Limits Lower Upper 3037.70 5292.04 2800.11 5529.63 2598.14 5731.60

(e) The last observation is yt = 3885; the last residual in series is e t = −881.87 (obtained from the computer package). Now So

Yt = Yt−1 + et − 0.3174et−1 . e30 Yˆ31 = Y30 + eˆ31 − 0.3174ˆ

= 3885 + 0 − 0.3174(−881.87) = 4164.9 ˆ Y32 = Yˆ31 + 0 − 0.3174(0) = 4164.9 Yˆ33 = Yˆ32 + 0 − 0.3174(0) = 4164.9

(f ) See the graph on the following page. 7.5 (a) The monthly data show strong seasonality and the seasonal pattern is reasonably stable. There is no trend in the data (this is a mature product). (b) The pattern in the ACF plot shows the dominance of the seasonality. The autocorrelations at lags 6, 18 and 30 are negative (because we are correlating

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Exercise 7.4(f ): Predicted number of strikes in USA. 95% prediction intervals shown.

the high periods with the low periods) and at lags 12, 14 and 36 they are positive (because we are correlating high periods with high periods). (c) The pattern in the PACF plot is not particularly revealing. However, there is little need to try to interpret this plot when the analysis clearly shows the dominance of the seasonality. The best approach would be to difference the series to reduce the effect of the seasonality and then see what is left over. (d) These graphs suggest a seasonal MA(1) because of the spike at lag 12 in the ACF and the decreasing spikes at lags 12 and 24 in the PACF. Overall, the suggested model is ARIMA(0,1,0)(0,1,1) 12 . (e) Using the backshift operator: (1 − B)(1 − B 12 )Yt = (1 − ΘB 12 )et . Rewriting gives Yt − Yt−12 − Yt−1 + Yt−13 = et − Θet−12 . 7.6 (a) ARIMA(3,1,0). (b) For the differenced data, the PACF has a significant spikes at lags 1, 2 and 3 and a spike at lag 17 which is marginally significant. The spike at lag 17 is probably due to chance. Therefore an AR(3) is an appropriate model for the differenced data. Consequently, an ARIMA(3,1,0) model is suitable for the original data.

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Chapter 7: The Box-Jenkins methodology for ARIMA models (c) Now

(Yt − Yt−1 ) = 0.42(Yt−1 − Yt−2 ) − 0.20(Yt−2 − Yt−3 ) − 0.30(Yt−3 − Yt−4 ) + et . Therefore

Yt = 1.42Yt−1 − 0.62Yt−2 − 0.10Yt−3 + 0.30Yt−4 + et

and

Yˆ1940 = 1.42(1797) − 0.62(1791) − 0.10(1627) + 0.30(1665) = 1778.1 Yˆ1941 = 1.42(1778.1) − 0.62(1797) − 0.10(1791) + 0.30(1627) = 1719.8 Yˆ1942 = 1.42(1719.8) − 0.62(1778.1) − 0.10(1797) + 0.30(1791) = 1697.3 7.7 (a) ARIMA(4,0,0). (b) The model was chosen because the last significant spike in the PACF was at lag 4. Note that the spikes at lags 2 and 3 were not significant. This makes no difference. It is the last significant spike which determines the order of the model. (c) The model is Yt = 146.1 + 0.891Yt−1 − 0.257Yt−2 + 0.392Yt−3 − 0.333Yt−4 + et . So Yˆ1969 Yˆ1970 Yˆ1971

= 146.1 + 0.891(545) − 0.257(552) + 0.392(534) − 0.333(512) = 528.7 = 146.1 + 0.891(528.7) − 0.257(545) + 0.392(552) − 0.333(534) = 515.7

= 146.1 + 0.891(515.7) − 0.257(528.7) + 0.392(545) − 0.333(552) = 499.5

7.8 (a) The centered 12-MA smooth is shown in the plot on the next page. The trend is generally linear and increasing with a flat period between 1990 and 1993. (b) The variation does not change much with the level, so transforming will not make much difference. (c) The data are not stationary. There is a trend and seasonality in the data. Differencing at lag 12 gives the data shown in the plot on page 131. These appear stationary although it is possible another difference at lag 1 is needed. (d) From the plots on page 131 it is clear there is a seasonal MA component of order 1. In addition there is a significant spike at lag 1 in both the ACF and PACF. Hence plausible models are ARIMA(1,0,0)(0,1,1) 12 and ARIMA(0,0,1)(0,1,1)12 . Comparing the two models we have the following results ARIMA(1,0,0)(0,1,1)12 ARIMA(0,0,1)(0,1,1)12

AIC=900.2 AIC=926.9

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US electricity generation

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Exercise 7.8(a): Total net generation of electricity in USA.

Hence the better model is the first one. Note that different packages will give different values for the AIC depending on how it is calculated. Therefore the same package should be used for all calculations. (e) The residuals from the ARIMA(1,0,0)(0,1,1) 12 are shown in the plots on page 132. Because there are significant spikes in the ACF and PACF, the model is not adequately describing the series. These plots suggest we need to add an MA(1) term to the model. So we fit the revised model ARIMA(1,0,1)(0,1,1) 12 . This time, the residual plots (not shown here) look like white noise. The AIC is 876.7. Part of the computer output for fitting the revised model is shown below. Parameter MA1,1 MA2,1 AR1,1

Estimate 0.74427 0.77650 0.99566

Approx. Std Error 0.05887 0.09047 0.0070613

T Ratio 12.64 8.58 141.00

Lag 1 12 1

So the fitted model is (1 − 0.996B)(1 − B 12 )Yt = (1 − 0.744B)(1 − 0.777B 12 )et .

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Exercise 7.8(e): Residuals from ARIMA(1,0,0)(0,1,1)12 model fitted to the electricity data.

Chapter 7: The Box-Jenkins methodology for ARIMA models

133

Output using SAS for Exercise 7.8: Parameter MA1,1 MA2,1 AR1,1

Estimate 0.86486 0.80875 0.27744

Approx. Std Error 0.06044 0.09544 0.10751

T Ratio 14.31 8.47 2.58

Lag 1 12 1

Variance Estimate = 41.8498466 Std Error Estimate = 6.46914574 AIC = 864.616345 SBC = 873.195782 Number of Residuals= 129 To Lag 6 12 18 24

Chi Square DF 1.60 3 7.67 9 15.32 15 18.67 21

Autocorrelations of Residuals Prob 0.659 0.568 0.429 0.607

0.028 -0.036 -0.020 -0.010 -0.014 0.095 0.004 -0.082 0.073 0.128 -0.095 0.072 0.126 0.016 -0.091 0.125 -0.105 -0.020 0.065 -0.048 0.051 0.005 0.069 -0.085

Note that the first term on the left is almost the same as differencing (1 − B). This suggests that we probably should have taking a first difference as well as a seasonal difference. We repeated the above analysis and arrived at the following model: ARIMA(1,1,1)(0,1,1)12 which has AIC=864.6. The computer output for the final model is shown above. The figures under the heading Chi Square concern the Ljung-Box test. Clearly the model passes the test (see Table E in Appendix III). (f ) Forecasts for the next 24 months are given on the following page.

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Part D. Solutions to exercises Month Nov 96 Dec 96 Jan 97 Feb 97 Mar 97 Apr 97 May 97 Jun 97 Jul 97 Aug 97 Sep 97 Oct 97 Nov 97 Dec 97 Jan 98 Feb 98 Mar 98 Apr 98 May 98 Jun 98 Jul 98 Aug 98 Sep 98 Oct 98

Obs 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166

Forecast 240.1614 262.5516 270.2423 244.0064 249.8899 232.7683 249.3720 270.7257 295.5439 295.6598 257.1358 246.4526 245.0224 267.1930 274.8228 248.5699 254.4488 237.3258 253.9291 275.2828 300.1009 300.2168 261.6929 251.0096

Std Error 6.4691 6.9981 7.1820 7.3027 7.4074 7.5069 7.6042 7.6999 7.7944 7.8878 7.9800 8.0712 8.4340 8.6077 8.7406 8.8622 8.9796 9.0948 9.2083 9.3205 9.4313 9.5407 9.6490 9.7560

Lower 95% 227.4821 248.8356 256.1659 229.6934 235.3718 218.0550 234.4680 255.6341 280.2671 280.2000 241.4952 230.6332 228.4920 250.3222 257.6914 231.2003 236.8491 219.5004 235.8811 257.0150 281.6160 281.5173 242.7812 231.8881

Upper 95% 252.8407 276.2677 284.3187 258.3194 264.4081 247.4816 264.2759 285.8173 310.8207 311.1196 272.7764 262.2719 261.5528 284.0638 291.9541 265.9395 272.0484 255.1513 271.9772 293.5506 318.5858 318.9163 280.6045 270.1311

7.9 (a) See the plot on the following page. Note that there is strong seasonality and a pronounced trend-cycle. One way to study the consistency of the seasonal pattern is to compute the seasonal sub-series and see how stable each month is. The results are given below. 1955: 1956: 1957: 1958: 1959: 1960: 1961: 1962: 1963: 1964: 1965: 1966: 1967: 1968: 1969: 1970:

Jan 94.7 94.1 95.9 95.0 93.9 95.7 95.2 93.5 94.6 93.6 95.6 97.0 93.9 91.2 96.0 95.8

Feb 94.0 93.5 96.8 94.8 94.5 95.7 94.8 93.7 93.4 93.2 92.7 93.7 93.6 91.7 94.3 93.0

Mar 96.5 96.8 99.0 96.1 96.4 95.1 96.5 95.6 95.5 94.6 94.0 95.2 94.1 94.5 94.1 92.0

Apr 101.3 103.1 97.7 100.4 100.9 98.9 101.3 100.7 99.1 98.6 96.7 97.0 99.0 99.0 96.8 96.0

May 102.4 104.1 99.5 101.7 102.1 100.8 101.7 102.3 100.8 100.2 99.4 98.4 102.5 101.9 100.7 100.2

Jun 103.7 102.8 101.1 102.1 103.3 102.5 103.7 102.5 104.1 103.5 103.7 104.1 105.7 103.1 104.5 103.7

Jul 104.5 103.7 102.0 103.6 104.7 104.6 105.2 104.4 106.1 106.6 108.2 105.9 109.2 105.7 106.3 106.0

Aug 104.3 103.6 103.3 104.9 106.0 106.0 105.3 106.4 107.4 107.5 108.0 107.2 109.9 106.0 107.2 105.8

Sep 104.1 103.6 105.1 104.4 104.4 104.0 104.3 103.5 104.1 103.6 104.7 104.2 104.9 103.5 103.7 102.7

Oct Nov Dec 101.2 98.3 95.4 101.7 98.6 96.8 103.2 99.8 96.7 101.1 96.7 95.3 100.9 98.7 96.1 100.1 98.0 96.7 101.2 97.7 96.5 101.0 97.6 96.9 100.7 97.9 97.4 101.7 97.9 96.9 100.5 98.4 99.6 99.7 97.1 96.8 99.8 98.3 93.8 100.2 100.7 99.1 102.5 100.2 99.4 98.9 97.1 96.5

These detrended data are relatively consistent from year to year with only minor

135

Chapter 7: The Box-Jenkins methodology for ARIMA models

240

Employment in motion picture industry o o oo o o o

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o o o o o

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Exercise 7.9(a): Employment in the motion picture industry

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100 95

ratios

105

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Part D. Solutions to exercises

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Exercise 7.9(a): Sub-series of detrended data

variations occurring here and there. For example, December 1967 and January and February 1968 were noticeably lower than surrounding years. Another way to look at seasonal patterns is via autocorrelation functions. Note that for the raw data, the ACF shows strong seasonality over several seasonal lags. This is further evidence of the consistency of the seasonal pattern. The plot on the previous page shows the detrended data. Again, the seasonal pattern is very consistent although the amplitude of the pattern each year varies. Unusual results in early 1968 and early 1970 are seen. (b) For the first 96 months, we identified an ARIMA(0,1,0)(0,1,1) 12 . For the second 96 months, we identified an ARIMA(0,1,0)(1,1,0) 12 : In practice, there is little difference between these models. This means that once the trend has been eliminated (by differencing), the seasonal patterns are very similar. (c) Using the above ARIMA(0,1,0)(0,1,1) 12 model, we obtained the following forecasts.

137

Chapter 7: The Box-Jenkins methodology for ARIMA models

110

Detrended employment in motion picture industry o o oo o

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Exercise 7.9(a): Seasonal differences of employment in the motion picture industry

138

Part D. Solutions to exercises Month Jan 1963 Feb 1963 Mar 1963 Apr 1963 May 1963 Jun 1963 Jul 1963 Aug 1963 Sep 1963 Oct 1963 Nov 1963 Dec 1963 Jan 1964 Feb 1964 Mar 1964 Apr 1964 May 1964 Jun 1964 Jul 1964 Aug 1964 Sep 1964 Oct 1964 Nov 1964 Dec 1964

Actual 167.2 165.0 168.8 175.0 177.9 183.7 187.2 189.3 183.4 177.3 172.3 171.4 164.9 164.4 166.9 174.2 177.5 183.6 189.5 191.6 185.1 181.9 175.4 174.2

Forecast 167.3286 166.7030 167.7141 176.5972 176.9498 179.2212 186.1708 188.7598 186.1678 177.2392 170.7711 168.8708 167.5935 163.9412 167.4086 174.2641 176.4075 180.0358 186.3499 191.2063 187.7172 178.9557 175.7857 172.6567

Upper (95%) 172.4228 171.7902 172.8012 181.6844 182.0369 184.3084 191.2579 193.8470 191.2550 182.3264 175.8583 173.9580 172.6807 169.0247 172.4920 179.3475 181.4909 185.1193 191.4333 196.2898 192.8007 184.0392 180.8692 177.7402

Lower (95%) 162.2345 161.6159 162.6269 171.5100 171.8626 174.1340 181.0836 183.6727 181.0806 172.1520 165.6839 163.7837 162.5063 158.8577 162.3251 169.1806 171.3240 174.9523 181.2664 186.1228 182.6338 173.8722 170.7022 167.5732

Error -0.1286 -1.7030 1.0859 -1.5972 0.9502 4.4788 1.0292 0.5402 -2.7678 0.0608 1.5289 2.5292 -2.6935 0.4588 -0.5086 -0.0641 1.0925 3.5642 3.1501 0.3937 -2.6172 2.9443 -0.3857 1.5433

(d) For the second half of the data we used the ARIMA(0,1,0)(1,1,0) 12 to obtain the forecasts at the top of the following page. The actual 1971–1972 figures are also shown. The source is “Employment and Earnings, US 1909–1978”, published by the Department of Labor, 1979. A good exercise would be to take these forecasts and check the MAPE for 1971 and 1972 separately. The MAPE for the first forecast year should be smaller than the MAPE for the second year. (e) If the objective is to forecast the next 12 months then the latest data is obviously the most relevant but to get seasonal indices we have to go back several years and to anticipate what the next move the large cycle is going to be, we really need to look at as much data as possible. So a good strategy would be i. ii. iii. iv.

study the trend-cycle by looking at the 12-month moving average; remove the trend-cycle and study the consistency of the seasonality; decide how much of the data series to retain for the ARIMA modeling; forecast the next 12 months and use some judgment as to how to modify the ARIMA forecasts on the basis of anticipated trend-cycle movements.

139

Chapter 7: The Box-Jenkins methodology for ARIMA models Month Jan 1971 Feb 1971 Mar 1971 Apr 1971 May 1971 Jun 1971 Jul 1971 Aug 1971 Sep 1971 Oct 1971 Nov 1971 Dec 1971 Jan 1972 Feb 1972 Mar 1972 Apr 1972 May 1972 Jun 1972 Jul 1972 Aug 1972 Sep 1972 Oct 1972 Nov 1972 Dec 1972

Actual 194.5 187.9 187.7 198.3 202.7 204.2 211.7 213.4 212.0 203.4 199.5 199.3 191.3 192.1 193.3 203.4 205.5 218.2 220.3 219.9 211.9 204.5 198.5 200.5

Forecast 196.0141 191.2939 189.9446 197.3595 205.7424 213.1446 217.8789 218.8543 213.2939 208.3371 204.7595 203.4670 196.2469 191.0587 189.3618 196.9907 205.3066 212.5618 217.4298 218.2314 213.0587 207.6473 204.3907 203.2051

Upper (95%) 201.4418 198.9698 199.3456 208.2149 217.8790 226.4396 232.2392 234.2061 229.5769 225.5009 222.7611 222.2690 217.0365 213.6617 213.6432 222.8417 232.6373 241.2960 247.5022 249.5848 245.6428 241.4174 239.3064 239.2300

Lower (95%) 190.5864 183.6180 180.5436 186.5042 193.6057 199.8495 203.5186 203.5025 197.0109 191.1733 186.7580 184.6650 175.4573 168.4557 165.0804 171.1396 177.9760 183.8276 187.3575 186.8780 180.4746 173.8773 169.4750 167.1802

Error -1.5141 -3.3939 -2.2446 0.9405 -3.0424 -8.9446 -6.1789 -5.4543 -1.2939 -4.9371 -5.2595 -4.1670 -4.9469 1.0413 3.9382 6.4093 0.1934 5.6382 2.8702 1.6686 -1.1587 -3.1473 -5.8907 -2.7051

Forecasts for Exercise 7.9(d)

7.10 (a) There is strong seasonality as can be seen from the time plot and the seasonal peaks in the ACF. (b) The trend in the series is small compared to the seasonal variation. However, there is a period of downward trend in the first four years, followed by an upward trend for four years. At the end the trend seems to have levelled off. (c) The one large spike in the PACF of Figure 7-34 suggests the series needs differencing at lag 1. This is also apparent from the slow decay in the ACF and the non-stationary mean in the time plot. (d) You would need to difference again at lag 1 and plot the ACF and PACF of the new series (differenced at lags 12 and 1). It is not possible to identify a model from Figures 7-33 and 7-34.

140

Part D. Solutions to exercises

Chapter 8: Advanced forecasting models 8.1 (a) The fitted model in Exercise 6.7 (using OLS) was Yt = 78.7 + 0.534xt + Nt . The computer output below shows the results for fitting the straight line regression with AR(1) errors. Hence the new model is Yt = 79.3 + 0.508xt + Nt

where

Nt = 0.72Nt−1 + et .

In this case, the error model makes very little difference to the parameters.

Output from SAS for Exercise 8.1: Parameter MU AR1,1 NUM1

Estimate 79.27236 0.72469 0.50801

Approx. Std Error 0.76093 0.14647 0.02318

T Ratio 104.18 4.95 21.91

Lag 0 1 0

Variable Shift SALES 0 SALES 0 ADVERT 0

Constant Estimate = 21.8242442 Variance Estimate = 1.11639088 Std Error Estimate = 1.056594 AIC = 74.2915405 SBC = 77.825702 Number of Residuals= 24

To Lag 6 12 18

Chi Square DF 3.46 5 9.31 11 16.39 17

Autocorrelation Check of Residuals Autocorrelations Prob 0.630 0.027 0.099 -0.037 0.111 -0.060 -0.274 0.593 0.055 0.126 0.229 -0.227 0.060 -0.095 0.497 -0.117 -0.238 -0.080 0.054 -0.108 0.101

(b) The ACF and PACF of the errors is plotted on the following page. An AR(1) model for the errors is appropriate since there is a single significant spike at lag 1 in the PACF and geometric decay in the ACF. This is confirmed by the Ljung-Box test in the computer output above. The Q ∗ values are given under the column Chi Square. None are significant showing the residuals from the full model are white noise.

141

Chapter 8: Advanced forecasting models

Errors from regression model o

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Exercise 8.1: Errors from regression model with AR(1) error term.

12

142

Part D. Solutions to exercises

Output from SAS for Exercise 8.2(a):

Parameter MU AR1,1 NUM1

Estimate 9.56328 0.78346 -0.02038

Constant Estimate

Approx. Std Error 0.40537 0.06559 0.01066

T Ratio 23.59 11.94 -1.91

Lag 0 1 0

Variable Shift HURON 0 HURON 0 YEAR 0

= 2.07087134

Variance Estimate = 0.51219788 Std Error Estimate = 0.71568001 AIC = 216.450147 SBC = 224.205049 Number of Residuals= 98

To Lag 6 12 18 24

Chi Square 8.35 15.01 16.36 25.47

Autocorrelation Check of Residuals Autocorrelations DF 5 11 17 23

Prob 0.138 0.222 -0.100 -0.133 -0.056 -0.007 -0.042 0.182 -0.051 0.009 0.175 0.017 -0.121 -0.107 0.499 -0.053 0.014 0.019 0.058 0.006 -0.067 0.326 -0.071 -0.166 -0.043 0.051 0.160 0.092

8.2 (a) To reduce numerical error, we subtracted 1900 from the year to create an explanatory variable. Hence the year ranged from -25 (1875) to 72 (1972). The computer output above shows the fitted model to be Yt = 9.56 − 0.02xt + Nt

where

Nt = 0.78Nt−1 + et

where xt is the year −1900. (b) The errors are shown in the plot on the following page. This demonstrates that a better model would have an AR(2) error term since the PACF has two significant spikes at lags 1 and 2. The spike at lag 10 is probably due to chance. The ACF shows geometric decay which is possible with an AR(2) model. So the full regression model is Yt = β 0 + β 1 xt + N t

where Nt = φ1 Nt−1 + φ2 Nt−2 + et .

Fitting this model gives the output shown on page 144. So the fitted model is Yt = 9.53 − 0.02xt + Nt

where Nt = Nt−1 − 0.29Nt−2 + et .

143

Chapter 8: Advanced forecasting models

Errors from regression model

2

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Exercise 8.2(b): Errors from regression model with AR(1) error term.

o

144

Part D. Solutions to exercises

Output from SAS for Exercise 8.2(b):

Parameter MU AR1,1 AR1,2 NUM1

Estimate 9.53078 1.00479 -0.29128 -0.02157

Constant Estimate

Std Error 0.30653 0.09839 0.10030 0.0082537

Approx. T Ratio Lag 31.09 0 10.21 1 -2.90 2 -2.61 0

Variable Shift HURON 0 HURON 0 HURON 0 YEAR 0

= 2.73048107

Variance Estimate = 0.4760492 Std Error Estimate = 0.68996319 AIC = 210.396534 SBC = 220.736404 Number of Residuals= 98

To Lag 6 12 18 24

Chi Square 0.60 5.35 6.21 10.49

Autocorrelation Check of Residuals Autocorrelations DF 4 10 16 22

Prob 0.964 0.018 -0.028 -0.003 0.040 0.054 -0.007 0.867 -0.032 -0.037 0.167 -0.007 -0.098 -0.055 0.986 -0.036 0.005 -0.025 0.035 -0.006 -0.063 0.981 -0.003 -0.141 -0.007 0.006 0.116 0.008

8.3 (a) ARIMA(0,1,1)(2,1,0)12 . This model would have been chosen by first identifying that differences at lags 12 and 1 are necessary to make the data stationary. Then looking at the ACF and PACF of the differenced data would have shown two significant spikes in the PACF at lags 12 and 24. There would have also been a significant spike in the ACF at lag 1 and geometric decay in the early lags of the PACF. (b) Since both parameter estimates are positive (and significantly different from zero), we can conclude that electricity consumption increases with both heating degrees and cooling degrees. Because b 2 is larger, we know that there is a greater increase in electricity usage for each heating degree than for each cooling degree. (c) To use this model for forecasting, we would first need forecasts of both X 1,t and X2,t into the future. These could be obtained by taking averages of these variables over the equivalent months of the previous few decades. Then the model can be used to forecast electricity demand over the next 12 months by

Chapter 8: Advanced forecasting models

145

forecasting the Nt series using the method discussed in chapter 7 and plugging the forecasts of X1,t , X2,t and Nt into the formula for Yt . (d) If the model was fitted using a standard regression package (thus modeling N t as white noise), then the seasonality and autocorrelation in the data would have been ignored. This would result in less efficient parameter estimates and invalid estimates of their standard errors. In particular, tests for significance would be incorrect, as would prediction intervals. Also, when producing forecasts of Y t , the forecasts of Nt would be all be zero. Hence, the model would not adequately allow for the seasonality or autocorrelation in the data. 8.4 (a) b = 3, r = 1, s = 2. (b) ARIMA(2,0,0) (c) ω0 = −0.53, ω1 = −0.37, ω2 = −0.51, δ1 = 0.57, δ2 = 0, θ1 = θ2 = 0, φ1 = 1.53, φ2 = −0.63.

(d) 27 seconds.

8.5 See the graphs on the following page. 8.6 (a) The three series are shown on page 147. For Set 1, four X t values are needed (since v1 , v2 , v3 and v4 are all non-zero). Therefore 27 Yt values can be produced. Similarly 26 Yt values for Set 2 and 24 Yt values for Set 3 can be calculated. (b) The first model is 2.0B Xt + N t . 1 − 0.7B The simplest way to generate data for this transfer function is to rewrite it as follows Yt =

(1 − 0.7B)Yt = (1 − 0.7B)B(2 − 1.4B)Xt + (1 − 0.7B)Nt so that

Yt = 0.7Yt−1 + 2.0Xt−1 − 1.4Xt−2 + Nt − 0.7Nt−1 .

Thus Yt values can only be generated for times t = 3, 4, . . . since we need at least two previous Xt values. However, for t = 3, we also need Y 2 . To start the process going, we have assumed here that Y 2 = 0. Other values could also have been used. The effect of this initialization is negligible after a few time periods. The second model is easier to generate as we can write it Yt = 1.2Xt + 2.0Xt−1 − 0.8Xt−2 + Nt .

146

Part D. Solutions to exercises

(b)

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Exercise 8.5: Impulse response weights for the four different transfer functions.

147

Chapter 8: Advanced forecasting models

t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Nt -0.8003 0.8357 1.4631 0.7332 0.3260 -0.7442 0.7362 1.1931 -1.4681 -0.5285 0.4314 -1.6341 0.8198 0.4183 -0.4065 -0.0615 0.1432 -1.0747 -0.5355 -0.1454 0.2088 -0.6854 0.1182 0.6971 0.3698 -0.0802 -0.9202 1.1483 -0.1663 -0.5461

Xt 50 90 50 30 80 80 30 70 60 10 40 20 40 20 10 30 60 70 40 70 10 30 30 40 30 100 60 90 60 100

Set 1 Yt

58.7 52.3 61.3 65.7 59.2 55.5 49.5 37.4 27.4 29.8 30.4 23.6 19.9 29.1 46.9 56.5 56.9 49.2 34.3 28.1 30.7 34.4 46.9 64.1 76.1 75.8 76.5

Set 2 Yt

58.3 51.3 62.7 66.2 56.5 56.5 50.4 35.4 29.8 29.4 29.6 23.9 20.1 27.9 47.5 56.9 57.2 48.3 35.1 28.7 30.4 33.9 46.1 66.1 74.8 75.5

Set 3 Yt

74.7 88.2 87.5 79.5 83.4 72.4 54.8 47.4 41.6 40.9 36.1 27.9 38.5 59.9 74.2 76.3 73.1 54.7 43.4 43.9 44.1 61.1 84.8 97.5

Set 4

Set 5

0.0 110.9 51.3 25.7 135.0 143.8 49.3 130.2 113.7 16.4 75.5 38.8 79.0 38.6 19.3 59.7 118.6 139.2 79.7 140.1 19.2 60.1 60.7 80.3 59.9 199.1 121.1 179.8 119.4

201.5 64.7 116.3 231.3 132.7 81.2 186.5 75.5 20.4 94.4 56.8 88.4 19.6 39.9 124.1 178.9 139.5 107.9 120.2 -0.7 88.1 84.7 92.4 147.9 247.1 149.1 203.8 167.5

Generated data for Exercise 8.6

148

Part D. Solutions to exercises

8.7 (a) The average cost of a night’s accommodation is C/R. (b) There are a number of ways this could be done. The simplest is to define the monthly CPI to be the same as that of the quarter. For example, January, February and March of 1980 would each have a CPI of 45.2; April, May and June 1980 would each have a CPI of 46.6; and so on. Other methods might involve fitting a smooth curve through the quarterly figures and using the curve to predict the CPI at other points along the time axis. (c) See figure below.

80

100

Consumer price index (Melbourne)

40

60

Average rate per room per night ($)

1980

1982

1984

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1990

1992

1994

1996

Exercise 8.7(c): Time plots of average room rate and CPI.

(d) Our preliminary model is Yt = a + (ν0 + ν1 B + · · · + ν6 B 6 )Xt + Nt where Yt denotes the average room rate, Xt denotes the CPI and Nt is an AR(1) process. The estimated errors from this model are shown in the figure on the previous page. They are clearly non-stationary and have some seasonality. So we difference both Yt and Xt and refit the model with Nt specified as an ARIMA(1,0,0)(1,0,0)12 . The parameter estimates are shown below (as given by SAS).

149

Chapter 8: Advanced forecasting models

Residuals from dynamic regression with AR(1) errors o o

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Exercise 8.7(d): Errors from regression model with AR(1) error term.

20

150

Part D. Solutions to exercises Parameter a ν0 ν1 ν2 ν3 ν4 ν5 ν6

Estimate 0.20200 0.20730 -0.41687 0.23165 0.32048 -0.72093 0.74707 -0.36272

s.e. 0.2848 0.2602 0.2634 0.2655 0.2716 0.2665 0.2633 0.2656

P -value 0.4791 0.4267 0.1154 0.3842 0.2396 0.0075 0.0051 0.1739

Thus the intercept and first four coefficients are not significant and can be omitted. Hence we select b = 4. We shall retain the last three coefficients for the moment. Since they show no clear pattern, we select r = 0 and s = 3 giving the model Yt = (ω0 + ω1 B + ω2 B 2 )B 4 Xt + Nt . Looking at the ACF and PACF of the error series (not shown) and trying a number of alternative models led us to the model ARIMA(2,1,0)(2,0,0) 12 for Nt . That is (1 − φ1 B − φ2 B 2 )(1 − Φ1 B 12 − Φ2 B 24 )(1 − B)Nt = et . The parameter values (all significant) were Parameter Estimate

ω0 0.52

ω1 0.61

ω2 -0.47

φ1 -0.49

φ2 -0.33

Φ1 0.37

Φ2 0.41

The model suggests that there is a lag of four months between changes in the CPI and changes in the price of travel accommodation. The seasonality inherent in the model may be due to seasonal price variation or due to the way CPI was estimated from quarterly data. (e) Forecasts of CPI were obtained using Holt’s method. These are only needed from November 1995 because of the time lag of 4 months. Actual data beyond June 1995 are given in the second column for comparison.

151

Chapter 8: Advanced forecasting models Month 1995 1995 1995 1995 1995 1995 1996 1996 1996 1996 1996 1996

Actual Yt 94.0 96.7 94.8 89.6 95.8 91.5 92.0 95.5 100.6 94.1 97.2 102.9

Xt−4 115.0 116.2 116.2 116.2 116.9 117.3 117.7 118.1 118.5 118.9 119.3 119.8

Xt−5 115.0 115.0 116.2 116.2 116.2 116.9 117.3 117.7 118.1 118.5 118.9 119.3

Xt−6 115.0 115.0 115.0 116.2 116.2 116.2 116.9 117.3 117.7 118.1 118.5 118.9

60 40

Perpetual speed score

80

Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun

Predicted Yt 90.4 91.8 92.0 91.6 93.4 90.3 90.4 92.6 94.7 90.7 91.5 93.0

0

20

40

60

80

100

120

Day

Exercise 8.8: Time plot of daily perceptual speed scores for a schizophrenic patient. The drug intervention is shown at day 61.

8.8 (a) See the figure above. (b) The step intervention model with an ARIMA(0,1,1) error was used: Yt = ωXt + Nt

where (1 − B)Nt = θet−1 + et

152

Part D. Solutions to exercises where Yt denotes the perceptual speed score and X t denotes the step dummy variable. The estimated coefficients were Parameter Estimate

ω -22.1

θ 0.76

(c) The drug has lowered the perceptual speed score by about 22. (d) The new model is Yt =

ω Xt + N t 1 − δB

where (1 − B)Nt = θet−1 + et

(An ARIMA(0,1,1) error was found to be the best model again.) Here the estimated coefficients were Parameter Estimate

ω -13.21

δ 0.54

θ 0.76

The following accuracy measures show that the delayed effect model fits the data better. Model MAPE MSE AIC

Step 15.1 92.5 542.8

Delayed step 15.0 91.1 538.4

The forecasts for the two models are very similar. This is because the effect of the step in the delayed step model is almost complete at the end of the series, 60 days after the drug intervention. (e) The best ARIMA model we found was an ARIMA(0,1,1) with θ = 0.69. This gave MAPE=15.4, MSE=100.8 and AIC=550.9. This model gives a flat forecast function (since we did not include a constant term). The forecast values are 33.9. Because the step effect is almost complete in the delayed step model, it also gives a virtually flat forecast function with forecast values of 34.1. Hence there is virtually no difference. If forecasts had been made earlier (for example, at day 80), there would have been a difference because the step effect would still be in progress and so the delayed step model would have showed a continuing decline in perceptual speed. The real advantage of the intervention model over the ARIMA model is that the intervention model provides a way of measuring and evaluating the effect of an intervention. (f ) If the drug varied from day to day and the reaction times depended on dose, then a better model would be a dynamic regression model with the the quantity of drug as an explanatory variable.

153

Chapter 8: Advanced forecasting models 8.9 (a)

(b)



Yt − Yt−1 Xt − Xt−1





   Yt−1 − Yt−2 Yt−2 − Yt−3 = Φ1 + Φ2 Xt−1 − Xt−2 Xt−2 − Xt−3   Yt−12 − Yt−13 + · · · + Φ12 + Zt. Xt−12 − Xt−13

Yt = Yt−1 − 0.38(Yt−1 − Yt−2 ) + 0.15(Xt−1 − Xt−2 )

− 0.37(Yt−2 − Yt−3 ) + 0.13(Xt−2 − Xt−3 ) + · · ·

= 0.62Yt−1 + 0.01Yt−2 + 0.15Xt−1 − 0.02Xt−2 + · · · (c) • Multivariate model assumes feedback. That is, X t depends on past values of Yt . But regression does not allow this. • Regression model does not assume Xt is random. • Regression model allows Yt to depend on Xt as well as past values Xt−1 , Xt−2 , . . .. Multivariate AR only allows dependence on past values of {Xt }. • For these data, it is unlikely room rates will substantially affect Y t although it is possible. Small values in lower left of coefficient matrices suggest that Xt is not affecting Yt . Yt should depend on Xt . So regression is probably better. 8.10 (a) An AR(3) model can be written using the same procedure as the AR(2) model described in Section 8/5/1. Thus we define X 1,t = Yt , X2,t = Yt−1 and X3,t = Yt−2 . Then write     φ1 φ2 φ3 at X t =  1 0 0  X t−1 +  0  0 1 0 0 and

Yt = [1 0 0]X t .

This is now in state space form    1 φ1 φ2 φ3 F =  1 0 0 ,G =  0 0 1 0 0

with    0 0 at 1 0  , H = [1 0 0], et =  0  and zt = 0. 0 1 0

(b) An MA(1) can be written as Yt = θat−1 + at where at is white noise. We can write this in state space form by letting F = 0, G = θ, e t = at−1 , zt = at and H = 1. Thus Yt = X t + a t and Xt = θat−1 .

154

Part D. Solutions to exercises (c) Holt’s method is defined in Chapter 4 as Lt = αYt + (1 − α)(Lt−1 + bt−1 ),

bt = β(Lt − Lt−1 ) + (1 − β)bt−1 ,

with the one-step forecast as Ft+1 = Lt + bt . Hence the one-step error is et = Yt − Lt−1 − bt−1 . The first row can be written Lt = α(Yt − Lt−1 − bt−1 ) + Lt−1 + bt−1 = αet + Lt−1 + bt−1

and the second row can be written bt = β(Lt − Lt−1 ) + (1 − β)bt−1 = bt−1 + β(Lt − Lt−1 − bt−1 ) = bt−1 + βαet

using the first equation. Now let Xt,1 = Lt and Xt,2 = bt . Then the state space form of the model is     1 1 α Xt = X t−1 + et 0 1 βα Yt = [1 1] X t−1 + et . (d) The state space form might be preferable because • it allows missing values to be handled easily; • it is easy to generalize to allow the parameters to change over time; • the Kalman recursion equations can be used to calculate the forecasts and likelihood.

Chapter 9: Forecasting the long term

155

Chapter 9: Forecasting the long term 9.1 There is little doubt that the trends in computer power and memory show a very clear exponential growth while that of price is declining exponentially. It is therefore a question of time until computers that cost only a few hundred dollars will exist that can perform an incredible array of tasks which until now have been the sole prerogative of humans, for example playing chess (a high-power judgmental and creative process). It is therefore up to our imaginations to come up with future scenarios when such computers will be used as extensively as electrical appliances are used today. The trick is to free our thinking process so that we can come up with scenarios that are not constrained by our perception of the present when computers are being used mostly to make calculations. 9.2 As the cost of computers (including all of the peripherals such as printers and scanners) is being reduced drastically, and at the same time we will be getting soon to devices that will perform a great number of functions now done by separate machines, it will become more practical and economical to work at home. Furthermore, the size of these all-purpose machines is being continuously reduced. In the next five to ten years we will be able to have everything that is provided to us now in an office at home with two machines: one a powerful all-inclusive computer and the other a printer-scanner-photocopier-fax machine. Moreover these two machines will be connected to any network we wish via modems so that we can communicate and get information from anywhere. 9.3 As it was also mentioned in Exercise 9.1, there is no doubt that the trend in computer and equipment prices are declining exponentially at a fast rate. This would make it possible for everyone to be able to afford them and be able to have an office not only at home but at any other place he or she wishes, including one’s car, a hotel room, a summer vacation residence, or a sail boat. 9.4 Statements like those referred to in Exercise 9.4 abound and demonstrate the shortsightedness of peoples’ ability to predict the future. As a matter of fact as late as the beginning of our century people did not predict all four major inventions of the Industrial Revolution (cars, telephones, electrical appliances and television) that have dramatically changed our lives. Moreover, they did not predict the huge impact of computers even as late as the beginning of the 1950s. This is why we must break from our present mode of thinking and see things in a different, new light. This is where scenarios and analogies can be extremely useful.

156

Part D. Solutions to exercises

Chapter 10: Judgmental forecasting and adjustments 10.1 Phillips’ problems have to do with the management bias of overoptimism, that is believing that all changes will be successful and that they can overcome peoples’ resistance to change. This is not true, but we tend to believe that most organisational changes are successful because we hear and we read about the successful ones while there is very little mention of those that fail. Introducing changes must be considered, therefore, in an objective manner and our ability to succeed estimated correctly. 10.2 The quote by Glassman illustrates the extent to which professional, expert investment managers underperform the average of the market. Business Week, Fortune and other business journals regularly publish summary statistics of the performance of mutual funds and other professionally-managed funds, benchmarking them with the Standard & Poor or other appropriate indexes. The instructor can therefore get some more recent comparisons than those shown in Chapter 10 and show them in class. 10.3 Assuming that the length of cycles varies considerably we have little way to say how long it will take until the expansion started in May 1991 will be interrupted. Unfortunately the length (and depth) of cycles varies a great deal making it extremely difficult to say how long an expansion will last. It will all depend on the specific situation involved that will require judgmental inputs, structured in such a way as to avoid biases and other problems. 10.4 There are twenty 8s that one will encounter when counting from 0–100. When given this exercise most people say nine or ten because they are not counting the eights coming from 81 to 87, and 89 (they usually count the 8s in 88 often one time).

Chapter 11: The use of forecasting methods in practice

157

Chapter 11: The use of forecasting methods in practice 11.1 The results of Table 1 are very similar to those of the previous M-Competitions. As a matter of fact the resemblance is phenomenal given the fact that the series used and the time horizon they refer to are completely different. 11.2 In our view the combined method will do extremely well. More specifically its accuracy will be higher than the individual methods being combined while its variance of forecasting error will be smaller than that of the methods involved. 11.3 It seems that proponents of new forecasting methods usually exaggerate their benefits. This has been the case with methods under the banner of neural networks, machine learning and expert systems. These methods did not do well in the M3-IJF Competition. In addition only few experts participated in the competition using such methods, even though more than a hundred were contacted (and invited to participate) and more than fifty expressed an interest in the M3-IJF Competition, indicating that they would “possibly” participate . In the final analysis it seems that it is not so simple to run a great number of series by such methods resulting in not too many participants from such methods.

158

Part D. Solutions to exercises

Chapter 12: Implementing forecasting: its uses, advantages, and limitations The exercises for Chapter 12 are general and can be answered by referring to the text of Chapter 12 which covers each one of the topics. Each instructor can therefore form his/her way of answering these exercises.

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