Ebook Pearson new international edition (9/E): Part 1

Part 1 book “Pearson new international edition” has contents: Introduction to forecasting, exploring data patterns and an introduction to forecasting techniques, moving averages and smoothing methods, time series and their components, simple linear regression.

Business Forecasting John E Hanke Dean Wichern

Ninth Edition

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Table of Contents

1 Introduction to Forecasting

1

John E Hanke/Dean Wichern

2 Exploring Data Patterns and an Introduction to Forecasting Techniques

15

John E Hanke/Dean Wichern

3 Moving Averages and Smoothing Methods

61

John E Hanke/Dean Wichern

4 Time Series and Their Components

119

John E Hanke/Dean Wichern

5 Simple Linear Regression

175

John E Hanke/Dean Wichern

6 Multiple Regression Analysis

235

John E Hanke/Dean Wichern

7 Regression with Time Series Data

295

John E Hanke/Dean Wichern

8 The Box-Jenkins (ARIMA) Methodology

355

John E Hanke/Dean Wichern

9 Judgmental Forecasting and Forecast Adjustments

437

John E Hanke/Dean Wichern

10 Managing the Forecasting Process

459

John E Hanke/Dean Wichern

Appendix: Tables

477

John E Hanke/Dean Wichern

Appendix: Data Sets and Databases

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INTRODUCTION TO FORECASTING

This text is concerned with methods used to predict the uncertain nature of businesstrends in an effort to help managers make better decisions and plans Such efforts ofteninvolve the study of historical data and the manipulation of these data in the search forpatterns that can be effectively extrapolated to produce forecasts

In this text, we regularly remind readers that sound judgment must be used alongwith numerical results if good forecasting is to result The examples and the cases at theend of the chapter emphasize this point There are more discussions of the role of judg-ment in this chapter

THE HISTORY OF FORECASTING

In a book on the history of risk, author Peter Bernstein (1996) notes that the development

of business forecasting in the seventeenth century was a major innovation He writes:

forecasting—long denigrated as a waste of time at best and a sin at worst—

became an absolute necessity in the course of the seventeenth century foradventuresome entrepreneurs who were willing to take the risk of shaping thefuture according to their own design

Over the next 300 years, significant advances in data-based forecasting methodsoccurred, with much of the development coming in the twentieth century Regressionanalysis, decomposition, smoothing, and autoregressive moving average methods areexamples of data-based forecasting procedures discussed in this text These procedureshave proved to be highly effective and routinely appear in the menus of readily available forecasting software

Along with the development of data-based methods, the role of judgment andjudgmental approaches to forecasting has grown significantly over the last 25 years

Without any data history, human judgment may be the only way to make predictionsabout the future In cases where data are available, judgment should be used to reviewand perhaps modify the forecasts produced by quantitative procedures

With the proliferation of powerful personal computers and the availability ofsophisticated software packages, forecasts of future values for variables of interest areeasily generated However, this ease of computation does not replace clear thinking

Lack of managerial oversight and improper use of forecasting techniques can lead tocostly decisions

From Chapter 1 of Business Forecasting, Ninth Edition John E Hanke, Dean W Wichern.

Copyright © 2009 by Pearson Education, Inc All rights reserved.

1A recent review of the current state of forecasting is available in a special issue of the International Journal

of Forecasting, edited by R J Hyndman and J K Ord (2006).

New forecasting procedures continue to be developed as the need for accurateforecasts accelerates.1Particular attention is being paid to the forecasting process

in organizations with the need to coordinate objectives, methods, assessment, andinterpretation

IS FORECASTING NECESSARY?

In spite of the inherent inaccuracies in trying to predict the future, forecasts ily drive policy setting and planning How can the Federal Reserve Board realisticallyadjust interest rates without some notion of future economic growth and inflationarypressures? How can an operations manager realistically set production schedules with-out some estimate of future sales? How can a company determine staffing for its callcenters without some guess of the future demand for service? How can a bank makerealistic plans without some forecast of future deposits and loan balances? Everyonerequires forecasts The need for forecasts cuts across all functional lines as well as alltypes of organizations Forecasts are absolutely necessary to move forward in today’sever-changing and highly interactive business environment

necessar-This text discusses various ways of generating forecasts that rely on logical ods of manipulating data that have been generated by historical events But it is ourbelief that the most effective forecaster is able to formulate a skillful mix of quantita-tive forecasting and good judgment and to avoid the extremes of total reliance oneither At one extreme, we find the executive who, through ignorance and fear of quan-titative techniques and computers, relies solely on intuition and feel At the otherextreme is the forecaster skilled in the latest sophisticated data manipulation tech-niques but unable or unwilling to relate the forecasting process to the needs of theorganization and its decision makers We view the quantitative forecasting techniquesdiscussed in most of this text to be only the starting point in the effective forecasting ofoutcomes important to the organization: Analysis, judgment, common sense, and busi-ness experience must be brought to bear on the process through which these importanttechniques have generated their results

meth-Another passage from Bernstein (1996) effectively summarizes the role of casting in organizations

fore-You do not plan to ship goods across the ocean, or to assemble merchandisefor sale, or to borrow money without first trying to determine what the futuremay hold in store Ensuring that the materials you order are delivered on time,seeing to it that the items you plan to sell are produced on schedule, and get-ting your sales facilities in place all must be planned before that moment whenthe customers show up and lay their money on the counter The successfulbusiness executive is a forecaster first; purchasing, producing, marketing, pric-ing, and organizing all follow

TYPES OF FORECASTS

When managers are faced with the need to make decisions in an atmosphere of tainty, what types of forecasts are available to them? Forecasting procedures mightfirst be classified as long term or short term Long-term forecasts are necessary to set

uncer-the general course of an organization for uncer-the long run; thus, uncer-they become uncer-the particularfocus of top management Short-term forecasts are needed to design immediate strate-gies and are used by middle and first-line management to meet the needs of the imme-diate future.

Forecasts might also be classified in terms of their position on a micro–macrocontinuum, that is, in terms of the extent to which they involve small details versuslarge summary values For example, a plant manager might be interested in forecastingthe number of workers needed for the next several months (a micro forecast), whereasthe federal government is forecasting the total number of people employed in theentire country (a macro forecast) Again, different levels of management in an organi-zation tend to focus on different levels of the micro–macro continuum Top manage-ment would be interested in forecasting the sales of the entire company, for example,whereas individual salespersons would be much more interested in forecasting theirown sales volumes

Forecasting procedures can also be classified according to whether they tend to bemore quantitative or qualitative At one extreme, a purely qualitative technique is onerequiring no overt manipulation of data Only the “judgment” of the forecaster is used

Even here, of course, the forecaster’s “judgment” may actually be the result of the tal manipulation of historical data At the other extreme, purely quantitative techniquesneed no input of judgment; they are mechanical procedures that produce quantitativeresults Some quantitative procedures require a much more sophisticated manipulation

men-of data than do others, men-of course This text emphasizes the quantitative forecasting niques because a broader understanding of these very useful procedures is needed inthe effective management of modern organizations However, we emphasize again thatjudgment and common sense must be used along with mechanical and data-manipula-tive procedures Only in this way can intelligent forecasting take place

tech-Finally, forecasts might be classified according to the nature of the output One

must decide if the forecast will be a single number best guess (a point forecast), a range

of numbers within which the future value is expected to fall (an interval forecast), or an entire probability distribution for the future value (a density forecast) Since unpre-

dictable “shocks” will affect future values (the future is never exactly like the past),nonzero forecast errors will occur even from very good forecasts Thus, there is someuncertainty associated with a particular point forecast The uncertainty surroundingpoint forecasts suggests the usefulness of an interval forecast However, if forecastsare solely the result of judgment, point forecasts are typically the only recourse Injudgmental situations, it is extremely difficult to accurately describe the uncertaintyassociated with the forecast

MACROECONOMIC FORECASTING CONSIDERATIONS

We usually think of forecasting in terms of predicting important variables for an vidual company or perhaps for one component of a company Monthly company sales,unit sales for one of a company’s stores, and absent hours per employee per month in afactory are examples

indi-By contrast, there is growing interest in forecasting important variables for theentire economy of a country Much work has been done in evaluating methods for

doing this kind of overall economic forecasting, called macroeconomic forecasting.

Examples of interest to the federal government of the United States are the ment rate, gross domestic product, and prime interest rate Economic policy is based, inpart, on projections of important economic indicators such as these For this reason,

unemploy-there is great interest in improving forecasting methods that focus on overall measures

of a country’s economic performance

One of the chief difficulties in developing accurate forecasts of overall economicactivity is the unexpected and significant shift in a key economic factor Significantchanges in oil prices, inflation surges, and broad policy changes by a country’s govern-ment are examples of shifts in a key factor that can affect the global economy

The possibility of such significant shifts in the economic scene has raised a keyquestion in macroeconomic forecasting: Should the forecasts generated by the fore-casting model be modified using the forecaster’s judgment? Current work on forecast-ing methodology often involves this question

Theoretical and practical work on macroeconomic forecasting continues.Considering the importance of accurate economic forecasting to economic policy for-mulation in this country and others, increased attention to this kind of forecasting can

be expected in the future A good introductory reference for macroeconomic ing is Pindyck and Rubinfeld (1998)

forecast-CHOOSING A FORECASTING METHOD

The preceding discussion suggests several factors to be considered in choosing a casting method The level of detail must be considered Are forecasts of specific detailsneeded (a micro forecast)? Or is the future status of some overall or summary factorneeded (a macro forecast)? Is the forecast needed for some point in the near future(a short-term forecast) or for a point in the distant future (a long-term forecast)? Towhat extent are qualitative (judgment) and quantitative (data-manipulative) methodsappropriate? And, finally, what form should the forecast take (point, interval, or den-sity forecast)?

fore-The overriding consideration in choosing a forecasting method is that the resultsmust facilitate the decision-making process of the organization’s managers Rarelydoes one method work for all cases Different products (for example, new versus estab-lished), goals (for example, simple prediction versus the need to control an importantbusiness driver of future values), and constraints (for example, cost, required expertise,and immediacy) must be considered when selecting a forecasting method With theavailability of current forecasting software, it is best to think of forecasting methods asgeneric tools that can be applied simultaneously Several methods can be tried in agiven situation The methodology producing the most accurate forecasts in one casemay not be the best methodology in another situation However, the method(s) chosenshould produce a forecast that is accurate, timely, and understood by management sothat the forecast can help produce better decisions

The additional discussion available in Chase (1997) can help the forecaster select

an initial set of forecasting procedures to be considered

FORECASTING STEPS

All formal forecasting procedures involve extending the experiences of the pastinto the future Thus, they involve the assumption that the conditions that gener-ated past relationships and data are indistinguishable from the conditions of thefuture

A human resource department is hiring employees, in part, on the basis of a pany entrance examination score because, in the past, that score seemed to be an impor-tant predictor of job performance rating To the extent that this relation continues to

com-hold, forecasts of future job performance—hence hiring decisions—can be improved by

using examination scores If, for some reason, the association between examination

score and job performance changes, then forecasting job performance ratings from

examination scores using the historical model will yield inaccurate forecasts and

poten-tially poor hiring decisions This is what makes forecasting difficult The future is not

always like the past To the extent it is, quantitative forecasting methods work well To

the extent it isn’t, inaccurate forecasts can result However, it is generally better to have

some reasonably constructed forecast than no forecast

The recognition that forecasting techniques operate on the data generated by

histor-ical events leads to the identification of the following five steps in the forecasting process:

1 Problem formulation and data collection

2 Data manipulation and cleaning

3 Model building and evaluation

4 Model implementation (the actual forecast)

5 Forecast evaluation

In step 1, problem formulation and data collection are treated as a single step

because they are intimately related The problem determines the appropriate data If a

quantitative forecasting methodology is being considered, the relevant data must be

available and correct Often accessing and assembling appropriate data is a challenging

and time-consuming task If appropriate data are not available, the problem may have

to be redefined or a nonquantitative forecasting methodology employed Collection

and quality control problems frequently arise whenever it becomes necessary to obtain

pertinent data for a business forecasting effort

Step 2, data manipulation and cleaning, is often necessary It is possible to have too

much data as well as too little in the forecasting process Some data may not be

rele-vant to the problem Some data may have missing values that must be estimated Some

data may have to be reexpressed in units other than the original units Some data may

have to be preprocessed (for example, accumulated from several sources and

summed) Other data may be appropriate but only in certain historical periods (for

example, in forecasting the sales of small cars, one may wish to use only car sales data

since the oil embargo of the 1970s rather than sales data over the past 60 years)

Ordinarily, some effort is required to get data into the form that is required for using

certain forecasting procedures

Step 3, model building and evaluation, involves fitting the collected data into a

forecasting model that is appropriate in terms of minimizing forecasting error The

sim-pler the model is, the better it is in terms of gaining acceptance of the forecasting

process by managers who must make the firm’s decisions Often a balance must be

struck between a sophisticated forecasting approach that offers slightly more accuracy

and a simple approach that is easily understood and gains the support of—and is

actively used by—the company’s decision makers Obviously, judgment is involved in

this selection process Since this text discusses numerous forecasting models and their

applicability, the reader’s ability to exercise good judgment in the choice and use of

appropriate forecasting models will increase after studying this material

Step 4, model implementation, is the generation of the actual model forecasts once

the appropriate data have been collected and cleaned and an appropriate forecasting

model has been chosen Data for recent historical periods are often held back and later

used to check the accuracy of the process

Step 5, forecast evaluation, involves comparing forecast values with actual

his-torical values After implementation of the forecasting model is complete, forecasts

are made for the most recent historical periods where data values were known butheld back from the data set being analyzed These forecasts are then compared withthe known historical values, and any forecasting errors are analyzed Some forecast-ing procedures sum the absolute values of the errors and may report this sum, orthey divide this sum by the number of forecast attempts to produce the averageforecast error Other procedures produce the sum of squared errors, which is thencompared with similar figures from alternative forecasting methods Some proce-dures also track and report the magnitude of the error terms over the forecastingperiod Examination of error patterns often leads the analyst to modify the forecasting model.

MANAGING THE FORECASTING PROCESS

The discussion in this chapter serves to underline our belief that management abilityand common sense must be involved in the forecasting process The forecaster should

be thought of as an advisor to the manager rather than as a monitor of an automaticdecision-making device Unfortunately, the latter is sometimes the case in practice,especially with the aura of the computer Again, quantitative forecasting techniquesmust be seen as what they really are, namely, tools to be used by the manager in arriv-ing at better decisions According to Makridakis (1986):

The usefulness and utility of forecasting can be improved if managementadopts a more realistic attitude Forecasting should not be viewed as a substi-tute for prophecy but rather as the best way of identifying and extrapolatingestablished patterns or relationships in order to forecast If such an attitude isaccepted, forecasting errors must be considered inevitable and the circum-stances that cause them investigated

Given that, several key questions should always be raised if the forecasting process is

to be properly managed:

• Why is a forecast needed?

• Who will use the forecast, and what are their specific requirements?

• What level of detail or aggregation is required, and what is the proper time horizon?

• What data are available, and will the data be sufficient to generate the neededforecast?

• What will the forecast cost?

• How accurate can we expect the forecast to be?

• Will the forecast be made in time to help the decision-making process?

• Does the forecaster clearly understand how the forecast will be used in theorganization?

• Is a feedback process available to evaluate the forecast after it is made and toadjust the forecasting process accordingly?

FORECASTING SOFTWARE

Today, there are a large number of computer software packages specifically designed

to provide the user with various forecasting methods Two types of computer packagesare of primary interest to forecasters: (1) general statistical packages that includeregression analysis, time series analysis, and other techniques used frequently by

2 At the time this text was written, the Institute for Forecasting Education provided reviews of

forecasting software on its website These reviews can be accessed at www.forecastingeducation.com/

forecastingsoftwarereviews.asp.

forecasters and (2) forecasting packages that are specifically designed for forecastingapplications In addition, some forecasting tools are available in Enterprise ResourcePlanning (ERP) systems

Graphical capabilities, interfaces to spreadsheets and external data sources, cally and statistically reliable methods, and simple automatic algorithms for the selectionand specification of forecasting models are now common features of business forecastingsoftware However, although development and awareness of forecasting software haveincreased dramatically in recent years, the majority of companies still use spreadsheets(perhaps with add-ins) to generate forecasts and develop business plans

numeri-Examples of stand-alone software packages with forecasting tools includeMinitab, SAS, and SPSS There are many add-ins or supplemental programs that pro-vide forecasting tools in a spreadsheet environment For example, the AnalysisToolPak add-in for Microsoft Excel provides some regression analysis and smoothingcapabilities There are currently several more comprehensive add-ins that provide a(almost) full range of forecasting capabilities.2

It is sometimes the case, particularly in a spreadsheet setting, that “automatic”

forecasting is available That is, the software selects the best model or procedure forforecasting and immediately generates forecasts We caution, however, that this con-venience comes at a price Automatic procedures produce numbers but rarely providethe forecaster with real insight into the nature and quality of the forecasts The genera-tion of meaningful forecasts requires human intervention, a give and take betweenproblem knowledge and forecasting procedures (software)

Many of the techniques in this text will be illustrated with Minitab 15 andMicrosoft Excel 2003 (with the Analysis ToolPak add-in) Minitab 15 was chosen for itsease of use and widespread availability Excel, although limited in its forecasting func-tionality, is frequently the tool of choice for calculating projections

ONLINE INFORMATION

Information of interest to forecasters is available on the World Wide Web Perhaps thebest way to learn about what’s available in cyberspace is to spend some time searching forwhatever interests you, using a browser such as Netscape or Microsoft Internet Explorer

Any list of websites for forecasters is likely to be outdated by the time this textappears; however, there are two websites that are likely to remain available for some

time B&E DataLinks, available at www.econ-datalinks.org, is a website maintained

by the Business and Economic Statistics Section of the American Statistical Association

This website contains extensive links to economic and financial data sources of interest

to forecasters The second site, Resources for Economists on the Internet, sponsored

by the American Economic Association and available at rfe.org, contains an extensive set

of links to data sources, journals, professional organizations, and so forth

FORECASTING EXAMPLES

Discussions in this chapter emphasize that forecasting requires a great deal of ment along with the mathematical manipulation of collected data The following exam-ples demonstrate the kind of thinking that often precedes a forecasting effort in a real

judg-firm Notice that the data values that will produce useful forecasts, even if they exist,may not be apparent at the beginning of the process and may or may not be identified

as the process evolves In other words, the initial efforts may turn out to be useless andanother approach required

The results of the forecasting efforts for the two examples discussed here are notshown, as they require topics that are described throughout the text Look for the tech-niques to be applied to these data For the moment, we hope these examples illustratethe forecasting effort that real managers face

Example 1 Alomega Food Stores

Alomega Food Stores is a retail food provider with 27 stores in a midwestern state The company engages in various kinds of advertising and, until recently, had never studied the effect its advertising dollars have on sales, although some data had been collected and stored for 3 years.

The executives at Alomega decided to begin tracking their advertising efforts along with the sales volumes for each month Their hope was that after several months the col- lected data could be examined to possibly reveal relationships that would help in determin- ing future advertising expenditures.

The accounting department began extending its historical records by recording the sales volume for each month along with the advertising dollars for both newspaper ads and

TV spots They also recorded both sales and advertising values that had been lagged for one and two months This was done because some people on the executive committee thought that sales might depend on advertising expenditures in previous months rather than in the month the sales occurred.

The executives also believed that sales experienced a seasonal effect For this reason, a dummy or categorical variable was used to indicate each month In addition, they wondered about any trend in sales volume.

Finally, the executives believed that Alomega’s advertising dollars might have an effect

on its major competitors’ advertising budgets the following month For each following month, it was decided that competitors’ advertising could be classified as a (1) small amount, (2) a moderate amount, or (3) a large amount.

After a few months of collecting data and analyzing past records, the accounting department completed a data array for 48 months using the following variables:

• Sales dollars

• Newspaper advertising dollars

• TV advertising dollars

• Month code where January ⫽ 1, February ⫽ 2, through December ⫽ 12

• A series of 11 dummy variables to indicate month

• Newspaper advertising lagged one month

• Newspaper advertising lagged two months

• TV advertising lagged one month

• TV advertising lagged two months

• Month number from 1 to 48

• Code 1, 2, or 3 to indicate competitors’ advertising efforts the following month Alomega managers, especially Julie Ruth, the company president, now want to learn anything they can from the data they have collected In addition to learning about the effects of advertising on sales volumes and competitors’ advertising, Julie wonders about any trend and the effect of season on sales However, the company’s production manager, Jackson Tilson, does not share her enthusiasm At the end of the forecasting planning meeting, he makes the following statement: “I’ve been trying to keep my mouth shut during this meeting, but this is really too much I think we’re wasting a lot of people’s time with all this data collection and fooling around with computers All you have

to do is talk with our people on the floor and with the grocery store managers to understand what’s going on I’ve seen this happen around here before, and here we go again Some

of you people need to turn off your computers, get out of your fancy offices, and talk with a few real people.”

Example 1.2 Large Web-based Retailer

One of the goals of a large Internet-based retailer is to be the world’s most centric company The company recognizes that the ability to establish and maintain long- term relationships with customers and to encourage repeat visits and purchases depends, in part, on the strength of its customer service operations For service matters that cannot be handled using website features, customer service representatives located in contact centers are available 24 hours a day to field voice calls and emails.

consumer-Because of its growing sales and its seasonality (service volume is relatively low in the summer and high near the end of the year), a challenge for the company is to appropriately staff its contact centers The planning problem involves making decisions about hiring and training at internally managed centers and about allocating work to outsourcers based on the volume of voice calls and emails The handling of each contact type must meet a targeted service level every week.

To make the problem even more difficult, the handling time for each voice call and email is affected by a number of contact attributes, including type of product, customer, and purchase type These attributes are used to classify the contacts into categories: in this case, one “primary” category and seven “specialty” categories Specific skill sets are needed to resolve the different kinds of issues that arise in the various categories Since hiring and training require a 6-week lead time, forecasts of service contacts are necessary in order to have the required number of service representatives available 24 hours a day, 7 days a week throughout the year.

Pat Niebuhr and his team are responsible for developing a global staffing plan for the contact centers His initial challenge is to forecast contacts for the primary and specialty cat- egories Pat must work with monthly forecasts of total orders (which, in turn, are derived from monthly revenue forecasts) and contacts per order (CPO) numbers supplied by the finance department Pat recognizes that contacts are given by

Contacts ⫽ Orders ⫻ CPO For staff planning purposes, Pat must have forecasts of contacts on a weekly basis.

Fortunately, there is a history of actual orders, actual contacts, actual contacts per order, and other relevant information, in some cases, recorded by day of the week This history is organized in a spreadsheet Pat is considering using this historical information to develop the forecasts he needs.

Summary

The purpose of a forecast is to reduce the range of uncertainty within which ment judgments must be made This purpose suggests two primary rules to which theforecasting process must adhere:

manage-1 The forecast must be technically correct and produce forecasts accurate enough tomeet the firm’s needs

2 The forecasting procedure and its results must be effectively presented to ment so that the forecasts are used in the decision-making process to the firm’sadvantage; results must also be justified on a cost-benefit basis

manage-Forecasters often pay particular attention to the first rule and expend less effort on thesecond Yet if well-prepared and cost-effective forecasts are to benefit the firm, thosewho have the decision-making authority must use them This raises the question ofwhat might be called the “politics” of forecasting Substantial and sometimes majorexpenditures and resource allocations within a firm often rest on management’s view

of the course of future events Because the movement of resources and power within

an organization is often based on the perceived direction of the future (forecasts), it isnot surprising to find a certain amount of political intrigue surrounding the forecastingprocess The need to be able to effectively sell forecasts to management is at least asimportant as the need to be able to develop the forecasts

The remainder of this text discusses various forecasting models and procedures.First, we review basic statistical concepts and provide an introduction to correlationand regression analysis.

CASES

CASE 1 MR TUX

John Mosby owns several Mr Tux rental stores, most

of them in the Spokane, Washington, area.3 His

Spokane store also makes tuxedo shirts, which he

dis-tributes to rental shops across the country Because

rental activity varies from season to season due to

proms, reunions, and other activities, John knows that

his business is seasonal He would like to measure this

seasonal effect, both to assist him in managing his

business and to use in negotiating a loan repayment

with his banker

Of even greater interest to John is finding a way

of forecasting his monthly sales His business

contin-ues to grow, which, in turn, requires more capital and

3 We are indebted to John Mosby, the owner of Mr Tux rental stores, for his help in preparing this case.

4 We are indebted to Marv Harnishfeger, executive director of Consumer Credit Counseling of Spokane, and Dorothy Mercer, president of its board of directors, for their help in preparing the case Dorothy is a former M.B.A student of JH who has consistently kept us in touch with the use of quantitative methods in the real world of business.

long-term debt He has sources for both types ofgrowth financing, but investors and bankers areinterested in a concrete way of forecasting futuresales Although they trust John, his word that thefuture of his business “looks great” leaves themuneasy

As a first step in building a forecasting model,John directs one of his employees, McKennah Lane,

to collect monthly sales data for the past severalyears Various techniques are used to forecast thesesales data for Mr Tux and John Mosby attempts tochoose the forecasting technique that will best meethis needs

CASE 2 CONSUMER CREDIT

COUNSELING

Consumer Credit Counseling (CCC), a private,

non-profit corporation, was founded in 1982.4The

pur-pose of CCC is to provide consumers with assistance

in planning and following budgets, with assistance in

making arrangements with creditors to repay

delin-quent obligations, and with money management

education

Private financial counseling is provided at no

cost to families and individuals who are experiencing

financial difficulties or who want to improve theirmoney management skills Money managementeducational programs are provided for schools, com-munity groups, and businesses A debt managementprogram is offered as an alternative to bankruptcy.Through this program, CCC negotiates with credi-tors on behalf of the client for special paymentarrangements The client makes a lump-sum pay-ment to CCC that is then disbursed to creditors

Minitab Applications

Minitab is a sophisticated statistical program that improves with each release

Described here is Release 15

Figure 1 shows you four important aspects of Minitab The menu bar is where youchoose commands For instance, click on Stat and a pull-down menu appears that con-tains all of the statistical techniques available The toolbar displays buttons for com-monly used functions Note that these buttons change depending on which Minitabwindow is open There are two separate windows on the Minitab screen: the data win-dow, where you enter, edit, and view the column data for each worksheet; and the ses-sion window, which displays text output, such as tables of statistics

Specific instructions will be given to enable you to enter data into the Minitabspreadsheet and to activate forecasting procedures to produce needed forecasts

CCC has a blend of paid and volunteer staff; in

fact, volunteers outnumber paid staff three to one

Seven paid staff provide management, clerical

sup-port, and about half of the counseling needs for

CCC Twenty-one volunteer counselors fulfill the

other half of the counseling needs of the service

CCC depends primarily on corporate funding to

support operations and services The Fair Share

Funding Program allows creditors who receive

pay-ments from client debt management programs to

donate back to the service a portion of the funds

returned to them through these programs

A major portion of corporate support comesfrom a local utility that provides funding to support

a full-time counselor position as well as office spacefor counseling at all offices

In addition, client fees are a source of funding

Clients who participate in debt management pay amonthly fee of $15 to help cover the administrativecost of this program (Fees are reduced or waivedfor clients who are unable to afford them.)

This background will be used as CCC faces cult problems related to forecasting important variables

diffi-Menu bar Toolbar

Session window

Data window

FIGURE 1 Basic Minitab Screen

Menu bar Toolbar Formula bar

For example, annual salaries for a number of employees could be entered into umn A and the average of these values calculated by Excel As another example,employee ages could be placed in column B and the relationship between age andsalary examined

col-There are several statistical functions available on Excel that may not be on thedrop-down menus on your screen To activate these functions, click on the following:Tools>Add-Ins

The Add-Ins dialog box appears Select Analysis ToolPak and click on OK

It is strongly recommended that an Excel add-in be used to help with the tude of statistical computations required by the forecasting techniques discussed inthis text

multi-References

Bernstein, P L Against the Gods: The Remarkable

Story of Risk New York: Wiley, 1996.

Carlberg, C “Use Excel’s Forecasting to Get

Terrific Projections.” Denver Business Journal 47

(18) (1996): 2B

Chase, C W., Jr “Selecting the Appropriate

Forecasting Method.” Journal of Business

Forecasting 15 (Fall 1997): 2.

Diebold, F X Elements of Forecasting, 3rd ed.

Cincinnati, Ohio: South-Western, 2004

Georgoff, D M., and R G Mardick “Manager’s

Guide to Forecasting.” Harvard Business Review

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EXPLORING DATA PATTERNS

AND AN INTRODUCTION TO

FORECASTING TECHNIQUES

One of the most time-consuming and difficult parts of forecasting is the collection of

valid and reliable data Data processing personnel are fond of using the expression

“garbage in, garbage out” (GIGO) This expression also applies to forecasting A

fore-cast can be no more accurate than the data on which it is based The most sophisticated

forecasting model will fail if it is applied to unreliable data

Modern computer power and capacity have led to an accumulation of an

incred-ible amount of data on almost all subjects The difficult task facing most forecasters

is how to find relevant data that will help solve their specific decision-making

problems

Four criteria can be applied when determining whether data will be useful:

1 Data should be reliable and accurate Proper care must be taken that data are

col-lected from a reliable source with proper attention given to accuracy

2 Data should be relevant The data must be representative of the circumstances for

which they are being used

3 Data should be consistent When definitions concerning data collection change,

adjustments need to be made to retain consistency in historical patterns This

can be a problem, for example, when government agencies change the mix or

“market basket” used in determining a cost-of-living index Years ago personal

computers were not part of the mix of products being purchased by consumers;

now they are

4 Data should be timely Data collected, summarized, and published on a timely

basis will be of greatest value to the forecaster There can be too little data (not

enough history on which to base future outcomes) or too much data (data from

irrelevant historical periods far in the past)

Generally, two types of data are of interest to the forecaster The first is data

col-lected at a single point in time, be it an hour, a day, a week, a month, or a quarter The

second is observations of data made over time When all observations are from the

same time period, we call them cross-sectional data The objective is to examine such

data and then to extrapolate or extend the revealed relationships to the larger

popu-lation Drawing a random sample of personnel files to study the circumstances of the

employees of a company is one example Gathering data on the age and current

maintenance cost of nine Spokane Transit Authority buses is another A scatter

dia-gram such as Figure 1 helps us visualize the relationship and suggests age might be

used to help in forecasting the annual maintenance budget

From Chapter 3 of Business Forecasting, Ninth Edition John E Hanke, Dean W Wichern.

Copyright © 2009 by Pearson Education, Inc All rights reserved.

Cross-sectional data are observations collected at a single point in time.

A time series consists of data that are collected, recorded, or observed over

suc-cessive increments of time

FIGURE 1 Scatter Diagram of Age and Maintenance Cost for

Nine Spokane Transit Authority Buses

Any variable that consists of data that are collected, recorded, or observed over

successive increments of time is called a time series Monthly U.S beer production is an

example of a time series

EXPLORING TIME SERIES DATA PATTERNS

One of the most important steps in selecting an appropriate forecasting method fortime series data is to consider the different types of data patterns There are typicallyfour general types of patterns: horizontal, trend, seasonal, and cyclical

When data collected over time fluctuate around a constant level or mean, a

horizontal pattern exists This type of series is said to be stationary in its mean Monthly

sales for a food product that do not increase or decrease consistently over an extendedperiod would be considered to have a horizontal pattern

When data grow or decline over several time periods, a trend pattern exists Figure 2

shows the long-term growth (trend) of a time series variable (such as housing costs)with data points one year apart A linear trend line has been drawn to illustrate thisgrowth Although the variable housing costs have not increased every year, the move-ment of the variable has been generally upward between periods 1 and 20 Examples ofthe basic forces that affect and help explain the trend of a series are population growth,price inflation, technological change, consumer preferences, and productivity increases

10 15 20

Many macroeconomic variables, like the U.S gross national product (GNP),

employment, and industrial production exhibit trendlike behavior Figure 10 contains

another example of a time series with a prevailing trend This figure shows the growth

of operating revenue for Sears from 1955 to 2004

The cyclical component is the wavelike fluctuation around the trend.

The trend is the long-term component that represents the growth or decline in

the time series over an extended period of time

FIGURE 2 Trend and Cyclical Components of an Annual

Time Series Such as Housing Costs

When observations exhibit rises and falls that are not of a fixed period, a cyclical

pattern exists The cyclical component is the wavelike fluctuation around the trend that

is usually affected by general economic conditions A cyclical component, if it exists,

typically completes one cycle over several years Cyclical fluctuations are often

influ-enced by changes in economic expansions and contractions, commonly referred to as

the business cycle Figure 2 also shows a time series with a cyclical component The

cyclical peak at year 9 illustrates an economic expansion and the cyclical valley at year

12 an economic contraction

When observations are influenced by seasonal factors, a seasonal pattern exists

The seasonal component refers to a pattern of change that repeats itself year after year.

For a monthly series, the seasonal component measures the variability of the series

each January, each February, and so on For a quarterly series, there are four seasonal

elements, one for each quarter Figure 3 shows that electrical usage for Washington

Electrical Usage for Washington Water Power: 1980 –1991

1,100

500 600 700 800 900 1,000

The concept of autocorrelation is illustrated by the data presented in Table 1 Notethat variables are actually the Y values that have been lagged by one and

two time periods, respectively The values for March, which are shown on the row fortime period 3, are March sales, ; February sales, ; and January sales,

Autocorrelation is the correlation between a variable lagged one or more time

periods and itself

The seasonal component is a pattern of change that repeats itself year after year.

EXPLORING DATA PATTERNS WITH AUTOCORRELATION ANALYSIS

When a variable is measured over time, observations in different time periods are quently related or correlated This correlation is measured using the autocorrelationcoefficient

fre-Equation 1 is the formula for computing the lag k autocorrelation coefficient (r k)

between observations, Y tand that are k periods apart.

(1)

where

Example 1

Harry Vernon has collected data on the number of VCRs sold last year by Vernon’s Music

Store The data are presented in Table 1 Table 2 shows the computations that lead to the

cal-culation of the lag 1 autocorrelation coefficient Figure 4 contains a scatter diagram of the

pairs of observations ( ) It is clear from the scatter diagram that the lag 1 correlation

will be positive.

The lag 1 autocorrelation coefficient (r1), or the autocorrelation between Y tand ,

is computed using the totals from Table 2 and Equation 1 Thus,

As suggested by the plot in Figure 4, positive lag 1 autocorrelation exists in this time

series The correlation between Y tand or the autocorrelation for time lag 1, is 572 This

means that the successive monthly sales of VCRs are somewhat correlated with each other.

This information may give Harry valuable insights about his time series, may help him

pre-pare to use an advanced forecasting method, and may warn him about using regression

analysis with his data.

Y t-1

r1 = a

n

t = 1 + 1 1Y t-1 - Y 21Y t - Y2 a

n

t = 1 1Y t- Y2 2

= 843 1,474 = .572

Y t-1

Y t , Y t - 1

Y t - k = the observation k time periods earlier or at time period t - k

Y t = the observation in time period t

Y = the mean of the values of the series

r k = the autocorrelation coefficient for a lag of k periods

r k =a

n

t = k + 1 1Y t -Y 21Y t - k -Y2a

Y Lagged Two Periods Y t-2

The second-order autocorrelation coefficient (r2), or the correlation between Y tand for Harry’s data, is also computed using Equation 1.

r2 = a

n

t = 2 + 1 1Y t - Y 21Y t - 2 - Y2 a

n

t = 1 1Y t- Y2 2

= 682 1,474 = .463

Y t-2

TABLE 2 Computation of the Lag 1 Autocorrelation Coefficient

for the Data in Table 1

The correlogram or autocorrelation function is a graph of the autocorrelations for

various lags of a time series

It appears that moderate autocorrelation exists in this time series lagged two time

peri-ods The correlation between Y tand , or the autocorrelation for time lag 2, is 463 Notice

that the autocorrelation coefficient at time lag 2 (.463) is less than the autocorrelation

coef-ficient at time lag 1 (.572) Generally, as the number of time lags (k) increases, the

magni-tudes of the autocorrelation coefficients decrease.

Figure 5 shows a plot of the autocorrelations versus time lags for the Harry Vernon

data used in Example 1 The horizontal scale on the bottom of the graph shows each time

lag of interest: 1, 2, 3, and so on.The vertical scale on the left shows the possible range of an

autocorrelation coefficient, -1 to 1 The horizontal line in the middle of the graph

repre-sents autocorrelations of zero.The vertical line that extends upward above time lag 1 shows

an autocorrelation coefficient of 57, or The vertical line that extends upward

above time lag 2 shows an autocorrelation coefficient of 46, or The dotted lines

and the T (test) and LBQ (Ljung-Box Q) statistics displayed in the Session window will be

discussed in Examples 2 and 3 Patterns in a correlogram are used to analyze key features

of the data, a concept demonstrated in the next section The Minitab computer package

(see the Minitab Applications section at the end of the chapter for specific instructions) can

be used to compute autocorrelations and develop correlograms

With a display such as that in Figure 5, the data patterns, including trend and sonality, can be studied Autocorrelation coefficients for different time lags for a vari-able can be used to answer the following questions about a time series:

sea-1 Are the data random?

2 Do the data have a trend (are they nonstationary)?

3 Are the data stationary?

4 Are the data seasonal?

If a series is random, the autocorrelations between Y tand for any time lag k

are close to zero The successive values of a time series are not related to each other

If a series has a trend, successive observations are highly correlated, and the correlation coefficients typically are significantly different from zero for the first sev-eral time lags and then gradually drop toward zero as the number of lags increases.The autocorrelation coefficient for time lag 1 will often be very large (close to 1) Theautocorrelation coefficient for time lag 2 will also be large However, it will not be aslarge as for time lag 1

auto-If a series has a seasonal pattern, a significant autocorrelation coefficient will occur

at the seasonal time lag or multiples of the seasonal lag The seasonal lag is 4 for terly data and 12 for monthly data

quar-How does an analyst determine whether an autocorrelation coefficient is cantly different from zero for the data of Table 1? Quenouille (1949) and others havedemonstrated that the autocorrelation coefficients of random data have a sampling dis-tribution that can be approximated by a normal curve with a mean of zero and anapproximate standard deviation of Knowing this, the analyst can compare thesample autocorrelation coefficients with this theoretical sampling distribution and deter-mine whether, for given time lags, they come from a population whose mean is zero.Actually, some software packages use a slightly different formula, as shown inEquation 2, to compute the standard deviations (or standard errors) of the autocorre-

signifi-lation coefficients This formula assumes that any autocorresignifi-lation before time lag k is different from zero and any autocorrelation at time lags greater than or equal to k is

zero For an autocorrelation at time lag 1, the standard error is used

n = the number of observations in the time series

k = the time lag

r i = the autocorrelation at time lag i

of the autocorrelation at time lag k

SE 1r k2 = the standard error 1estimated standard deviation2

Although testing each r kto see if it is individually significantly different from 0 is

useful, it is also good practice to examine a set of consecutive r k ’s as a group We can

use a portmanteau test to see whether the set, say, of the first 10 r kvalues, is

signifi-cantly different from a set in which all 10 values are zero

One common portmanteau test is based on the Ljung-Box Q statistic (Equation

3) This test is usually applied to the residuals of a forecast model If the

autocorre-lations are computed from a random (white noise) process, the statistic Q has a

chi-square distribution with m (the number of time lags to be tested) degrees of

free-dom For the residuals of a forecast model, however, the statistic Q has a chi-square

distribution with the degrees of freedom equal to m minus the number of

parame-ters estimated in the model The value of the Q statistic can be compared with the

chi-square table (Table 4 in Appendix: Tables) to determine if it is larger than we

would expect it to be under the null hypothesis that all the autocorrelations in the

set are zero Alternatively, the p-value generated by the test statistic Q can be

com-puted and interpreted The Q statistic is given in Equation 3 It will be demonstrated

in Example 3

(3)

where

Are the Data Random?

A simple random model, often called a white noise model, is displayed in Equation 4.

Observation Y t is composed of two parts: c, the overall level, and ␧t, which is the

ran-dom error component It is important to note that the ␧tcomponent is assumed to be

uncorrelated from period to period

(4)

Are the data in Table 1 consistent with this model? This issue will be explored in

Examples 2 and 3

Example 2

A hypothesis test is developed to determine whether a particular autocorrelation coefficient

is significantly different from zero for the correlogram shown in Figure 5 The null and

alter-native hypotheses for testing the significance of the lag 1 population autocorrelation

r k = the sample autocorrelation function of the residuals lagged k time periods

m = the number of time lags to be tested

k = the time lag

n = the number of observations in the time series

has a t distribution with Here, , so for a 5% significance level, the decision rule is as follows:

If t 2.2 or t 2.2, reject H0and conclude the lag 1 autocorrelation is significantly different from 0.

The critical values ⫾2.2 are the upper and lower 025 points of a t distribution with 11

value of the test statistic becomes

Using the decision rule above, cannot be rejected, since -2.2 < 1.98 < 2.2.

Notice the value of our test statistic, , is the same as the quantity in the Lag 1 row under the heading T in the Minitab output in Figure 5 The T values in the Minitab out- put are simply the values of the test statistic for testing for zero autocorrelation at the various lags.

To test for zero autocorrelation at time lag 2, we consider

and the test statistic

Using Equation 2,

and

This result agrees with the T value for Lag 2 in the Minitab output in Figure 5.

Using the decision rule above, cannot be rejected at the 05 level, since

- 2.2 ⬍ 1.25 ⬍ 2.2 An alternative way to check for significant autocorrelation is to construct, say, 95% confidence limits centered at 0 These limits for time lags 1 and 2 are as follows:

Autocorrelation significantly different from 0 is indicated whenever a value for r kfalls side the corresponding confidence limits The 95% confidence limits are shown in Figure 5 by the dashed lines in the graphical display of the autocorrelation function.

out-Example 3

Minitab was used to generate the time series of 40 pseudo-random three-digit numbers shown in Table 3 Figure 6 shows a time series graph of these data Because these data are random (independent of one another and all from the same population), autocorrelations for

lag 2: 0 ; t.025* SE 1r22 or 0 ; 2.21.3712 : 1-.816, 8162 lag 1: 0 ; t.025* SE 1r12 or 0 ; 2.21.2892 : 1-.636, 6362

n - 1 = 12 - 1 = 11

df = n - 1

Y t

all time lags should theoretically be equal to zero Of course, the 40 values in Table 3 are only

one set of a large number of possible samples of size 40 Each sample will produce different

autocorrelations Most of these samples will produce sample autocorrelation coefficients that

are close to zero However, it is possible that a sample will produce an autocorrelation

coef-ficient that is significantly different from zero just by chance.

Next, the autocorrelation function shown in Figure 7 is constructed using Minitab Note

that the two dashed lines show the 95% confidence limits Ten time lags are examined, and all

the individual autocorrelation coefficients lie within these limits There is no reason to doubt

that each of the autocorrelations for the first 10 lags is zero However, even though the

individ-ual sample autocorrelations are not significantly different from zero, are the magnitudes of

the first 10 r k’s as a group larger than one would expect under the hypothesis of no

autocorre-lation at any lag? This question is answered by the Ljung-Box Q (LBQ in Minitab) statistic.

If there is no autocorrelation at any lag, the Q statistic has a chi-square distribution

with, in this case, Consequently, a large value for Q (in the tail of the chi-square

dis-tribution) is evidence against the null hypothesis From Figure 7, the value of Q (LBQ) for

10 time lags is 7.75 From Table 4 in Appendix: Tables, the upper 05 point of a chi-square

distribution with 10 degrees of freedom is 18.31 Since 7.75 < 18.31, the null hypothesis

can-not be rejected at the 5% significance level These data are uncorrelated at any time lag, a

result consistent with the model in Equation 4.

Do the Data Have a Trend?

If a series has a trend, a significant relationship exists between successive time seriesvalues The autocorrelation coefficients are typically large for the first several time lagsand then gradually drop toward zero as the number of lags increases

A stationary time series is one whose basic statistical properties, such as the mean

and variance, remain constant over time Consequently, a series that varies about a fixedlevel (no growth or decline) over time is said to be stationary A series that contains a

trend is said to be nonstationary The autocorrelation coefficients for a stationary series

decline to zero fairly rapidly, generally after the second or third time lag On the otherhand, sample autocorrelations for a nonstationary series remain fairly large for severaltime periods Often, to analyze nonstationary series, the trend is removed before addi-tional modeling occurs

A method called differencing can often be used to remove the trend from a

nonsta-tionary series The VCR data originally presented in Table 1 are shown again in

Figure 8, column A The Y tvalues lagged one period, , are shown in column B.The differences, (column A - column B), are shown in column C Forexample, the first value for the differences is Note theupward growth or trend of the VCR data shown in Figure 9, plot A Now observethe stationary pattern of the differenced data in Figure 9, plot B Differencing thedata has removed the trend

Example 4

Maggie Trymane, an analyst for Sears, is assigned the task of forecasting operating revenue for 2005 She gathers the data for the years 1955 to 2004, shown in Table 4 The data are plotted as a time series in Figure 10 Notice that, although Sears operating revenues were declining over the 2000–2004 period, the general trend over the entire 1955–2004 time frame is up First, Maggie computes a 95% confidence interval for the autocorrelation coef- ficients at time lag 1 using where, for large samples, the standard normal

.025 point has replaced the corresponding t distribution percentage point:

Next, Maggie runs the data on Minitab and produces the autocorrelation function shown in Figure 11 Upon examination, she notices that the autocorrelations for the first

FIGURE 8 Excel Results for Differencing the VCR Data of Example 1

Y t

FIGURE 9 Time Series Plots of the VCR Data and the

Differenced VCR Data of Example 1

FIGURE 10 Time Series Plot of Sears Operating Revenue

for Example 4

TABLE 4 Yearly Operating Revenue for Sears, 1955–2004, for Example 4

Year Y t Year Y t Year Y t Year Y t Year Y t

Source: Industry Surveys, various years.

four time lags are significantly different from zero (.96, 92, 87, and 81) and that the values

then gradually drop to zero As a final check, Maggie looks at the Q statistic for 10 time lags.

The LBQ is 300.56, which is greater than the chi-square value 18.3 (the upper 05 point of a chi-square distribution with 10 degrees of freedom) This result indicates the autocorrela- tions for the first 10 lags as a group are significantly different from zero She decides that the data are highly autocorrelated and exhibit trendlike behavior.

Maggie suspects that the series can be differenced to remove the trend and to create a stationary series She differences the data (see the Minitab Applications section at the end

of the chapter), and the results are shown in Figure 12 The differenced series shows no dence of a trend, and the autocorrelation function, shown in Figure 13, appears to support this conclusion Examining Figure 13, Maggie notes that the autocorrelation coefficient at time lag 3, 32, is significantly different from zero (tested at the 05 significance level) The autocorrelations at lags other than lag 3 are small, and the LBQ statistic for 10 lags is also relatively small, so there is little evidence to suggest the differenced data are autocorrelated Yet Maggie wonders whether there is some pattern in these data that can be modeled by one of the more advanced forecasting techniques.

evi-FIGURE 11 Autocorrelation Function for Sears Operating

Revenue for Example 4

FIGURE 12 Time Series Plot of the First Differences of the

Sears Operating Revenue for Example 4

FIGURE 13 Autocorrelation Function for the First Differences of

the Sears Operating Revenue for Example 4

Are the Data Seasonal?

If a series is seasonal, a pattern related to the calendar repeats itself over a particularinterval of time (usually a year) Observations in the same position for different sea-sonal periods tend to be related If quarterly data with a seasonal pattern are analyzed,first quarters tend to look alike, second quarters tend to look alike, and so forth, and asignificant autocorrelation coefficient will appear at time lag 4 If monthly data areanalyzed, a significant autocorrelation coefficient will appear at time lag 12 That is,January will correlate with other Januarys, February will correlate with otherFebruarys, and so on Example 5 discusses a series that is seasonal

Example 5

Perkin Kendell is an analyst for the Coastal Marine Corporation He has always felt that sales were seasonal Perkin gathers the data shown in Table 5 for the quarterly sales of the Coastal Marine Corporation from 1994 to 2006 and plots them as the time series graph shown in Figure 14 Next, he computes a large-sample 95% confidence interval for the auto- correlation coefficient at time lag 1:

Then Perkin computes the autocorrelation coefficients shown in Figure 15 He notes that the autocorrelation coefficients at time lags 1 and 4 are significantly different from zero ( ) He concludes that Coastal Marine sales are seasonal

on a quarterly basis.

CHOOSING A FORECASTING TECHNIQUE

This text is devoted mostly to explaining various forecasting techniques and strating their usefulness First, the important job of choosing among several forecastingtechniques is addressed

demon-Some of the questions that must be considered before deciding on the most priate forecasting technique for a particular problem follow:

appro-• Why is a forecast needed?

• Who will use the forecast?

• What are the characteristics of the available data?

• What time period is to be forecast?

• What are the minimum data requirements?

• How much accuracy is desired?

• What will the forecast cost?

To select the appropriate forecasting technique properly, the forecaster must be

able to accomplish the following:

• Define the nature of the forecasting problem

• Explain the nature of the data under investigation

• Describe the capabilities and limitations of potentially useful forecasting techniques

• Develop some predetermined criteria on which the selection decision can be made

A major factor influencing the selection of a forecasting technique is the

identifica-tion and understanding of historical patterns in the data If trend, cyclical, or seasonal

FIGURE 14 Time Series Plot of Quarterly Sales for Coastal

Marine for Example 5

FIGURE 15 Autocorrelation Function for Quarterly Sales for

Coastal Marine for Example 5

patterns can be recognized, then techniques that are capable of effectively extrapolatingthese patterns can be selected.

Forecasting Techniques for Stationary Data

A stationary series was defined earlier as one whose mean value is not changing over

time Such situations arise when the demand patterns influencing the series are tively stable It is important to recognize that stationary data do not necessarily varyrandomly about a mean level Stationary series can be autocorrelated, but the nature

rela-of the association is such that the data do not wander away from the mean for anyextended period of time In its simplest form, forecasting a stationary series involvesusing the available history of the series to estimate its mean value, which then becomesthe forecast for future periods More-sophisticated techniques allow the first few fore-casts to be somewhat different from the estimated mean but then revert to the meanfor additional future periods Forecasts can be updated as new information becomesavailable Updating is useful when initial estimates are unreliable or when the stability

of the average is in question In the latter case, updating provides some degree ofresponsiveness to a potential change in the underlying level of the series

Stationary forecasting techniques are used in the following circumstances:

• The forces generating a series have stabilized, and the environment in which the

series exists is relatively unchanging Examples are the number of breakdowns per

week on an assembly line having a uniform production rate, the unit sales of aproduct or service in the maturation stage of its life cycle, and the number of salesresulting from a constant level of effort

• A very simple model is needed because of a lack of data or for ease of explanation or

implementation An example is when a business or organization is new and very

few historical data are available

• Stability may be obtained by making simple corrections for factors such as

popula-tion growth or inflapopula-tion Examples are changing income to per capita income and

changing dollar sales to constant dollar amounts

• The series may be transformed into a stable one Examples are transforming a

series by taking logarithms, square roots, or differences

• The series is a set of forecast errors from a forecasting technique that is considered

adequate (See Example 7.)

Techniques that should be considered when forecasting stationary series includenaive methods, simple averaging methods, moving averages, and autoregressive mov-ing average (ARMA) models (Box-Jenkins methods)

Forecasting Techniques for Data with a Trend

Simply stated, a trend in a time series is a persistent, long-term growth or decline For a

trending time series, the level of the series is not constant It is common for economictime series to contain a trend

Forecasting techniques for trending data are used in the following circumstances:

• Increased productivity and new technology lead to changes in lifestyle Examples

are the demands for electronic components, which increased with the advent of thecomputer, and railroad usage, which decreased with the advent of the airplane

• Increasing population causes increases in demand for goods and services Examples

are the sales revenues of consumer goods, demand for energy consumption, and use

of raw materials

• The purchasing power of the dollar affects economic variables due to inflation.

Examples are salaries, production costs, and prices

• Market acceptance increases An example is the growth period in the life cycle of a

new product

Techniques that should be considered when forecasting trending series include

moving averages, Holt’s linear exponential smoothing, simple regression, growth

curves, exponential models, and autoregressive integrated moving average (ARIMA)

models (Box-Jenkins methods)

Forecasting Techniques for Seasonal Data

A seasonal series was defined earlier as a time series with a pattern of change that

repeats itself year after year One way to develop seasonal forecasts is to estimate

sea-sonal indexes from the history of the series For example, with monthly data, there is an

index for January, an index for February, and so forth These indexes are then used to

include seasonality in forecasts or to remove such effects from the observed values The

latter process is referred to as seasonally adjusting the data

Forecasting techniques for seasonal data are used in the following circumstances:

• Weather influences the variable of interest Examples are electrical consumption,

summer and winter activities (e.g., sports such as skiing), clothing, and agricultural

growing seasons

• The annual calendar influences the variable of interest Examples are retail sales

influenced by holidays, three-day weekends, and school calendars

Techniques that should be considered when forecasting seasonal series include

classical decomposition, Census X-12, Winter’s exponential smoothing, multiple

regression, and ARIMA models (Box-Jenkins methods)

Forecasting Techniques for Cyclical Series

The cyclical effect was defined earlier as the wavelike fluctuation around the trend

Cyclical patterns are difficult to model because their patterns are typically not stable

The up-down wavelike fluctuations around the trend rarely repeat at fixed intervals of

time, and the magnitude of the fluctuations also tends to vary Decomposition methods

can be extended to analyze cyclical data However, because of the irregular behavior of

cycles, analyzing the cyclical component of a series, if it exists, often requires finding

coincident or leading economic indicators

Forecasting techniques for cyclical data are used in the following circumstances:

• The business cycle influences the variable of interest Examples are economic,

mar-ket, and competition factors

• Shifts in popular tastes occur Examples are fashions, music, and food.

• Shifts in population occur Examples are wars, famines, epidemics, and natural

disasters

• Shifts in product life cycle occur Examples are introduction, growth, maturation

and market saturation, and decline

Techniques that should be considered when forecasting cyclical series include

clas-sical decomposition, economic indicators, econometric models, multiple regression,

and ARIMA models (Box-Jenkins methods)

Other Factors to Consider When Choosing a Forecasting Technique

The time horizon for a forecast has a direct bearing on the selection of a forecasting

technique For short- and intermediate-term forecasts, a variety of quantitative

tech-niques can be applied As the forecasting horizon increases, however, a number of

these techniques become less applicable For instance, moving averages, exponentialsmoothing, and ARIMA models are poor predictors of economic turning points,whereas econometric models are more useful Regression models are appropriate forthe short, intermediate, and long terms Means, moving averages, classical decomposi-tion, and trend projections are quantitative techniques that are appropriate for theshort and intermediate time horizons The more complex Box-Jenkins and economet-ric techniques are also appropriate for short- and intermediate-term forecasts.Qualitative methods are frequently used for longer time horizons.

The applicability of forecasting techniques is generally something a forecasterbases on experience Managers frequently need forecasts in a relatively short time.Exponential smoothing, trend projection, regression models, and classical decomposi-tion methods have an advantage in this situation (See Table 6.)

Computer costs are no longer a significant part of technique selection Desktopcomputers (microprocessors) and forecasting software packages are becoming com-monplace for many organizations Due to these developments, other criteria will likelyovershadow computer cost considerations

Ultimately, a forecast will be presented to management for approval and use in theplanning process Therefore, ease of understanding and interpreting the results is animportant consideration Regression models, trend projections, classical decomposi-tion, and exponential smoothing techniques all rate highly on this criterion

It is important to point out that the information displayed in Table 6 should beused as a guide for the selection of a forecasting technique It is good practice to try

TABLE 6 Choosing a Forecasting Technique

Method

Pattern of Data

Time Horizon

Type of Model

Minimal Data Requirements Nonseasonal Seasonal

Pattern of the data: ST, stationary; T, trending; S, seasonal; C, cyclical Time horizon: S, short term (less than three months); I, intermediate term; L, long term Type of model: TS, time series; C, causal

Seasonal: s, length of seasonality Variable: V, number of variables

more than one forecasting method for a particular problem, holding out some recentdata, and then to compute forecasts of these holdout observations using the differentmethods The performance of the methods for these holdout test cases can bedetermined using one or more of the accuracy measures defined in Equations 7through 11, discussed below Assuming an adequate fit to the data, the most accuratemethod (the one with the smallest forecast error) is a reasonable choice for the “best”

method It may not be the best method in the next situation

Empirical Evaluation of Forecasting Methods

Empirical research has found that the forecast accuracy of simple methods is often asgood as that of complex or statistically sophisticated techniques (see Fildes et al., 1997;

Makridakis et al., 1993; and Makridakis and Hibon, 2000) Results of the M3–IJFCompetition, where different experts using their favorite forecasting methodologyeach generated forecasts for 3,003 different time series, tended to support this finding(Makridakis and Hibon, 2000) It would seem that the more statistically complex atechnique is, the better it should predict time series patterns Unfortunately, estab-lished time series patterns can and do change in the future Thus, having a model thatbest represents the historical data (the thing complex methods do well) does not neces-sarily guarantee more accuracy in future predictions Of course, the ability of the fore-caster also plays an important role in the development of a good forecast

The M3–IJF Competition was held in 1997 The forecasts produced by the variousforecasting techniques were compared across the sample of 3,003 time series, with theaccuracy assessed using a range of measures on a holdout set The aim of the 1997study was to check the four major conclusions of the original M-Competition on alarger data set (see Makridakis et al., 1982) Makridakis and Hibon (2000) summarizedthe latest competition as follows:

1 As discussed previously, statistically sophisticated or complex methods do not essarily produce more accurate forecasts than simpler methods

nec-2 Various accuracy measures produce consistent results when used to evaluate ferent forecasting methods

dif-3 The combination of three exponential smoothing methods outperforms, on age, the individual methods being combined and does well in comparison withother methods

aver-4 The performance of the various forecasting methods depends on the length of theforecasting horizon and the kind of data (yearly, quarterly, monthly) analyzed

Some methods perform more accurately for short horizons, whereas others aremore appropriate for longer ones Some methods work better with yearly data, andothers are more appropriate for quarterly and monthly data

As part of the final selection, each technique must be evaluated in terms of its ability and applicability to the problem at hand, its cost effectiveness and accuracycompared with competing techniques, and its acceptance by management Table 6 sum-marizes forecasting techniques appropriate for particular data patterns As we havepointed out, this table represents a place to start—that is, methods to consider for datawith certain characteristics Ultimately, any chosen method should be continuouslymonitored to be sure it is adequately doing the job for which it was intended

reli-MEASURING FORECAST ERROR

Because quantitative forecasting techniques frequently involve time series data, a

mathematical notation is developed to refer to each specific time period The letter Y

will be used to denote a time series variable unless there is more than one variable