Spss TRENDS 13.0 User Manual

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SPSS® Trends 13.0

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Preface

SPSS 13.0 is a comprehensive system for analyzing data. The Trends optional add-on mod The Trends add-on module must be used with the SPSS 13.0 Base system and is completely integrated into that system.

Installation

To install the Trends add-on module, run the License Authorization Wizard using the authorization code that you received from SPSS Inc. For more information, see the installa

Compatibility

SPSS is designed to run on many computer systems. See the installation instructions that cam requirements.

ule provides the additional analytic techniques described in this manual.

tion instructions supplied with the SPSS Base system.

e with your system for specific information on minimum and recommended

Serial N

Your ser this serial number when you contact SPSS Inc. for information regarding support, payment, or an upgraded system. The serial number was provided with your Base system

Customer Service

If you have any questions concerning your shipment or account, contact your local office your serial number ready for identification.

umbers

ial number is your identification number with SPSS Inc. You will need

, listed on the SPSS Web site at http://www.spss.com/worldwide. Please have

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Training Seminars

SPSS Inc. provides both public and onsite training seminars. All seminars feature hands-on workshops. Seminars will be offered in major cities on a regular basis. For more informa

tion on these seminars, contact your local office, listed on the SPSS

Web site at http://www.spss.com/worldwide.

Technical S

The service

upport

s of SPSS Technical Support are available to registered customers. Customers may contact Technical Support for assistance in using SPSS or for installation help for one of the supported hardware environments. To reach Technical Support, s

ee the SPSS Web site at http://www.spss.com, or contact your local office, listedontheSPSSWebsiteathttp://www.spss.com/worldwide.Bepreparedto identify yourself, your organization, and the serial number of your system.

Additional Publications

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SS Web site.

The SPSS Statistical Procedures Companion, by Marija Norušis, has been published by Prentice Hall. A new version of this book, updated for SPSS 13.0, is planned

.TheSPSS Advanced Statistical Procedures Companion, also based on SPSS

13.0, is forthcoming. The SPSS Guide to Data Analysis forSPSS13.0isalsoin development. Announcements of publications available exclusively through Prentice Hall wi your home country, and then click

ll be available on the SPSS Web site at http://www.spss.com/estore (select

Books).

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Attn.: Director of Product Planning, 233 South Wacker Drive, 11th Floor, Chicago, IL 60606-6412.

About This Manual

This manual documents the graphical user interface for the procedures included in the Trends add-on module. Illustrations of dialog boxes are taken from SPSS for Windows. Di

alog boxes in other operating systems are similar. Detailed information about the command syntax for features in this module is provided in the SPSS Command Syntax Reference, available from the Help menu.

Contacting SPSS

If you would like to be on our mailing list, contact one of our offices, listed on our Web site at http://www.spss.com/worldwide.

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Contents

Part I: User

's Guide

1Overview 1

TimeSeriesAnalysis........................................... 2

ReasonsforAnalyzingTimeSeries ............................ 2

A Model-B

HowTrendsCanHelp.......................................... 4

ModelIdentification........................................ 4

Parameter

Diagnosis ............................................... 6

2TrendsPr

Defining

Data Tran

New Serie

Historic

Reusing M

Changing

Performa

alandValidationPeriods ................................. 9

uildingStrategy................................... 3

Estimation ...................................... 5

ocedures Common Features 7

TimeSeriesData....................................... 7

sformations.......................................... 7

sNames............................................ 8

odels............................................. 10

SettingswithCommandSyntax.......................... 11

nceConsiderations ................................... 12

3 Exponential Smoothing 15

Custom E

xponentialSmoothingModels ........................... 17

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ExponentialSmoothingParameters .............................. 18

SavingPredictedValuesandResiduals ........................... 20

EXSMOOTHCommandAdditionalFeatures......................... 21

4 Autoregres

AutoregressionOptions ....................................... 25

SavingPredictedValuesandResiduals ........................... 27

Autoregression Performance and Embedded Missing Data. . . . . . . . . . . . . 28

AREGCommandAdditionalFeatures.............................. 28

sion 23

5ARIMA 31

ARIMA Opt

Saving Pr

ARIMA Per

ARIMA Com

ions.............................................. 34

edictedValuesandResiduals ........................... 36

formanceandEmbeddedMissingData................... 37

mandAdditionalFeatures ............................ 38

6 Seasonal Decomposition 39

Seasona

SEASON C

lDecompositionSave.................................. 41

ommandAdditionalFeatures........................... 41

7 Spectral Plots 43

SPECTR

ACommandAdditionalFeatures........................... 46

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Part II: Exa

mples

8 Exponential Smoothing 49

Using Expo

RelatedProcedures.......................................... 90

RecommendedReadings ...................................... 90

nential Sm oothing to Predict Future Sales . . . . . . . . . . . . . . . . . 49

Prelimina

UnderstandingYourData................................... 50

Building and Analyzing Exponential Smoothing Models . . . . . . . . . . . . 53

Test i ng t h e

UsingtheModeltoPredictFutureSales ....................... 84

Summary............................................... 89

9 Autoregre

Determining Significan t Predictors in the Presence of Autocorrelation . . . . 91

Preliminaries............................................ 91

AnalysisfromOrdinaryLeast-SquaresRegression................ 92

Applying A

Summary.............................................. 105

Related Pr

ocedures......................................... 105

ries............................................ 49

Predictive Ability of the Model . . . . . . . . . . . . . . . . . . . . . . 78

ssion 91

utoregressiontotheProblem ....................... 99

10 ARIMA 107

TheARIMAModel.......................................... 107

Autoregression(ARIMA).................................. 107

Differen

cing(ARIMA) .................................... 108

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Moving-average(ARIMA)................................. 109

SeasonalOrders........................................ 110

StepsinUsingARIMA ....................................... 110

Identification........................................... 110

Estimation............................................. 111

Diagnosis ............................................. 11

Preliminaries .............................................. 112

Using Seasonal ARIMA with Predictors to Model Catalog Sales . . . . . . . . 113

PlottingtheCatalogSalesSeries............................ 114

Identifying a

CreatingtheModel...................................... 124

ModelDiagnosis........................................ 126

Adding Predic

Testing the Predictive Ability of the Model . . . . . . . . . . . . . . . . . . . . . 129

Summary.............................................. 136

Using Intervention Analysis to Determine Change in Market Share . . . . . . 137

PlottingtheMarketShareSeries............................ 137

InterventionAnalysisStrategy.............................. 139

Identifying a M

DeterminingtheInterventionPeriod.......................... 144

CreatingInterventionVariables............................. 148

Running the Ana

ModelDiagnosis........................................ 152

AssessingtheIntervention ................................ 155

Summary.............................................. 156

RelatedProcedures......................................... 157

RecommendedReadings ..................................... 157

Model...................................... 116

torstotheModel............................. 128

odel...................................... 140

lysis..................................... 151

11 Seasonal Decomposition 159

RemovingSeasonalityfromSalesData........................... 159

Preliminaries........................................... 159

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DeterminingandSettingthePeriodicity....................... 160

RunningtheAnalysis..................................... 166

Understandi

Summary.............................................. 169

Related Proc

ngtheOutput................................. 167

edures ......................................... 170

12 Spectral Plots 171

Using Spect

Running the

Understanding the Periodogram and Spectral Density . . . . . . . . . . . . 173

Summary.............................................. 176

RelatedProcedures......................................... 176

ral Plots to Verify Expectations about Periodicity . . . . . . . . . . 171

Analysis..................................... 171

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Appendices

A Durbin-Watson Significance Tables 177

B Guide to ACF/PACF Plots 201

Bibliogr

aphy 207

Index 209

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Part 1: User's Guide

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Overview

SPSS Trends performs comprehensive forecasting and time series analyses. Its graphical u that you will find Trends easy to use.

The range of analytical techniques available in Trends extends from simple, basic

tools to mo

Plots. With facilities i n the SPSS Base system, you can easily produce a variety of

series and autocorrelation plots that you can enhance using the SPSS Chart Editor.

Smoothing

high-quality forecasts with a minimum of effort.

Decomposition. Yo u can break down a series into its components, saving the seasonal

factors an further analysis.

ser interface, comprehensive manual, and online Help system ensure

re sophisticated types of analysis. These include:

You can use simple but efficient smoothing techniques that can yield

d trend, cycle, and error components automatically—ready to use in

Chapter

Regression. You can build regression models using a variety of techniques, including

those in t fitting. Trends adds a special facility for regression with autocorrelated errors.

ARIMA Modeling. You can apply the powerful techniques of ARIMA modeling in

a fully in quicklyleadyoutothebestmodel.

Spectral Analysis. You can examine a time series as a combination of periodic cycles

of vario

he SPSS Base system, such as ordinary least-squares regression and curve

teractive environment in which identification, estimation, and diagnosis

us lengths.

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Chapter 1

Time Series A

A time serie regularly over a period of time. In a series of inventory data, for example, the observations might represent daily inventory levels for several months. A series showing the over a few years. A series of total sales figures might consist of one observation per month for many years. What each of these examples has in common is that some variable w the form of the data for a typical time series is a single sequence or list of observations representing measurements taken at regular intervals.

Tab l e 1- 1

Daily inventory time series

Time Week Day Inventor

nalysis

s is a set of observations obtained by measuring a single variable

market share of a product might consist of weekly market share taken

as observed at regular, known intervals over a certain length of time. Thus,

level

1 Monday 160 1Tuesday 1 Wednesday 129 1 Thursday 122 1 Friday 108 2 Monday

...

12 Friday 120

135

150

Reasons for Analyzing Time Series

One of the most important reasons for doing time series analysis is to forecast future

s of the series. The parameters of the model that explained the past values may

value also predict whether and how much the next few values will increase or decrease. The ability to make such predictions successfully is obviously important to any

ness or scientific field.

busi

Another reason for analyzing time series data is to evaluate the effect of some event

that intervenes and changes the normal behavior of a series. Intervention analysis

ines the pattern of a time series before and after the occurrence of such an event.

exam The goal is to see if the outside event h ad a significant impact on the series pattern.

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If it did, there is a significant upward or downward shift in the values of the series after the occurrence of the event. For this reason, such series are called interrupted

time series

. Weekly numbers of automobile fatalities before and after a new seat belt law, monthly totals of armed robberies before and after a new gun law, and daily measurements of productivity before and after initiation of an incentive plan are examples o

f interrupted time series. What they have in common is a hypothetical interruption in their usual pattern after the specific time when some outside event occurred. Since the time of the outside event is known and the pattern before and after the e

vent is observable, you can evaluate the impact of the interruption.

A Model-Building Strategy

No matter what the primary goal of the time series analysis, the approach basically starts wi for building a model is the one developed by Box and Jenkins (Box and Jenkins,

1976), who defined three major stages of model building: identification, estimation, and diag usefulness of this strategy specifically for ARIMA model building, the general principles can be extended to all model building.

This tentative model type includes initial judgements about the number and kind of parameters involved and how they are combined. In making these judgements, you should the series and its autocorrelation function to find out whether the series shows any upward or downward trend, whether some sort of data transformation might simplify analy

its parameters. This stage usually involves using a computerized model-fitting routine to est can then be used to see how well they would have predicted the observed values. If the parameter estimates are unsatisfactory on statistical grounds, you return to the iden behavior of the series.

the d residual, or error, series. This information tells you whether the model can be used

th building a model that will explain the series. The most popular strategy

nostic checking. Although Box and Jenkins originally demonstrated the

ication involves selecting a tentative model type with which to work.

Identif

be parsimonious. The methods usually employed at this stage include plotting

sis, and whether any kind of seasonal pattern is apparent.

Estimation is the process of fitting the tentative model to the data and estimating

imate the parameters and test them for significance. The estimated parameters

tification stage, since the tentative model could not satisfactorily explain the

Diagnosis is the stage in which you examine how well the tentative model fits

ata. Methods used at this stage include plots and statistics describing the

Overview

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Chapter 1

with confidence or whether you should return to the first stage and try to identify a better model.

How Trends Can Help

SPSS Trends is designed to help you accomplish the goals of model identification, model parameter estimation, and model diagnosis.

Model Identification

The most useful tools for identifying a model are plots of the series itself or of various correlation functions. The SPSS Base system provides many plots that are helpful for analyzing

Plotting the Series. Withthe Sequence Charts procedure in the SPSS Base system, you

can plot the values of your series horizontally or vertically. You have the option of plotting adjacent (or seasonally adjacent) points in the series.

time series, such as sequence charts and autocorrelation plots.

the series itself, a log transformation of the series, or the differences between

Plotting Correlation Functions. The Base system provides facilities for plotting

correlat transformation of the function, or the differences between adjacent (or seasonally adjacent) points. Confidence limits are included on the plots, and the values and standar facilities are available:

 The Autocorrelations procedure displays and plots the autocorrelation function

 The Cross-Correlations procedure displays and plots the cross-correlation

ion functions. As with the series plots, you can show the function itself, a log

d errors of the correlation function are displayed in the Viewer. The following

and the p lags. It also displays the Box-Ljung statistic and i ts probability level at each lagintheViewer.

functi

artial autocorrelation function among the values of a series at different

ons of two or more time series at different lags.

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Parameter Estimation

SPSS Trends includes techniques for estimating the coefficients of your model. You can group these techniques loosely under the general areas of smoothing, regression methods, Bo their component frequencies.

Smoothing. The Exponential Smoothing procedure uses exponential smoothing

methods to e Forecasts and forecast error values for one or more time series are produced using the most recent data in the series, previous forecasts and their errors, and estimates of trend and s or let Trends find them for you. The output includes statistics arranged to help you evaluate the estimates.

Trends al estimate multiplicative or additive seasonal factors for periodic time series. New series containing seasonally adjusted values, seasonal factors, trend and cycle componen file so that you can perform further analyses.

x-Jenkins or ARIMA analysis, and decomposition of cyclic data into

stimateuptothreeparametersfor a wide variety of common models.

easonality. You can specify your own estimates for any of the parameters

so includes the Seasonal Decomposition procedure, which lets you

ts, and error components can be automatically added to your working data

Overview

Regression Methods. The Regression procedure in the SPSS Base system is useful

when you

want to analyze time series using ordinary least-squares regression.

Additional procedures for regression methods include:

 The Curve Estimation procedure, which is part of the Base system, fits selected

curves t

o time series and produces forecasts, forecast error values, and confidence interval values. The curve is chosen from a variety of trend-regression models that assume that the observed series is some function of the passage of time.

 The Autoregression procedure, which is part of Trends, allows you t o estimate

sion models reliably when the error from the regression is correlated

regres between one time point and the next—a common situation in time series analysis. Autoregression offers two traditional methods (Prais-Winsten and

ane-Orcutt) as well as an innovative maximum-likelihood method that is

Cochr able to handle missing data embedded in the series.

enkins Analysis.

Box-J

The ARIMA procedure lets you estimate nonseasonal and seasonal univariate ARIMA models. You can include predictor variables in the model to evaluate the effect of some outside event or influence while estimating the

icients of the ARIMA process. ARIMA produces maximum-likelihood estimates

coeff

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Chapter 1

and can process time series with missing observations. It uses the traditional ARIMA model syntax, so you can describe your model just as it would be described in a book on ARIM

A analysis. Summary statistics for the parameter estimates help you to evaluate the model. New series containing forecasts as well as their errors and confidence limits are automatically created.

Diagnosis

Seasonal-A

djustment Methods.

The Seasonal Decomposition procedure lets you estimate multiplicative or additive seasonal factors for periodic time series using the ratio-to-moving-average (Census I) method of seasonal decomposition. Seasonal Decomposi

tion automatically creates new series in your working data file containing seasonally adjusted values, seasonal factors, trend and cycle components, and error components so that you can perform further analyses.

Frequenc

y-Component Analysis.

The Spectral Plots procedure lets you decompose a time series into its harmonic components, a set of regular periodic functions at different wavelengths or periods. By noting the prominent frequencies in this model-fr

ee analysis, you can detect features of a periodic or cyclic series that would be obscured by o ther methods. Spectral Plots provides statistics, plots, and methods of tailoring them for univariate and bivariate spectral analysis, including periodo

grams, spectral density estimates, gain and phase spectra, popular spectral windows for smoothing the periodogram, and optional user-defined filters. Plots can be produced by period, frequency, or both.

The ability to diagnose how well the model fits the data is a vital part of time series analysis. Several facilities are available to assist you in evaluating models:

 The automatic residual and confidence-interval series generated along with the

forecasts help you to assess the accuracy of your models.

 Standard errors and other statistics help you to judge the significance of the

cients estimated for your model.

coeffi

 In reg

ressionanalysisandelsewhere,youfrequentlyneedtodeterminewhether the residuals from a model are normally distributed. The SPSS Base system offers Normal P-P and Normal Q-Q plots, which compare the observed values of

es against the values that would be observed if the series were normally

aseri distributed. They give you quick and effective visual checks for normality.

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Chapter

Trends Proce Features

Defining T

ime Series Data

A time seri regularly over a period of time. The form of the data for a typical time series is a single sequence or list of observations representing measurements taken at regular intervals.

When you de corresponds to a separate variable. For example, to define a time series in the Data Editor, click the observat A property of time series data is that the observations are taken at equally spaced time intervals.

If you ope arranged in a column in the spreadsheet. If you already have a spreadsheet with time series arranged in rows, you can open it anyway and use Transpose on the Data menu to flip t

es is a set of observations obtained by measuring a single variable

fine time series data for use with SPSS Trends, each series

Variable View tab and enter a variable name in any blank row. Each

ion in a time series corresponds to a case in SPSS (a row in the Data Editor).

n a spreadsheet containing time series data, each series should be

he rows into columns.

dures Common

Data Transformations

A number of data transformation procedures provided in the SPSS Base system are

in time series analysis.

useful

 The De

fine Dates procedure (on the Data menu) generates date variables used to establish periodicity and to distinguish between historical, validation, and forecasting periods. Trends is designed to work with the variables created by the Def

ine Dates procedure.

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Chapter 2



TheCreateTimeSeriesprocedure(ontheTransformmenu)createsnewtime series vari

ables as functions of existing time series variables. It includes functions that use neighboring observations for smoothing, averaging, and differencing. For example, you can perform a difference transformation on a nonstationary series to pr

 The Replac

oduce a series that is stationary and suitable for analysis.

e Missing Values procedure (on the Transform menu) replaces systemand user-missing values with estimates based on one of several methods. Missing data at the beginning or end of a series pose no particular problem; they simply shorten th

e useful length of the series. Gaps in the middle of a series (embedded missing data) can be a much more serious problem. In particular, Autoregression (other than the maximum-likelihood method), Exponential Smoothing, Seasonal Decompos

ition, and Spectral Plots can not be run in the presence of embedded

missing data.

See the SP transformations for time series.

New Serie

All of th generate new series containing such things as predicted values and residuals. Each procedure reports the names of any new series that it creates. The first three letters of the seri

 fit con  err is a

 ucl and lcl contain upper and lower confidence limits.  sep is the standard error of the fit or predicted series.  sas, saf,andstc are series components extracted by the Seasonal Decomposition

SS Base User’s Guide for detailed information concerning data

s Names

e Trends procedures, with the exception of Spectral Plots, can automatically

es name indicate the type of series:

tains the predicted value according to the current model.

residual or error series. (Normally the fit series plus the err series equals

the original series.)

procedure.

Page 23

Historical a

It is often u a validation period. You develop a model on the basis of the observations in the historical period and then test it to see how well it works in the validation period. When you are efficient than comparing models based on the entire sample.

The facilities in the Select Cases dialog box (available through the Data menu) and theSavedi make it easy to set aside part of your data for validation purposes.

Select Cases. Specifies a range of observations for analysis. The selection Based on

time or cas

you have attached them to your time series or using observation numbers if you have not. You normally define a historical period in this way.

Save. Spe

that save new series containing such things as fit values and residuals allow you to predict values for observations past the end of the series being analyzed. To define a validati Trends then uses the model developed from the historical period to “forecast” values throughthevalidationperiodsothatyoucancomparetheseforecaststotheactual data. Fo generating one-step-ahead forecasts, refer to “Forecasts” below.

Trends Procedu

res Common Features

nd Validation Periods

sefultodivideyourtimeseriesintoahistorical or estimation period and

not sure which model to choose, this technique is sometimes more

alog box (available through the main dialog box for many procedures)

e range

cifies a range of observations for forecasts or validation. Trends procedures

on period, select the default

recasts created in this way are n-step-ahead forecasts. For information on

allows you to specify a range of observations using date variables if

Predict from estimation period through last case.

sts

Foreca

sts are ubiquitous in time series analysis—both real forecasts and the

Foreca validation “forecasts” discussed above. It is often useful to distinguish between “one-step-ahead” forecasts and “n-step-ahead” forecasts. One-step-ahead forecasts use—an

d require—information in the time period immediately preceding the period being forecast, while n-step-ahead forecasts are based on older information. You can produce either type of forecast in Trends.

orecasts—that is, forecasts for observations beyond the end of existing

Real f series—are always n-step-ahead forecasts. To generate these forecasts, specify the forecast range in a Save dialog box using the

atically extend the series to allow room for the forecast observations. (This

autom

Predict through alternative. Trends will

Page 24

Chapter 2

type of forecast can be generated by ARIMA and Exponential Smoothing and by Curve Estimation in the Base system.)

Validation f

orecasts can be either one- or n-step-ahead. To generate n-step-ahead validation forecasts, simply specify the historical period in the Select Cases dialog box and the validation period in the Save dialog box, as discussed above. If you need one-st

ep-ahead validation forecasts, you must use a certain amount of SPSS

command syntax:

1. Specify the historical period in the Select Cases dialog box.

2. Estimate the model in which you are interested. Instead of executing it directly, click

Paste button to paste its command syntax into a syntax window.

the

3. Execute the command from the syntax window by clicking the toolbar.

4. Go back t

o the Select Cases dialog box and specify both the historical and validation

periods. Generally this means to select

5. Activate the syntax window and edit the command that you executed in step 3. Leave the command name ( with the s

ingle specification execute the command by clicking Run. Trends generates a fit variable through both the historical and validation periods, based on the coefficients estimated in step 3 for the h

istorical period.

Reusing Models

When you click OK or Paste in SPSS, current dialog box settings are saved. When you retu still there (unless you have opened a different data file). This persistence of dialog box settings is especially helpful as you develop models for time series data, since you can

 You ca

 You can switch variables to repeat your analysis or chart with different variables

rn to a dialog box you have used once, all of your previous specifications are

selectively modify model specifications as needed:

n change one or more specifications in the dialog box (or in any subdialog

box) and repeat the analysis with the new specifications.

but with the same specifications.

Run button on the

All cases.

EXSMOOTH, ARIMA,orAREG), but replace all of its specifications

/APPLY FIT (for EXSMOOTH use /APPLY). Then

Page 25

Trends Procedu



You can use the Select Cases facility to restrict the analysis to a range of cases or

to process a

ll cases instead of restricting the analysis to a previously specified

res Common Features

range. You can then repeat an analysis or chart with identical specifications but a different range of observations.

 You can use a transformation procedure, such as Replace Missing Values, or edit

data values

in the Data Editor and then repeat an analysis or chart with identical

specifications but modified data.

Reusing Command Syntax

If you are using command syntax instead of the dialog boxes, you can still reuse and selectively modify models using the APPLY subcommand. When it is used, the APPLY subcomman

d is usually the first specification after the name of the command or after the name of a command and a series. It means run this command as before, with the following changes. If you want to change any specifications from the previous model, co

ntinue the command with a slash and enter only those specifications you

want to add or change.

For commands that estimate coefficients that you can apply to prediction (

AREG,and

EXSMOOTH), you have the option of applying the coefficients estimated for

ARIMA,

a previous model to a new model, either as initial estimates or as “final” coefficients to be used in calculating predicted values and residuals.

You can a

lso apply specifications or coefficients from an earlier model rather than

from the previous specification of the same command by specifying the model name.

See the SPSS Command Syntax Reference for a complete discussion of the

subcomm

and and models.

APPLY

Changing Settings with Command Syntax

Several SPSS commands determine settings that affect the operation of Trends

ures. In particular, the

proced operation of most subsequent analytical procedures in Trends. If you execute such commands from a syntax window and later execute Trends procedures from the

boxes, you cannot necessarily assume that the settings you established in the

dialog syntax window remain in effect. The following are areas where this might occur:

 The Select Cases dialog box can generate a USE command.

TSET, USE,andPREDICT commands modify the

Page 26

Chapter 2



The Save dialog box for any Trends procedure and for Curve Estimation in the Base system

 Trends dial

can generate a

og boxes routinely generate a are specified in the dialog box. Never assume that your survive the use of a dialog box without inspecting the journal file for a command gen

erated by the dialog box.

The existence and name of the journal file can be verified on the General tab in the Options dialog box (Edit menu).

Performance Considerations

New Series

PREDICT command.

TSET command to reflect settings that

TSET specifications

TSET

Many Trend

s procedures automatically generate new series. This facility can be a

great aid—but not always. Possible difficulties with saving new series include:

 Trendsmustreadandwritetheentirefileanextratimetoaddthenewseries.  Your file becomes larger—in some cases, dramatically so—and subsequent

processin

 Most of th

g therefore takes longer.

e time, you do not need most of the new series. Merely keeping track

of their names becomes a problem.

When you use procedures on the

Analyze

Time Seri

submenu, it’s a good idea always to open the Save subdialog box and give a moment’s thought to the creation of new variables. The default for these procedures is to add new varia are not yet certain of the models you want to use, the

bles permanently to your data file. If you are doing preliminary analysis and

Replace existing alternative for

new variables gives you the benefits of residuals and predicted values but does not keep all into the Save subdialog box and choose the

of them around. Once you have settled on a model, you may want to go back

Add to file alternative.

Page 27

Trends Procedu

General Techniques for Efficiency

res Common Features

For any iterative procedures in Trends, you may find it useful to:

 Relax the convergence c riteria for the procedure. These are specified in the

Options dial

 Perform exp  Restore the

og box for the specific procedure.

loratory analysis to determine the best model.

stricter convergence criteria for the final estimation of your

coefficients.

The general

point is that some estimation algorithms used in Trends require a lot of processing and will take a long time if you use them blindly. Take advantage of the interactive character of Trends. Loosen things up for speed while you are exploring your data,

and then—when you are ready to estimate your final coefficients—exploit

the full accuracy of the Trends algorithms.

Page 28

Page 29

Chapter

Exponential

This procedure produces fit/forecast values and residuals for one or more time series, using an algorithm that smoothes out irregular components of time series data. A variety of models differing in trend (none, linear, or exponential) and seasonality (none, additive, or multiplicative) are available.

Example. Inventory-intensive businesses often employ statistical techniques for

projecting future inventory. The Exponential Smoothing procedure can be used both to develop a model of the inventory time series and to produce fast forecasts basedonthatmodel.

Statistics. Initial seasonal indices, initial level parameter, initial trend parameter, error

degrees of freedom. For each model: model parameters, sum of squared errors.

Data. The variables and any seasonal factors should be numeric. Assumptions. The variables should not contain any embedded missing data.

Creating an Exponential Smoothing Model

Smoothing

E From the menus choose:

Analyze

Time Series

Exponential Smoothing

Page 30

Chapter 3

Figure 3-1

Exponential Smoothing dialog box

E Select o

ne or more variables from the available list and move them into the Variables

list. Note that the list includes only numeric variables.

E Select the type of model to use. These models differ in their trend and seasonal

components.

 Simple. The simple model assumes that the series has no trend and no seasonal

variation.

 Holt. The Holt model assumes that the series has a linear trend and no seasonal

on.

variati

 Winter

The Winters model assumes that the series has a linear trend and multiplicative seasonal variation (its magnitude increases or decreases with the overall level of the series). The Winters model is unavailable unless you have defined

 Custom

dates with the Define Dates dialog box, available from the Data menu.

A custom model allows you to specify the trend and seasonality

components.

For seas

onal models, the series must contain at least four full seasons of data.

The following options are also available:

 For models with seasonal components, you can specify a set of seasonal factors or

allow T

rends to determine them for you.

Page 31



Click Save to specify how new variables are to be saved or to select options

for predict

 Click Param

ing cases.

to set the parameter values for the model, or specify a grid

eters

search to determine the best-fitting parameter values.

Custom Exponential Smoothing Models

Figure 3-2

Exponential Smoothing Custom Model dialog box

Trend Component. Choices for the custom trend component include no trend, linear

trend, ex

 None. Th

 Linear.

 Exponential. The mean level of the series increases or decreases exponentially

 Damped. The mean level of the series increases or decreases with time, but the

ponential trend, and damped trend.

e level of the series does not show any overall linear trend.

The mean level of the series increases or decreases linearly (at a constant

rate) with time.

with time.

rate of c

hange declines. A trend that is dying out.

Exponential Sm

oothing

Seasonal Component. Choices for the custom seasonal component include no

seasonality, additive seasonality, and multiplicative seasonality.

 None. Theseriesdoesnotshowperiodicvariation at the periodicity defined for it.  Additive. The series has seasonal variation that is additive: the magnitude of

seasonal variation does not depend on the overall level of the series.

 Multiplicative. The series shows periodic variation at the periodicity defined for it,

and the s

ize of the periodic variation depends on the overall level of the series.

Page 32

Chapter 3

The Holt model in the main Exponential Smoothing dialog box is equivalent to selecting Linear for Trend Component and None for Seasonal Component in the Custom dial Component and Multiplicative for Seasonal Component.

og box. The Winters model is equivalent to selecting

Linear for Trend

Exponentia

Figure 3-3

Exponential Smoothing Parameters dialog box

Alpha. Exponential smoothing parameter that controls the relative weight given to

recent observations, as opposed to the overall series mean. When alpha equals 1, the single most recent observation is used exclusively; when alpha equals 0, old observations count just as heavily as recent ones. Alpha is used for all models.

l Smoothing Parameters

Gamma. Exponential smoothing parameter that controls the relative weight given to

recent observations in estimating the present series trend. It ranges from 0 to 1, with higher values giving more weight to recent values. Gamma is used only for exponential smoothing models with a linear or exponential trend, or a damped trend and no seasonal component. It is not used for the simple model.

Delta. Exponential smoothing parameter that controls the relative weight given to

recent observations in estimating the present seasonality. It ranges from 0 to 1, with values near 1 giving higher weight to recent values. Delta is used for all exponential

Page 33

Exponential Sm

oothing

smoothing models with a seasonal component. It is not used for the simple or Holt models.

Phi. Expone

ntial smoothing parameter that controls the rate at which a trend is "damped," or reduced in magnitude over time. It ranges from 0 to 1 (but cannot equal 1), with values near 1 representing more gradual damping. Phi is used for exponentia

l smoothing models with a damped trend. It is not used for the simple,

Holt, or Winters models.

Value. Values for alpha, gamma and delta can be any value between and including 0

and 1. For p

Grid Search. The parameter is assigned a starting value in the Start box, an increment

hi, the value can be any value between but not including 0 and 1.

value in the By box, and an ending value in the Stop box. Enter these values after selecting

Grid Search. The ending value must be greater than the starting value, and

the increment value must be less than their difference.

Initial Values. You can specify the starting and trend values used in smoothing the

series by

 Automat

selecting one of the following:

ic.

The program calculates suitable starting and trend values from the

data. This is usually desirable.

 Custom. Enter a number in the Start text box and, for models with a trend, a

number in the Trend text box. Poor choice of initial values can result in an inferior

solution.

Display only 10 best models f or grid search. The parameter value(s) and sum of squared

errors (SSE) are displayed for only the 10 parameter combinations with the lowest SSE rega

rdless of the number of parameter combinations tested. If this option is not

selected, all tested parameter combinations are displayed.

Page 34

Chapter 3

Saving Predi

Figure 3-4

Exponential Smoothing Save dialog box

Create Variables. Allows you to choose how to treat new variables.

 Addtofile. The new series are saved as regular variables in your working data

file. Variable names are formed from a three-letter prefix, an underscore, and a number.

 Replace existing. The new series are saved as temporary variables in the working

data file, and any existing temporary variables created by Trends commands are dropped. Variable names are formed from a three-letter prefix, a pound sign (#), and a number.

 Do not create. Thenewseriesarenotaddedtotheworkingdatafile.

cted Values and Residuals

Predict Cases. Allows you to use the estimated model to predict values of the

dependent variable:

 Predict from estimation period through last case. Predicts values for all cases from

the estimation period through the end of the file but does not create new cases. If you are analyzing a range of cases that start after the beginning of the file, cases prior to that range are not predicted. If no estimation period has been defined, all cases are used to predict values.

 Predict through. Predicts values through the specified date, time, or observation

number, based on the cases in the estimationperiod.Thiscanbeusedtoforecast values beyond the last case in the time series. The text boxes that are available

Page 35

for specifying the end of the prediction period depend on the currently defined date variab

les.

EXSMOOTH Command Additional Features

The SPSS command language also allows you to:

 Specify initial seasonal factor estimates for seasonal models. Seasonal factors

can be specified numerically by providing as many additive or multiplicative numbersast

See the SPSS Command Syntax Reference for complete syntax information.

he seasonal periodicity.

Exponential Sm

oothing

Page 36

Page 37

Chapter

Autoregress

The Autoregression procedure estimates true regression coefficients from time series with first(Prais-Winsten and Cochrane-Orcutt) transform the regression equation to remove the autocorrelation. The third (maximum likelihood) uses the same algorithm that the ARIMA p (ML) estimation is more demanding computationally but gives better results—and it can tolerate missing data in the series.

Example.

per-capita income and the price level of the spirits in question for that year? A standard regression analysis may not be valid in this case because the variables represen any two trending series will correlate simply because of the trends, regardless of whether they are causally related or not. Autoregression allows you to remove the autocor relationships between dependent variables and candidate regressors.

order autocorrelated errors. It offers three algorithms. Two algorithms

rocedure uses for estimating autocorrelation. Maximum-likelihood

Is the consumption of alcoholic spirits in any given year related to real

t time series. Most time series have some trend, either up or down, and

relation inherent in many time series and ascertain any statistically significant

ion

Statistics. For the Prais-Winsten and Cochrane-Orcutt estimation methods: rho

value w iteration; R, R and regression statistics for the ordinary least-square and final Prais-Winsten or Cochra coefficients, adjusted sum of squares, and Marquardt constant at each iteration. For the final maximum-likelihood parameter estimates: regression statistics, correlation matri residual variance, model standard error, log-likelihood, Akaike’s information criterion, and Schwartz’s Bayesian criterion.

Metho

(equivalenttoanARIMA(1,0,0)model).

ith standard error, Dubin-Watson statistic, and mean squared error at each

,adjustedR2, standard error of the estimate, analysis-of-variance table,

ne-Orcutt estimates. For the maximum-likelihood method: rho, regression

x, covariance matrix, residual sum of squares, adjusted residual sum of squares,

ds.

Prais-Winsten, Cochrane-Orcutt, and Exact maximum-likelihood

Page 38

Chapter 4

Data. The dependent variable and any independent variables should be numeric. Assumptions. Unless the maximum-likelihood method is used, the dependent variable

and any inde

pendent variables should not contain any embedded missing data. The

time series to be modeled should be stationary.

 Stationary. A condition that must be met by the time series to which you fit an

ARIMA model

. Pure MA series will be stationary; however, AR and ARMA series might not be. A stationary series has a constant mean and a constant variance over time.

To C reate a

E From the m

Analyze

Time Series

Autoregression...

Figure 4-1

Autoregre

E Select a variable from the available list and move it into the Dependent list. Note that

n Autoregression Model

enus choose:

ssion dialog box

the list includes only numeric variables.

E Select one or more variables from the available list and move them into the

Independent(s) list.

Page 39

Autoregressio

Select one of the Method options to choose an estimation technique. Available methods are Exact maximum-likelihood, Cochrane-Orcutt, and Prais-Winsten.

 Exact maximum-likelihood. An estimation technique that for a given model and set

of data finds the parameter estimates that are "most likely" to have produced the observed da

ta. This method can handle missing data within the series and can be

used when one of the independent variables is the lagged dependent variable.

 Cochrane-Orcutt. A simple and widely used estimation procedure for estimating a

regression equation whose errors follow a first-order autoregressive process. It cannot be u

 Prais-Wi

sed when a series contains embedded missing values.

nsten.

A generalized least-squares method for estimating a regression equation whose errors follow a first-order autoregressive process. It cannot be used when a series contains embedded missing values. Generally, the Prais-Win

sten method is preferable to the Cochrane-Orcutt method.

The following options are also available:

 Deselect Include constant in model if you do not want to estimate a constant term.  Click Save to specify how new variables are to be saved or to select options

for predic

 Click Opt

ting cases.

to select the initial value of the autoregressive parameter, set

ions

convergence criteria, or choose how to display parameters in the output.

Autoregression Options

Figure 4-2

Autoregression Options dialog box

Page 40

Chapter 4

Initial value of autoregressive parameter. The value from which the iterative search for

the optimal value of rho begins. You can specify any number less than 1 and greater than -1, alt

Convergence Criteria. Allowsyoutospecifythecriteriausedtodeterminewhen

hough negative values of rho are uncommon in this procedure.

iteration ceases.

 Maximum iterations. The iterative algorithm stops after the number of specified

iterations even if the algorithm has not converged. You can specify a positive integer in

 Sum of squa

this text box.

res change.

The iterative algorithm stops if the adjusted sums of squares does not decrease by 0.001% from one iteration to the next. You can choose a smaller or larger value f or more or less precision in the parameter estimates

. For greater precision, it may be necessary to increase the maximum

iterations.

Display. A

 Initial a

llows you to control the information that is printed in the output.

nd final parameters with iteration summary.

Displays the initial and final parameter estimates, goodness-of-fit statistics, the number of iterations, and the reason that iteration terminated.

 Initial and final parameters with iteration details. Displays the initial and final

paramete

r estimates, the parameter estimates after each iteration, goodness-of-fit

statistics, the number of iterations, and the reason that iteration terminated.

 Final parameters only. Displays the final parameter estimates and goodness-of-fit

statistics.

Page 41

Saving Predi

Figure 4-3

Save dialog box

Create Variables. Allows you to choose how to treat new variables.

 Addtofile. The new series are saved as regular variables in your working data

file. Variable names are formed from a three-letter prefix, an underscore, and a number.

 Replace existing. The new series are saved as temporary variables in the working

data file, and any existing temporary variables created by Trends commands are dropped. Variable names are formed from a three-letter prefix, a pound sign (#), and a number.

 Do not create. Thenewseriesarenotaddedtotheworkingdatafile.

cted Values and Residuals

Autoregressio

Confidence intervals. Displays confidence limits for the forecast. As long as the

process remains the same, you should expect N% of the series to remain between the upper and lower confidence limits. Choose a level from the drop-down list.

Predict Cases. Allows you to use the estimated model to predict values of the

dependent variable.

 Predict from estimation period through last case. Predicts values for all cases from

the estimation period through the end of the file but does not create new cases. If you are analyzing a range of cases that start after the beginning of the file, cases

Page 42

Chapter 4

prior to that range are not predicted. If no estimation period has been defined, all cases ar

 Predict thr

eusedtopredictvalues.

ough.

Predicts values through the specified date, time, or observation number, based on the cases in the estimationperiod.Thiscanbeusedtoforecast values beyond the last case in the time series. The text boxes that are available for specify

ing the end of the prediction period depend on the currently defined

date variables.

Autoregression Performance and Embedded Missing Data

When you request maximum-likelihood estimation with the Autoregression procedure, Trends uses the same algorithms as in ARIMA. This means that Autor maximum-likelihood estimation is requested, but it may take a while.

To reduce processing time when your series has embedded missing data, you can follow th Options dialog box does not have controls for initial values, you need to use command syntax to apply initial estimates the second time you run the Autoregression procedu procedure). The command is:

egression can process series with embedded missing data when

e same steps outlined for ARIMA. However, since the Autoregression

re (see step 3 of the process for improving the performance of the A RIMA

AREG /AP

Again, i

PLY INITIAL.

f processing time is not a consideration, you can simply use the Autoregression procedure directly on the series with embedded missing data. Alternatively, if you do not need the best-quality estimates, you can stop after

fying the model and estimating the coefficients for the series without missing

identi data (see step 2 of the process for improving the performance of the ARIMA procedure) and get results as good as you would with most other packages.

AREG Command Additional Features

The SPSS command language also allows you to:

 Use the final estimate of rho from a previous execution of Autoregression as the

l estimate for iteration.

initia

 Exerc

ise more precise control over convergence criteria.

Page 43

Autoregressio

See the SPSS Command Syntax Reference for complete syntax information.

Page 44

Page 45

ARIMA

This procedure estimates nonseasonal and seasonal univariate ARIMA (Autoregre models) with or without fixed regressor variables. The procedure produces maximum-likelihood estimates and can process time series with missing observations.

Chapter

ssive Integrated Moving Average) models (also known as “Box-Jenkins”

Example. Y

to know if and when random fluctuations in product quality exceed their usual acceptable levels. You’ve tried modeling product quality scores with an exponential smoothin data—that the model does little more than predict the overall mean and hence is of little use. ARIMA models are well suited for describing complex time series. After building with the upper and lower confidence intervals produced by the model. Scores that fall outside of the confidence intervals may indicate a true decline in product quality.

Statist

moving average), regression coefficients, adjusted sum of squares, and Marquardt constant. For the final maximum-likelihood parameter estimates: residual sum of square log-likelihood, Akaike’s information criterion, Schwartz’s Bayesian criterion, regression statistics, correlation matrix, and covariance matrix.

Data. T Assumptions. The time series to be modeled should be stationary.

 Stationary. A condition that must be met by the time series to which you fit an

ou are in charge of quality control at a manufacturing plant and need

g model but found—presumably because of the highly erratic nature of the

an appropriate ARIMA model, you can plot the product quality scores along

ics.

For each iteration: seasonal and nonseasonal lags (autoregressive and

s, adjusted residual sum of squares, residual variance, model standard error,

he dependent variable and any independent variables should be numeric.

ARIMA m series might not be. A stationary series has a constant mean and a constant variance over time.

odel. Pure MA series will be stationary; however, AR and ARMA

Page 46

Chapter 5

Obtaining an ARIMA Analysis

E From the menus choose:

Analyze

Time Series

ARIMA

Figure 5-1

ARIMA dialog box

E Select one variable from the available list a nd move it into the Dependent list. Note

that the list includes only numeric variables.

E Specify values for the parameters of the ARIMA model: Autoregressive, Difference,

and Movi

ng Average. These parameters are commonly referred to as p, d, and q,

respectively.

 Autoregressive. The number of autoregressive parameters in the model. Specify a

gative integer. Each parameter measures the independent effect of values

non-ne with a specified lag. Thus, an autoregressive order of 2 means that a series value is affected by the preceding two values (independently of one another).

Page 47

ARIMA



Difference. d is the number of times the series must be differenced to make

it stationa

ry. Specify a non-negative integer. Enter 0 if the process is already

stationary.

 Moving Average. q is the order of moving average of the process. Specify a

non-negative integer. Enter 0 for an autoregressive process, 1 for a first-order moving aver

Transform. To analyze the dependent variable in a logarithmic scale, select one of

age, 2 for a second-order moving average, etc.

the alternatives o n the drop-down list. If you select a log transformation, ARIMA transform

s the predicted values (fit) and confidence intervals (lcl and ucl) that it creates back into the original metric but leaves the residuals (err) in the log metric for diagnostic purposes.

Seasonal.

The seasonal column contains text boxes in which you can specify the corresponding parameters (sp, sd, and sq) of the process at seasonal lags. These values can be 0 or a positive integer, usually 1. The Seasonal boxes are unavailable if no period

icity is defined. The current periodicity is displayed at the bottom of the

dialog box. Optionally, you can:

 Select one or more independent variables and move them into the Independent(s)

list. Note that the list includes only numeric variables.

 Deselect Include constant in model if you do not want to estimate a constant term.  Click Save to specify how new variables are to be saved or to select options

for predi

 Click Op

cting cases.

to select convergence criteria, set initial values for the model, select

tions

the forecasting method, or choose how to display parameters in the output.

Page 48

Chapter 5

ARIMA Option

Figure 5-2

ARIMA Options dialog box

Convergence Criteria. Allowsyoutospecifythecriteriausedtodeterminewhen

iteration ceases.

 Maximum iterations. The iterative algorithm stops after the number of specified

iterations even if the algorithm has not converged. You can specify a positive integer in this text box.

 Parameter change tolerance. Iteration stops if no parameter changes by more than

0.001 from one iteration to the next. You can choose a smaller or larger value for more or less precision in the parameter estimates. For greater precision, it may also be necessary to increase the maximum iterations.

 Sum of squares change. The iterative algorithm stops if the adjusted sums of

squares does not decrease by 0.001% from one iteration to the next. You can choose a smaller or larger value f or more or less precision in the parameter estimates. For greater precision, it may be necessary to increase the maximum iterations.

Page 49

ARIMA

Initial Values for Estimation. Allows you to select how initial values for the model are

determined.

 Automatic. ARIMA chooses initial values.  Apply from previous model. The parameter estimates from the previous execution

of ARIMA (in the same session) are used as initial estimates. This can save time if the data a

Forecasting Method. Allows you to specify the forecasting method to use.

 Unconditional least squares. The forecasts are unconditional least squares

forecasts.

 Condition

nd model are similar to the last one used.

They are also called finite memory forecasts.

al least squares.

The forecasts are conditional least squares forecasts.

They are also called infinite memory forecasts

 Use model constant for initialization. The forecasts are computed by assuming

that the unobserved past errors are 0 and that the unobserved past values of the response s

 Use begin

eries are equal to the mean.

ning series values for initialization.

The beginning series values are used

to initialize the recursive conditional least squares forecasting algorithm.

Display. A

 Initial a

llows you to control the information that is printed in the output.

nd final parameters with iteration summary.

Displays the initial and final parameter estimates, goodness-of-fit statistics, the number of iterations, and the reason that iteration terminated.

 Initial and final parameters with iteration details. Displays the initial and final

parameter

estimates, the parameter estimates after each iteration, goodness-of-fit

statistics, the number of iterations, and the reason that iteration terminated.

 Final parameters only. Displays the final parameter estimates and goodness-of-fit

statistics.

Page 50

Chapter 5

Saving Predi

Figure 5-3

Save dialog box

Create Variables. Allows you to choose how to treat new variables.

 Addtofile. The new series are saved as regular variables in your working data

file. Variable names are formed from a three-letter prefix, an underscore, and a number.

 Replace existing. The new series are saved as temporary variables in the working

data file, and any existing temporary variables created by Trends commands are dropped. Variable names are formed from a three-letter prefix, a pound sign (#), and a number.

 Do not create. Thenewseriesarenotaddedtotheworkingdatafile.

cted Values and Residuals

Confidence intervals. Displays confidence limits for the forecast. As long as the

process remains the same, you should expect N% of the series to remain between the upper and lower confidence limits. Choose a level from the drop-down list.

Predict Cases. Allows you to use the estimated model to predict values of the

dependent variable.

 Predict from estimation period through last case. Predicts values for all cases from

the estimation period through the end of the file but does not create new cases. If you are analyzing a range of cases that start after the beginning of the file, cases

Page 51

prior to that range are not predicted. If no estimation period has been defined, all cases ar

 Predict thr

eusedtopredictvalues.

ough.

ing the end of the prediction period depend on the currently defined

date variables.

ARIMA Performance and Embedded Missing Data

In the presence of embedded missing data, the SPSS Trends ARIMA procedure uses a technique called Kalman filtering, which requires considerably more calculation than the si a single embedded missing value increases ARIMA processing time greatly—in extreme cases, by a factor of 10.

If you wan

can use the following procedure to reduce processing time:

1. Make a copy of the series with valid data interpolated in place of the embedded missing data.

mpler technique used when no embedded missing data are present. Even

t to use ARIMA on a series that contains embedded missing data, you

ARIMA

2. Identify the correct model and estimate the coefficients for the series without missing data. AR

IMA can use a much faster algorithm when no embedded missing data

are present.

3. Once you have found the correct model, run ARIMA on the original series to get the best possible estimates for the coefficients, using Kalman filtering to handle the missing

previous model

data. This time, open the ARIMA Options dialog box and select

for Initial Values for Estimation. This should reduce the number

Apply from

of iterations needed this time.

Most ARIMA packages allow only the first two steps. You can always stop there with Tr

ends ARIMA too, but you have the option of using the Kalman filtering

algorithm to get the best possible estimates.

Page 52

Chapter 5

Note that the results obtained by following the steps above are the same as the results you would obtain if you used the ARIMA procedure directly on the series with embedded mi The only difference is processing time.

ssing data, without first estimating initial values from interpolated data.

ARIMA Comma

The SPSS co

 Specify co

parameters (either regular or s easonal) are estimated only for specified orders.

 Exercise more precise control over convergence criteria.

See the SPS

nd Additional Features

mmand language also allows you to:

nstrained models in which autoregressive or moving average

S Command Syntax Reference for complete syntax information.

Page 53

Chapter

Seasonal Dec

The Seasonal Decomposition procedure decomposes a series into a seasonal component, a combined trend and cycle component, and an “error” component. The procedure is an implementation of the Census Method I, otherwise known as the ratio-to-moving-average method.

Example. A scientist is interested in analyzing monthly measurements of the ozone

level at a particular weather station. The goal is to determine if there is any trend in the data. In order to uncover any real trend, the scientist first needs to account for the variation in readings due to seasonal effects. The Seasonal Decomposition procedure can be used to remove any systematic seasonal variations. The trend analysis is then performed on a seasonally adjusted series.

Statistics. The set of seasonal factors. Data. The variables should be numeric. Assumptions. The variables should not contain any embedded missing data. At least

one periodic date component must be defined.

omposition

Estimating Seasonal Factors

E From the menus choose:

Analyze

Time Series

Seasonal Decomposition...

Page 54

Chapter 6

Figure 6-1

Seasonal Decomposition dialog box

E Selectone or more variables from the available list and move them into the Variable(s)

list. Note that the list includes only numeric variables.

Model. The Seasonal Decomposition procedure offers two different approaches for

modeling the seasonal factors: multiplicative or additive.

 Multiplicative. The seasonal component is a factor by which the seasonally

adjusted series is multiplied to yield the original series. In effect, Trends estimates seasonal components that are proportional to the overall level of the series. Observations without seasonal variation have a seasonal component of 1.

 Additive. The seasonal adjustments are added to the seasonally adjusted series

to obtain the observed values. This adjustment attempts to remove the seasonal effect from a series in order to look at other characteristics of interest that may be "masked" by the seasonal component. In effect, Trends estimates seasonal components that do not depend on the overall level of the series. Observations without seasonal variation have a seasonal component of 0.

Moving Average Weight. The Moving Average Weight options allow you to specify

how to treat the series when computing moving averages. These options are available only if the periodicity of the series is even. If the periodicity is odd, all points are weighted equally.

Page 55



All points equal. Moving averages are calculated with a span equal to the

periodicit the periodicity is odd.

 Endpoints weighted by .5. Moving averages for series with even periodicity are

calculated with a span equal to the periodicity plus 1 and with the endpoints of the span w

Optionally, you can:

 Click Save to specify how new variables should be saved.

Seasonal De

Figure 6-2

SeasonSavedialogbox

Seasonal Decom

y and with all points weighted equally. This method is always used if

eightedby0.5.

composition Save

position

Create Variables. Allows you to choose how to treat new variables.

 Add to file. The new series created by Seasonal Decomposition are saved as

regular variables in your working datafile.Variablenamesareformedfroma three-letter prefix, an underscore, and a number.

 Replace existing. The new series created by Seasonal Decomposition are saved as

temporary variables in your working data file. At the same time, any existing temporary variables created by the Trends procedures are dropped. Variable names are formed from a three-letter prefix, a pound sign (#), and a number.

 Do not create. Thenewseriesarenotaddedtotheworkingdatafile.

SEASON Command Additional F eatures

The SPSS command language also allows you to:

 Specify any periodicity within the SEASON command rather than select one of the

alternatives offered by the Define Dates procedure.

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Chapter 6

See the SPSS Command Syntax Reference for complete syntax information.

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Chapter

Spectral Plo

The Spectral Plots procedure is used to identify periodic behavior in time series. Instead of analyzing the variation from one time point to the next, it analyzes the variation of the series as a whole into periodic components of different frequencies. Smooth series have stronger periodic components at low frequencies; random variation (“white noise”) spreads the component strength over all frequencies.

Series that include missing data cannot be analyzed with this procedure.

Example. The rate at which new houses are constructed is an important barometer of

the state of the economy. Data for housing starts t ypically exhibit a strong seasonal component. But are there longer cycles present in the data that analysts need to be aware of when evaluating current figures?

Statistics. Sine and cosine transforms, periodogram value, and spectral density

estimate for each frequency or period component. When bivariate analysis is selected: real and imaginary parts of cross-periodogram, cospectral density, quadrature spectrum, gain, squared coherency, and phase spectrum for each frequency or period component.

Plots. For univariate and bivariate analyses: periodogram and spectral density.

For bivariate analyses: squared coherency, quadrature spectrum, cross amplitude, cospectral density, phase spectrum, and gain.

Data. The variables should be numeric.

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Chapter 7

Assumptions. The variables should not contain any embedded missing data. The

timeseriestobeanalyzedshouldbestationaryandanynon-zeromeanshouldbe subtracted

 Stationar

out from the series.

A condition that must be met by the time series to which you fit an ARIMA model. Pure MA series will be stationary; however, AR and ARMA series might not be. A stationary series has a constant mean and a constant variance o

ver time.

Obtaining a Spectral Analysis

E From the menus choose:

Graphs

Time Serie

Spectral.

Figure 7-1

Spectral Plots dialog box

E Selec

t one or more variables from the available list and move them to the Variable(s)

list. Note that the list includes only numeric variables.

E Select one of the Spectral Windo w options to choose how to smooth the periodogram

in order t o obtain a spectral density estimate. Available smoothing options are

-Hamming, Tukey, Parzen, Bartlett, Daniell (Unit), and None.

Tukey

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Spectral Plots



Tukey-Hamming. The weights are Wk = .54Dp(2 pi fk) + .23Dp (2 pi fk + pi/p) +

.23Dp (2 pi f

k - pi/p), for k = 0, ..., p, where p is the integer part of half the span

and Dp is the Dirichlet kernel of order p.

 Tukey. The weights are Wk = 0.5Dp(2 pi fk) + 0.25Dp (2 pi fk + pi/p) + 0.25Dp(2

pi fk - pi/p), for k = 0, ..., p, where p is the integer part of half the span and Dp is the Dir

 Parzen. Th

ichlet kernel of order p.

e weights are Wk = 1/p(2 + cos(2 pi fk)) (F[p/2] (2 pi fk))**2, for k= 0, ... p, where p is the integer part of half the span and F[p/2] is the Fejer kernel of order p/2.

 Bartlett. The shape of a spectral window for which the weights of the upper half

of the wind

ow are computed as Wk = Fp (2*pi*fk), for k = 0, ... p, where p is the integer part of half the span and Fp is the Fejer kernel of order p. The lower half is symmetric with the upper half.

 Daniell (Unit). The shape of a spectral window for which the weights are all

equal to 1

 None. No s

moothing. If this option is chosen, the spectral density estimate is

the same as the periodogram.

Bivariat

e analysis – first variable with each.

If you have selected two or more variables,

you can select this option to request bivariate spectral analyses.

 The first variable in the Variable(s) list is treated as the independent variable, and

all remai

 Each ser

ning variables are treated as dependent variables.

ies after the first is analyzed with the first series independently of other

series named. Univariate analyses of each series are also performed.

Span. The

range of consecutive values across which the smoothing is carried out. Generally, an odd integer is used. Larger spans smooth the spectral density plot more than smaller spans.

Center v

ariables.

Adjusts the series to have a mean of 0 before calculating the

spectrum and to remove the large term that may be associated with the series mean.

Plot. Periodogram and spectral density are available for both univariate and bivariate

analyse

 Period

s. All other choices are available only for bivariate analyses.

ogram.

Unsmoothed plot of spectral amplitude (plotted on a logarithmic scale) against either frequency or period. Low-frequency variation characterizes a smooth series. Variation spread evenly across all frequencies indicates "white

noise.

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Chapter 7



Spectral density. A periodogram that has been smoothed to remove irregular

variation.

 Cospectral

density.

The real part of the cross-periodogram, which is a measure of

the correlation of the in-phase frequency components of two time series.

 Quadrature spectrum. The imaginary part of the cross-periodogram, which is a

measure of the correlation of the out-of-phase frequency components of two time series. The

 Cross ampl

components are out of phase by pi/2 radians.

itude.

The square root of the sum of the squared cospectral density

and the squared quadrature spectrum.

 Gain. The quotient of dividing the cross amplitude by the spectral density for one

of the series. Each of the two series has its own gain value.

 Squared coherency. Theproductofthegainsofthetwoseries.  Phase spectrum. A measure of the extent to which each frequency component

of one seri

By frequency. All plots are produced by frequency, ranging from frequency 0 (the

es leads or lags the other.

constant or mean term) to frequency 0.5 (the term for a cycle of two observations).

By period.

All plots are produced by period, ranging from 2 (the term for a cycle of two observations) to a period equal to the number of observations (the constant or mean term). Period is displayed on a logarithmic scale.

SPECTRA Command Additional Features

The SPSS command language also allows you to:

 Save computed spectral analysis variables to the working data file for later use.  Specify custom weights for the spectral window.  Produce plots by both frequency and period.  Print a complete listing of each value shown in the plot.

See the SPSS Command Syntax Reference for complete syntax information.

Page 61

Part 2: Examples

Page 62

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Chapter

Exponential

Smoothing

Using Exponential Smoothing to Predict Future Sales

A catalog company is interested in forecasting monthly sales of its men’s clothing

s end, the company collected monthly sales of men’s clothing for a

onential Smoothing procedure to predict monthly sales of men’s

ples that follow, it is more convenient to use variable names rather than

Prelimina

E From the menus choose:

line. To thi 10-year period. This information is collected in catalog_seasfac.sav, found in the \tutorial\sample_files\ subdirectory of the directory in which you installed SPSS.

Use the Exp

clothing for the following year.

ries

In the exam variable labels.

Edit

Options...

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Chapter 8

Figure 8-1

Options dialog box

Unders

E Select Display names in the Variable Lists group. E Click OK.

tanding Your Data

st step in analyzing a time series is to plot it. Visual inspection of a time series

The fir can often be a powerful guide in choosing an appropriate exponential smoothing model. In particular:

 Does the series have an overall trend? Does the trend appear constant or does

it appear to be dying out with time?

 Does the series show seasonality? Do the seasonal fluctuations seem to grow with

r do they appear constant over successive periods?

time o

Page 65

To obtain a plot of men’s clothing sales over time:

E From the menus choose:

Graphs

Sequence...

Figure 8-2

Sequence Charts dialog box

Exponential Sm

oothing

E Select E Select E Click T

men and move it into the Variables list. date and move it into the Time Axis Labels list.

ime Lines

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Chapter 8

Figure 8-3

Sequence Charts Time Axis Reference Lines dialog box

E Select Line at each change of. E Select YEAR_ and move it into the Reference Variable list.

These cho

ices result in a vertical reference line at the start of each year, which is

useful for identifying annual seasonality.

E Click Continue. E Click OK in the Sequence Charts dialog box.

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Figure 8-4

Sales of men’s clothing (in U.S. dollars)

Exponential Sm

oothing

The ser

ies shows a global upward trend; that is, the series values tend to increase over

time. The upward trend is seemingly constant, which indicates a linear trend.

The series also has a distinct seasonal pattern with annual highs in December. This

is easy

to see because of the vertical reference lines positioned at the start of each year. The seasonal variations appear to grow with the upward series trend, which suggests multiplicative rather than additive seasonality.

Building and Analyzing Exponential Smoothing Models

Building a best-fit exponential smoothing model involves determining the model type—does the model need to include trend and/or seasonality—and then obtaining

st-fit parameters for the chosen model.

the be

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Chapter 8

The plot of men’s clothing sales over time suggested a model with both a linear trend component and a multiplicative seasonality component. This implies a Winters model. Firs

t, however, we will explore a simple model (no trend and no seasonality) and then a Holt model (incorporates linear trend but no seasonality). This will give you practice in identifying when a model is not a good fit to the data, which is an essential

skill in successful model building.

Building and Analyzing a Simple Model

To build an exponential smoothing model:

E From the menus choose:

Analyze

Time Series

Exponential Smoothing...

Figure 8-5

Exponential Smoothing dialog box

E Select E Select E Click Pa

men and move it into the Variables list.

Simple in the Model group.

rameters

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Exponential Sm

Figure 8-6

Exponential Smoothing Parameters dialog box

E Select Grid Search in the General (Alpha) group. Leave the Start, Stop, and By text

boxes wi The

th their default values of 0, 1, and 0.1, respectively.

Grid Search option provides a convenient method for determining the best-fit

oothing

model parameters by calculating goodness-of-fit measures for each of the grid values. The curr

ent selections result in values of alpha ranging from 0 to 1 in increments of

0.1. Leave the default choice for displaying only the 10 best-fit models. The analysis is

rformed for each of the points in the grid, but the output is limited to displaying

still pe results for the 10 best models.

E Click Continue. E Click OK in the Exponential Smoothing dialog box.

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Chapter 8

Figure 8-7

Exponential smoothing , no trend or seasonality

The model output lists the 10 best-fitting values of alpha, along with the associated sums of squared errors (SSE) for each value. Each value of alpha corresponds to a different model, with models ranked according to their SSE value. A lower rank implies a smaller SSE and thus a model that gives a better fit to the data. The SSE measure of error is lowest when alpha is 0.1, indicating that an alpha of 0.1 gives the best fit to the data for this class of models. This low value of alpha indicates that the overall level of the series is best predicted when all observations have roughly equal weight.

To see how well the simple model fits the data, you’ll need to reopen the Sequence Charts dialog box. A convenient shortcut for reopening previous dialog boxes is as follows:

E Click the Dialog Recall toolbar icon. E Select the entry for the desired dialog box, which in this case is Sequence Charts.

Page 71

Figure 8-8

Sequence Charts dialog box

. This will restore the dialog box to its default settings, thus removing

E Click R

eset

the timeline settings used previously.

E Select men and FIT_1 and move them into the Variables list.

Exponential Sm

oothing

E Click OK.

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Chapter 8

Figure 8-9

Predictions from exponential smoothing, no trend or seasonality

Notice that the model, as expected, does not account for the observed seasonality. In addition, it predicts an initial downward trend in contrast to the data, which exhibits a continuing upward trend.

Examining the autocorrelations and partial autocorrelations for the residuals of the best-fit simple model provides more quantitative insight than viewing the sequence charts. Significant structure in either of these correlation functions would imply that the underlying model is incomplete.

E From the menus choose:

Graphs

Time Series

Autocorrelations...

Page 73

Exponential Sm

Figure 8-10

Autocorrelations dialog box

E Select the error variable ERR_1 associated with the fit from the simple model. E Click OK.

oothing

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Chapter 8

Figure 8-11

Residual autocorrelation plot for simple model

The autocorrelation function shows a significant peak at a lag of 12. This is no surprise, since the simple model doesn’t account for seasonality and there is a strong annual seasonal component in the data.

Page 75

Exponential Sm

Figure 8-12

Residual autocorrelation statistics for simple model

oothing

Notice also that the Box-Ljung statistic at a lag of 12 is statistically significant. In

eassociatedp value is 0 to the stated precision. This underscores the fact that

fact, th the lag 12 autocorrelation is significant and represents structure not accounted for in the present model.

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Chapter 8

Figure 8-13

Residual partial autocorrelation plot for simple model

The partial autocorrelation function removes the indirect effect of all intervening lags, providing the best measure of a direct relationship between time series values separated by a given lag. The partial autocorrelation function for the simple model shows the same significant peak at a lag of 12 as the autocorrelation function. This provides definitive proof that the residuals of the simple model contain the structure of the annual seasonality of the time series.

Summary

Given the poor fit of the simple model to the data and the presence of significant structure in the model’s residuals, you can conclude that the simple model is not a good choice for modeling the data.

The initial plot of the time series suggested a model with both trend and seasonality, neither of which is present in the simple model. Next, we will use the Holt model to extend the present model with a trend component.

Page 77

Building and Analyzing a Holt Model

Figure 8-14

Exponential

E To build a Holt model, open the Exponential Smoothing dialog box. E Select Holt in the Model group. E Click Parameters.

Smoothing dialog box

Exponential Sm

oothing

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Chapter 8

Figure 8-15

Exponential Smoothing Parameters dialog box

E SelectGrid Search in both the General (Alpha) and Trend (Gamma) groups. Leave the

Start, S

top, and By text boxes with their default values.

The current grid search results in evaluating 66 models, one for each of the possible combinations of alpha and gamma—11 values of alpha and 6 values of gamma.

E Click Continue. E Click OK in the Exponential Smoothing dialog box.

Figure 8-16

Exponential smoothing, linear trend and no seasonality

Page 79

Exponential Sm

oothing

The SSE measure of error is lowest when alpha is 0.1 and gamma is 0. The low values of alpha and gamma mean that all observations have roughly equal weight in modeling bo

th the overall level of the series and the series trend. The SSE for the best-fitting Holt model is 3612687888, which is slightly better than the SSE for the best-fitting simple model of 3716115994. The Holt model then appears to give a better fit

Figure 8-17

Sequence Charts dialog box

tothedatathanthesimplemodel.

E To see how well the Holt model fits the data, open the Sequence Charts dialog box. E Deselect the variable associated with the simple model (FIT_1) from the Variables list

and select the variable corresponding to the Holt model (FIT_2).

E Click OK.

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Chapter 8

Figure 8-18

Predictions from exponential smoothing, linear trend and no seasonality

Notice that the model does a good job of capturing the trend component of the data. As expected, though, it does not account for the observed seasonality. As in the case of the simple model, examination of the autocorrelations and partial autocorrelations of the residuals provides more quantitative insight.

Page 81

Exponential Sm

Figure 8-19

Autocorrelations dialog box

E Examine the autocorrelations of the residuals by opening the Autocorrelations

oothing

dialog box.

E Deselect the variable associated with the simple model (ERR_1) from the Variables

list and

E Click O

select the variable corresponding to the Holt model (ERR_2).

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Chapter 8

Figure 8-20

Residual autocorrelation plot for Holt model

The autocorrelation function shows a significant peak at a lag of 12, just as in the case of the simple model. This is no surprise, since the Holt model (like the simple model) doesn’t account for seasonality.

Page 83

Figure 8-21

Residual partial autocorrelation plot for Holt model

Exponential Sm

oothing

The partial autocorrelation function for the Holt model exhibits the same significant peak at a lag of 12 as the autocorrelation function. This shows that the residuals of the Holt model, like those of the simple model, contain the structure of the annual seasonality of the time series.

Summary

Given the presence of significant structure in the residuals of the Holt model corresponding to the seasonality component, you can conclude that the Holt model, although better than the simple model, is not a good choice for modeling the data.

Recall that the initial plot of men’s clothing sales over time suggested a model

incorporating a linear trend and multiplicative seasonality—a Winters model.

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Chapter 8

Building and Analyzing a Winters Model

Figure 8-22

Exponential

E To build a Winters model, open the Exponential Smoothing dialog box. E Select Winters in the Model group. E Select Seasonal_Factors_Men and move it into the Seasonal Factors list. E Click Parameters.

Smoothing dialog box

Page 85

Exponential Sm

Figure 8-23

Exponential Smoothing Parameters dialog box

E Select Grid Search in the General (Alpha), Trend (Gamma), and Seasonal (Delta)

groups.

E Click C

Leave the Start, Stop, and By text boxes with their default values.

ontinue

oothing

in the Exponential Smoothing dialog box.

E Click O

Figure 8-24

Exponential smoothing, linear trend and multiplicative seasonality

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Chapter 8

The SSE measure o f error is lowest when alpha is 0.1, gamma is 0, and delta is 0. The low values of alpha, gamma, and delta mean that all observations have roughly equal weigh

t in modeling the overall level of the series, the series trend, and the seasonality. The SSE for the best-fit Winters model is significantly less than the corresponding values for both the Holt (SSE of 3612687888) and simple (SSE of 371611599

4) models. This is the first indication that the Winters model provides a

substantially better fit to the data.

Figure 8-2

Sequence C

E To see how well the Wintersmodel fits the data, open the Sequence Charts dialog box. E Deselect the variable associated with the Holt model (FIT_2) from the Variables list

and se

E Clic

harts dialog box

lect the variable corresponding to the Winters model (FIT_3).

OK.

Page 87

Exponential Sm

Figure 8-26

Predictions from exponential smoothing, linear trend and multiplicative seasonality

oothing

Notice that the model does a good job of capturing both the trend and the seasonality of the data. The data set covers a period of 10 years and includes 10 seasonal peaks occurring in December of each year. The 10 peaks present in the predicted results match up well with the 10 annual peaks in the real data.

The results also underscore the limitations of the Exponential Smoothing procedure, since there is significant structure that is not accounted for. If you are primarily interested in modeling a long-term trend with seasonal variation, then exponential smoothing may be a good choice. To model a more complex structure, consider using the ARIMA procedure. An examination of the autocorrelations and partial autocorrelations of the residuals will provide stronger evidence for or against the Winters model.

Page 88

Chapter 8

Figure 8-27

Autocorrelations dialog box

E Examine the autocorrelations of the residuals by opening the Autocorrelations

dialog box.

E Deselect the variable associated with the Holt model (ERR_2) from the Variables list

and sele

ct the variable corresponding to the Winters model (ERR_3).

E Click O

Page 89

Figure 8-28

Residual autocorrelation plot for Winte r s model

Exponential Sm

oothing

The Winters model has adequately described the annual seasonal component, so there is no longer a significant peak at a lag of 12 in the autocorrelation function; the peak is within the confidence bounds.

Page 90

Chapter 8

Figure 8-29

Residual autocorrelation statistics for Winters model

Notice also that the Box-Ljung statistic a t a lag of 12 is no longer statistically significant.

Page 91

Figure 8-30

Residual partial autocorrelation plot for Winters model

Exponential Sm

oothing

The partial autocorrelation function no longer has a significant peak at a lag of 12. The pea

k at a lag of 11, however, is outside the confidence limits and appears

significant. This would suggest periodic behavior with an 11-month cycle.

The Box-Ljung statistic at a lag of 11, however, isn’t statistically significant. In

on, analysis of the catalog company’s records shows no good business reason to

additi expect an 11-month cycle. With reasonable confidence, you can then reject the lag 11 peak as irrelevant.

Summa

ry of Model Building

models incorporating different assumptions about trend and seasonality have

Three been considered, and the results clearly indicate a Winters model over a Holt or simple model. Not only does the best-fit Winters model give a good visual fit to

ata, but the residuals, as analyzed through their autocorrelations and partial

the d autocorrelations, show no significant structure.

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Chapter 8

Although the initial plot of the time series suggested linear trend and multiplicative seasonality, a more thorough analysis would also consider a custom model with linear trend and ad

ditive seasonality. The results of that analysis show, however, that the SSE for the best-fit Winters model is lower, giving us confidence that the Winters model is truly the best exponential smoothing model for our data.

Testing the Predictive Ability of the Model

You can judge the forecasting performance of your model by using holdouts. A holdout is a historical series point that is not used in the computation of the model parameter model to predict values that you actually know, you get an idea of how well the model forecasts. This method can be illustrated by holding out the data from January, 1998, through D model, and the model i s then used to forecast sales in 1998.

To perform a holdout analysis, first select the modeling period:

s, thus removing its effect on the computation of forecasts. By forcing the

ecember, 1998. The data prior to January, 1998, are used to build the

E From the menus choose:

Data

Select Cases...

Page 93

Figure 8-31

Select Cases dialog box

Basedontimeorcaserangein the Select group.

E Selec

Exponential Sm

oothing

E Click R

Figure 8-32

Select Cases Range dialog box

E Enter 1989 for the year associated with the first case and 1 for the month associated

ange

with the first case.

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Chapter 8

Enter 1997 for the year associated with the last case and 12 for the month associated with the last case.

These choice

E Click Conti E Click OK in t

Figure 8-33

Exponential Smoothing dialog box

E Open the Exponential Smoothing dialog box. E Click OK.

s will result in a model based on the period 01/1989 through 12/1997.

nue

he Select Cases dialog box.

This results in rerunning the Winters model using the data from 01/1989 to 12/1997 to determi

ne the best-fit parameters. The analysis also includes predictions of sales of men’s clothing during the holdout period (01/1998 to 12/1998) using the parameters from the best-fit model.

Page 95

Exponential Sm

oothing

Comparison of the model predictions for the holdout period with the actual data is best done by limiting the cases to the holdout period itself.

Figure 8-34

Select Cases dialog box

E Open t E Click

he Select Cases dialog box.

Range.

Page 96

Chapter 8

Figure 8-35

Select Cases Range dialog box

E Enter 1998 for the year associated with the first case and 1 for the month associated

with the first case.

E Enter 1998 for the year associated with the last case and 12 for the month associated

with the last case.

E Click Continue. E Click OK in the Select Cases dialog box.

Page 97

Exponential Sm

Figure 8-36

Sequence Charts dialog box

E Open th E Desele

e Sequence Charts dialog box.

ct the variable associated with the original Winters model (FIT_3) from the Variables list and select the variable corresponding to the Winters model with holdouts (FIT_4).

oothing

E Click OK.

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Chapter 8

Figure 8-37

Forecast from Winters model w ith holdouts

han the value for August, 1998, the forecast values show good agreement

Other t with the known values, indicating that the model has satisfactory predictive ability. A review of the company records shows that a 20%-off promotion was in effect for the mai

ling of August, 1998, which may explain why sales were unusually high for that month. You expect that the model will behave well in the absence of special events such as occasional promotions but may need to be modified if accounting

ecialeventsisimportant.

for sp

Using the Model to P redict Future Sales

You’ve determined that a Winters model is the best exponential smoothing model for

data and obtained satisfactory results regarding its predictive ability. Now it’s

your time to use the model to predict future sales.

Page 99

Figure 8-38

Select Cases dialog box

To pred

ict future values of sales of men’s clothing, first include all cases by

performing the following steps:

Exponential Sm

oothing

E Open the Select Cases dialog box. E Select All Cases in the Select group. E Click OK.

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Chapter 8

Figure 8-39

Exponential Smoothing dialog box

E Open the E Click Sa

Figure 8-40

Exponential Smoothing Save dialog box

E Select Addtofilein the Create Variables group.

Exponential Smoothing dialog box.

This causes the forecast values to be saved as regular variables in your working data file.

E Select Predict through in the Predict Cases group. E Enter 1999 for the year and 12 for the month.