Introduction to Time Series and Forecasting

428 Pages·2016·3.96 MB·English

by  Peter J. Brockwell

Springer Texts in Statistics
Peter J. Brockwell
Richard A. Davis
Introduction
to Time Series
and Forecasting
Third Edition
Springer Texts in Statistics
SeriesEditors:
R.DeVeaux
S.Fienberg
I.Olkin
Moreinformationaboutthisseriesathttp://www.springer.com/series/417

Peter J. Brockwell • Richard A. Davis
Introduction to Time Series and Forecasting
Third Edition
123
PeterJ.Brockwell RichardA.Davis
DepartmentofStatistics DepartmentofStatistics
ColoradoStateUniversity ColumbiaUniversity
FortCollins,CO,USA NewYork,NY,USA
Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com.
ISSN1431-875X ISSN2197-4136 (electronic)
SpringerTextsinStatistics
ISBN978-3-319-29852-8 ISBN978-3-319-29854-2 (eBook)
DOI10.1007/978-3-319-29854-2
LibraryofCongressControlNumber:2016939116
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To Pam and Patti

Preface
Thisbookisaimedatthereaderwhowishestogainaworkingknowledgeoftimeseries
and forecasting methods as applied in economics, engineering, and the natural and
social sciences. Unlike our more advanced book, Time Series: Theory and Methods,
Brockwell and Davis (1991), this one requires only a knowledge of basic calculus,
matrixalgebraandelementarystatisticsatthelevel,forexample,ofMendenhalletal.
(1990). It is intended for upper-level undergraduate students and beginning graduate
students.
Theemphasisisonmethodsandtheanalysisofdatasets.Theprofessionalversion
of the time series package ITSM2000, for Windows-based PC, enables the reader to
reproduce most of the calculations in the text (and to analyze further data sets of the
reader’s own choosing). It is available for download, together with most of the data
setsusedinthebook,fromhttp://extras.springer.com. AppendixEcontainsadetailed
introduction tothepackage.
Verylittlepriorfamiliaritywithcomputingisrequiredinordertousethecomputer
package. Thebookcanalsobeusedinconjunction withothercomputerpackagesfor
handlingtimeseries.Chapter14ofthebookbyVenablesandRipley(2003)describes
howtoperformmanyofthecalculationsusingSandR.ThepackageITSMRofWeigt
(2015)canbeusedinRtoreproducemanyofthefeaturesofITSM2000.Thepackage
Yuima, also for R, can be used for simulation and estimation of the Lévy-driven
CARMA processes discussed in Section 11.5 (see Iacus and Mercuri (2015)). Both
ofthesepackages canbedownloaded fromhttps://cran.rproject.org/web/packages.
There are numerous problems atthe end ofeach chapter, many ofwhich involve
useoftheprogramstostudythedatasetsprovided.
Tomaketheunderlyingtheoryaccessibletoawideraudience,wehavestatedsome
of the key mathematical results without proof, but have attempted to ensure that the
logical structure ofthedevelopment isotherwise complete. (References toproofsare
providedfortheinterested reader.)
There is sufficient material here for a full-year introduction to univariate and
multivariatetimeseriesandforecasting. Chapters1through6havebeenusedforsev-
eral years in introductory one-semester courses in univariate timeseries at Columbia
University, Colorado StateUniversity, andRoyalMelbourne Institute ofTechnology.
Thechapteronspectralanalysiscanbeexcludedwithoutlossofcontinuitybyreaders
whoaresoinclined.
In viewof theexplosion ofinterest infinancial timeseries inrecent decades, the
thirdeditionincludesanewchapter(Chapter7)specificallydevotedtothistopic.Some
ofthebasictoolsrequiredforanunderstandingofcontinuous-timefinancialtimeseries
models(Brownianmotion,Lévyprocesses, andItôcalculus)havealsobeenaddedas
vii
viii Preface
AppendixD,andanewSection11.5providesanintroductiontocontinuousparameter
ARMA(orCARMA)processes.
ThediskettecontainingthestudentversionofthepackageITSM2000isnolonger
included with the book since the professional version (which places no limit on the
lengthoftheseriestobestudied)cannowbedownloadedfromhttp://extras.springer.
comasindicatedabove.AtutorialfortheuseofthepackageisprovidedasAppendixE
andasearchable file,ITSM_HELP.pdf,givingmoredetailed instructions, isincluded
withthepackage.
Wearegreatlyindebted tothereadersofthefirstandsecondeditionsofthebook
and especially toMatthew Calder, coauthor of thecomputer package ITSM2000and
toAnthonyBrockwell,bothofwhommademanyvaluablecommentsandsuggestions.
We also wish to thank Colorado State University, Columbia University, the National
Science Foundation, Springer-Verlag, and our families for their continuing support
duringthepreparation ofthisthirdedition.
FortCollins,CO,USA PeterJ.Brockwell
NewYork,NY,USA RichardA.Davis
April,2016
Contents
Preface vii
1. Introduction 1
1.1. ExamplesofTimeSeries 1
1.2. ObjectivesofTimeSeriesAnalysis 5
1.3. SomeSimpleTimeSeriesModels 6
1.3.1. SomeZero-MeanModels 6
1.3.2. ModelswithTrendandSeasonality 7
1.3.3. AGeneralApproachtoTimeSeriesModeling 12
1.4. StationaryModelsandtheAutocorrelation Function 13
1.4.1. TheSampleAutocorrelation Function 16
1.4.2. AModelfortheLakeHuronData 18
1.5. Estimation and Elimination of Trend and Seasonal
Components 20
1.5.1. EstimationandEliminationofTrendintheAbsence
ofSeasonality 21
1.5.2. EstimationandEliminationofBothTrendand
Seasonality 26
1.6. TestingtheEstimatedNoiseSequence 30
Problems 34
2. Stationary Processes 39
2.1. BasicProperties 39
2.2. LinearProcesses 44
2.3. Introduction toARMAProcesses 47
2.4. PropertiesoftheSampleMeanandAutocorrelation Function 50
2.4.1. Estimationofμ 50
2.4.2. Estimationofγ(·)andρ(·) 51
2.5. ForecastingStationary TimeSeries 55
2.5.1. PredictionofSecond-OrderRandomVariables 57
2.5.2. ThePredictionOperator P(·|W) 58
2.5.3. TheDurbin–Levinson Algorithm 60
2.5.4. TheInnovations Algorithm 62
2.5.5. RecursiveCalculationoftheh-StepPredictors 65
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