TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pagei Springer Texts in Statistics Advisors: GeorgeCasella StephenFienberg IngramOlkin Springer NewYork Berlin Heidelberg Barcelona HongKong London Milan Paris Singapore Tokyo TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pageii TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pageiii Peter J. Brockwell Richard A. Davis Introduction to Time Series and Forecasting Second Edition With126Illustrations IncludesCD-ROM 1 3 TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pageiv PeterJ.Brockwell RichardA.Davis DepartmentofStatistics DepartmentofStatistics ColoradoStateUniversity ColoradoStateUniversity FortCollins,CO80523 FortCollins,CO80523 USA USA [email protected] [email protected] EditorialBoard GeorgeCasella StephenFienberg IngramOlkin DepartmentofStatistics DepartmentofStatistics DepartmentofStatistics Griffin-FloydHall CarnegieMellonUniversity StanfordUniversity UniversityofFlorida Pittsburgh,PA15213-3890 Stanford,CA94305 P.O.Box118545 USA USA Gainesville,FL32611-8545 USA LibraryofCongressCataloging-in-PublicationData Brockwell,PeterJ. Introductiontotimeseriesandforecasting/PeterJ.BrockwellandRichardA.Davis.—2nded. p. cm.—(Springertextsinstatistics) Includesbibliographicalreferencesandindex. ISBN0-387-95351-5(alk.paper) 1.Time-seriesanalysis. I.Davis,RichardA. II.Title. III.Series. QA280.B757 2002 519.5(cid:2)5—dc21 2001049262 Printedonacid-freepaper. ©2002,1996Springer-VerlagNewYork,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionof thepublishers(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010,USA),exceptforbrief excerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownor hereafterdevelopedisforbidden. Theuseofgeneraldescriptivenames,tradenames,trademarks,etc.,inthispublication,eveniftheformerare notespeciallyidentified,isnottobetakenasasignthatsuchnames,asunderstoodbytheTradeMarksand MerchandiseMarksAct,mayaccordinglybeusedfreelybyanyone. ProductionmanagedbyMaryAnnBrickner;manufacturingsupervisedbyJoeQuatela. TypesetbyTheBartlettPress,Inc.,Marietta,GA. PrintedandboundbyR.R.DonnelleyandSons,Harrisonburg,VA. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 ISBN0-387-95351-5 SPIN10850334 Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pagev To Pam and Patti TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pagevi TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pagevii Preface This book is aimed at the reader who wishes to gain a working knowledge of time seriesandforecastingmethodsasappliedineconomics,engineeringandthenatural and social sciences. Unlike our earlier book, Time Series: Theory and Methods, re- ferredtointhetextasTSTM,thisonerequiresonlyaknowledgeofbasiccalculus, matrix algebra and elementary statistics at the level (for example) of Mendenhall, WackerlyandScheaffer(1990).Itisintendedforupper-levelundergraduatestudents andbeginninggraduatestudents. The emphasis is on methods and the analysis of data sets. The student version ofthetimeseriespackageITSM2000,enablingthereadertoreproducemostofthe calculationsinthetext(andtoanalyzefurtherdatasetsofthereader’sownchoosing), isincludedontheCD-ROMwhichaccompaniesthebook.Thedatasetsusedinthe bookarealsoincluded.ThepackagerequiresanIBM-compatiblePCoperatingunder Windows95,NTversion4.0,oralaterversionofeitheroftheseoperatingsystems. TheprogramITSMcanberundirectlyfromtheCD-ROMorinstalledonaharddisk as described at the beginning of Appendix D, where a detailed introduction to the packageisprovided. Verylittlepriorfamiliaritywithcomputingisrequiredinordertousethecomputer package. Detailed instructions for its use are found in the on-line help files which are accessed, when the program ITSM is running, by selecting the menu option Help>Contents and selecting the topic of interest. Under the heading Data you willfindinformationconcerningthedatasetsstoredontheCD-ROM.Thebookcan also be used in conjunction with other computer packages for handling time series. Chapter 14 of the book by Venables and Ripley (1994) describes how to perform manyofthecalculationsusingS-plus. Therearenumerousproblemsattheendofeachchapter,manyofwhichinvolve useoftheprogramstostudythedatasetsprovided. To make the underlying theory accessible to a wider audience, we have stated some of the key mathematical results without proof, but have attempted to ensure that the logical structure of the development is otherwise complete. (References to proofsareprovidedfortheinterestedreader.) TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pageviii viii Preface Since the upgrade to ITSM2000 occurred after the first edition of this book appeared, we have taken the opportunity, in this edition, to coordinate the text with thenewsoftware,tomakeanumberofcorrectionspointedoutbyreadersofthefirst editionandtoexpandonseveralofthetopicstreatedonlybrieflyinthefirstedition. AppendixD,thesoftwaretutorial,hasbeenrewritteninordertobecompatible withthenewversionofthesoftware. Some of the other extensive changes occur in (i) Section 6.6, which highlights the role of the innovations algorithm in generalized least squares and maximum likelihood estimation of regression models with time series errors, (ii) Section 6.4, wherethetreatmentofforecastfunctionsforARIMAprocesseshasbeenexpanded and(iii)Section10.3,whichnowincludesGARCHmodelingandsimulation,topics ofconsiderableimportanceintheanalysisoffinancialtimeseries.Thenewmaterial hasbeenincorporatedintotheaccompanyingsoftware,towhichwehavealsoadded the option Autofit. This streamlines the modeling of time series data by fitting maximumlikelihoodARMA(p,q)modelsforaspecifiedrangeof(p,q)valuesand automaticallyselectingthemodelwithsmallestAICCvalue. Thereissufficientmaterialhereforafull-yearintroductiontounivariateandmul- tivariatetimeseriesandforecasting.Chapters1through6havebeenusedforseveral yearsinintroductoryone-semestercoursesinunivariatetimeseriesatColoradoState University and Royal Melbourne Institute of Technology. The chapter on spectral analysiscanbeexcludedwithoutlossofcontinuitybyreaderswhoaresoinclined. WearegreatlyindebtedtothereadersofthefirsteditionandespeciallytoMatthew Calder, coauthor of the new computer package, and Anthony Brockwell for their many valuable comments and suggestions. We also wish to thank Colorado State University, the National Science Foundation, Springer-Verlag and our families for theircontinuingsupportduringthepreparationofthissecondedition. FortCollins,Colorado PeterJ.Brockwell August2001 RichardA.Davis TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pageix Contents Preface vii 1. Introduction 1 1.1. ExamplesofTimeSeries 1 1.2. ObjectivesofTimeSeriesAnalysis 6 1.3. SomeSimpleTimeSeriesModels 7 1.3.1.SomeZero-MeanModels 8 1.3.2.ModelswithTrendandSeasonality 9 1.3.3.AGeneralApproachtoTimeSeriesModeling 14 1.4. StationaryModelsandtheAutocorrelationFunction 15 1.4.1.TheSampleAutocorrelationFunction 18 1.4.2.AModelfortheLakeHuronData 21 1.5. EstimationandEliminationofTrendandSeasonalComponents 23 1.5.1.EstimationandEliminationofTrendintheAbsenceof Seasonality 24 1.5.2. Estimation and Elimination of Both Trend and Seasonality 31 1.6. TestingtheEstimatedNoiseSequence 35 Problems 40 2. Stationary Processes 45 2.1. BasicProperties 45 2.2. LinearProcesses 51 2.3. IntroductiontoARMAProcesses 55 2.4. PropertiesoftheSampleMeanandAutocorrelationFunction 57 2.4.1.Estimationofµ 58 2.4.2.Estimationofγ(·)andρ(·) 59 2.5. ForecastingStationaryTimeSeries 63 2.5.1.TheDurbin–LevinsonAlgorithm 69 2.5.2.TheInnovationsAlgorithm 71 2.5.3.PredictionofaStationaryProcessinTermsofInfinitely ManyPastValues 75 TheBartlettPress,Inc. brockwel 8·i·2002 1:59p.m. Pagex x Contents 2.6. TheWoldDecomposition 77 Problems 78 3. ARMA Models 83 3.1. ARMA(p,q)Processes 83 3.2. TheACFandPACFofanARMA(p,q)Process 88 3.2.1.CalculationoftheACVF 88 3.2.2.TheAutocorrelationFunction 94 3.2.3.ThePartialAutocorrelationFunction 94 3.2.4.Examples 96 3.3. ForecastingARMAProcesses 100 Problems 108 4. Spectral Analysis 111 4.1. SpectralDensities 112 4.2. ThePeriodogram 121 4.3. Time-InvariantLinearFilters 127 4.4. TheSpectralDensityofanARMAProcess 132 Problems 134 5. Modeling and Forecasting with ARMA Processes 137 5.1. PreliminaryEstimation 138 5.1.1.Yule–WalkerEstimation 139 5.1.2.Burg’sAlgorithm 147 5.1.3.TheInnovationsAlgorithm 150 5.1.4.TheHannan–RissanenAlgorithm 156 5.2. MaximumLikelihoodEstimation 158 5.3. DiagnosticChecking(cid:1) (cid:2) 164 5.3.1.TheGraphof Rˆt,t (cid:4)1,...,n 165 5.3.2.TheSampleACFoftheResiduals 166 5.3.3.TestsforRandomnessoftheResiduals 166 5.4. Forecasting 167 5.5. OrderSelection 169 5.5.1.TheFPECriterion 170 5.5.2.TheAICCCriterion 171 Problems 174 6. Nonstationary and Seasonal Time Series Models 179 6.1. ARIMAModelsforNonstationaryTimeSeries 180 6.2. IdentificationTechniques 187