Time Series Analysis and Its Applications: With R Examples PDF
Preview Time Series Analysis and Its Applications: With R Examples
Springer Texts in Statistics Robert H. Shumway David S. Stoff er Time Series Analysis and Its Applications With R Examples Fourth Edition Springer Texts in Statistics SeriesEditors RichardDeVeaux StephenE.Fienberg IngramOlkin Moreinformationaboutthisseriesat http://www.springer.com/series/417 Robert H. Shumway • David S. Stoffer Time Series Analysis and Its Applications With R Examples Fourth Edition 123 RobertH.Shumway DavidS.Stoffer DepartmentofStatistics DepartmentofStatistics UniversityofCalifornia,Davis UniversityofPittsburgh Davis,CA,USA Pittsburgh,PA,USA ISSN1431-875X ISSN2197-4136 (electronic) SpringerTextsinStatistics ISBN978-3-319-52451-1 ISBN978-3-319-52452-8 (eBook) DOI10.1007/978-3-319-52452-8 LibraryofCongressControlNumber:2017930675 ©SpringerInternationalPublishingAG1999,2012,2016,2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Fourth Edition The fourtheditionfollowsthegenerallayoutofthethirdeditionbutincludessome modernizationoftopicsaswellasthecoverageofadditionaltopics.Theprefaceto thethirdedition—whichfollows—stillapplies,soweconcentrateonthedifferences between the two editions here. As in the third edition, R code for each example is given in the text, even if the code is excruciatingly long. Most of the examples with seemingly endless coding are in the latter chapters. The R package for the text, astsa, is still supportedand details may be foundin AppendixR. The global temperaturedeviationserieshavebeenupdatedto2015andareincludedinthenewest versionofthepackage;thecorrespondingexamplesandproblemshavebeenupdated accordingly. Chapter1ofthiseditionissimilartothepreviousedition,butwehaveincluded thedefinitionoftrendstationarityandtheconceptofprewhiteningwhenusingcross- correlation.TheNewYorkStockExchangedataset,whichfocusedonanoldfinancial crisis,wasreplacedwithamorecurrentseriesoftheDowJonesIndustrialAverage, which focuses on a newer financial crisis. In Chap. 2, we rewrote some of the regressionreview,changingthesmoothingexamplesfromthemortalitydataexample totheSouthernOscillationIndexandfindingElNiño.Wealsoexpandedonthelagged regressionexampleandcarrieditontoChap.3. InChap.3,weremovednormalityfromdefinitionofARMAmodels;whilethe assumption is not necessary for the definition, it is essential for inferenceand pre- diction.WeaddedasectiononregressionwithARMAerrorsandthecorresponding problems;thissection waspreviouslyin Chap. 5.Someof the exampleshavebeen modifiedandwe addedsome examplesinthe seasonalARMA section.Finally,we includedadiscussionoflaggedregressionwithautocorrelatederrors. In Chap. 4, we improved and added some examples. The idea of modulated seriesisdiscussedusingtheclassicstarmagnitudedataset.Wemovedsomeofthe filteringsectionforwardforeasieraccesstoinformationwhenneeded.Weremoved the reliance on spec.pgram (from the stats package) to mvspec (from the astsa package)sowecanavoidhavingtospendpagesexplainingthequirksofspec.pgram, v vi PrefacetotheFourthEdition whichtendedtotakeoverthenarrative.Thesectiononwaveletswasremovedbecause therearesomanyaccessibletextsavailable.Thespectralrepresentationtheoremsare discussedinalittlemoredetailusingexamplesbasedonsimpleharmonicprocesses. The general layout of Chap. 5 and of Chap. 7 is the same, although we have revisedsomeoftheexamples.Aspreviouslymentioned,wemovedregressionwith ARMAerrorstoChap.3. Chapter 6 sees the biggest change in this edition. We have added a section on smoothing splines, and a section on hidden Markov models and switching autore- gressions. The Bayesian section is completely rewritten and is on linear Gaussian state space modelsonly.The nonlinearmaterialin the previousedition is removed because it was old, and the newer materialis in Douc, Moulines, and Stoffer [53]. Manyoftheexampleshavebeenrewrittentomakethechaptermoreaccessible. The appendices are similar, with some minor changes to Appendix A and AppendixB.WeaddedmaterialtoAppendixC,includingadiscussionofRiemann– Stieltjesandstochasticintegration,aproofofthefactthatthespectraofautoregressive processesaredenseinthespaceofspectraldensities,andaproofofthefactthatspec- traareapproximatelytheeigenvaluesofthecovariancematrixofastationaryprocess. Wetweaked,rewrote,improved,orrevisedsomeoftheexercises,buttheoverall orderingandcoverageisroughlythesame.And,ofcourse,wemovedregressionwith ARMAerrorsproblemstoChap.3andremovedtheChap.4waveletproblems.The exercisesforChap.6havebeenupdatedaccordinglytoreflectthenewandimproved versionofthechapter. Davis,CA,USA RobertH.Shumway Pittsburgh,PA,USA DavidS.Stoffer December2016 Preface to the Third Edition Thegoalsofthisbookaretodevelopanappreciationfortherichnessandversatility of modern time series analysis as a tool for analyzing data, and still maintain a commitmenttotheoreticalintegrity,asexemplifiedbytheseminalworksofBrillinger [33]andHannan[86]andthetextsbyBrockwellandDavis[36]andFuller[66].The advent of inexpensive powerful computing has provided both real data and new software that can take one considerably beyond the fitting of simple time domain models, such as have been elegantly described in the landmark work of Box and Jenkins[30].Thisbookisdesignedtobeusefulasatextforcoursesintimeserieson severaldifferentlevelsandasareferenceworkforpractitionersfacingtheanalysisof time-correlateddatainthephysical,biological,andsocialsciences. We have used earlier versions of the text at both the undergraduateand gradu- ate levelsoverthepastdecade.Ourexperienceis thatan undergraduatecoursecan beaccessibletostudentswithabackgroundinregressionanalysisandmayinclude Sects. 1.1–1.5, Sects. 2.1–2.3, the results and numerical parts of Sects. 3.1–3.9, and briefly the results and numerical parts of Sects. 4.1–4.4. At the advanced un- dergraduateormaster’slevel,wherethestudentshavesomemathematicalstatistics background,moredetailedcoverageofthesamesections,withtheinclusionofextra topics from Chaps. 5 or 6, can be used as a one-semester course. Often, the extra topicsarechosenbythestudentsaccordingtotheirinterests.Finally,atwo-semester upper-levelgraduatecourseformathematics,statistics,andengineeringgraduatestu- dents can be crafted by addingselected theoreticalappendices.For the upper-level graduatecourse,weshouldmentionthatwearestrivingforabroaderbutlessrigorous levelofcoveragethanthatwhichisattainedbyBrockwellandDavis[36],theclassic entryatthislevel. Themajordifferencebetweenthisthirdeditionofthetextandthesecondeditionis thatweprovideRcodeforalmostallofthenumericalexamples.AnRpackagecalled astsaisprovidedforusewiththetext;seeSect.R.2fordetails.Rcodeisprovided simplytoenhancetheexpositionbymakingthenumericalexamplesreproducible. vii viii PrefacetotheThirdEdition We have tried, where possible, to keep the problem sets in order so that an instructormayhaveaneasytimemovingfromthesecondeditiontothethirdedition. However, some of the old problems have been revised and there are some new problems.Also,some ofthe data sets havebeenupdated.We addedonesectionin Chap.5onunitrootsandenhancedsomeofthepresentationsthroughoutthetext.The expositiononstate-spacemodeling,ARMAXmodels,and(multivariate)regression with autocorrelated errors in Chap. 6 have been expanded. In this edition, we use standard R functionsas much as possible, but we use our own scripts (includedin astsa) whenwefeelitisnecessarytoavoidproblemswithaparticularRfunction; theseproblemsarediscussedindetailonthewebsiteforthetextunderRIssues. We thank John Kimmel, Executive Editor, Springer Statistics, for his guidance inthepreparationandproductionofthiseditionofthetext.WearegratefultoDon Percival,UniversityofWashington,fornumeroussuggestionsthatledtosubstantial improvement to the presentation in the second edition, and consequently in this edition. We thank Doug Wiens, University of Alberta, for help with some of the R code in Chaps. 4 and 7, and for his many suggestions for improvement of the exposition. We are grateful for the continued help and advice of Pierre Duchesne, UniversityofMontreal,andAlexanderAue,UniversityofCalifornia,Davis.Wealso thankthemanystudentsandotherreaderswhotookthetimetomentiontypographical errors and other corrections to the first and second editions. Finally, work on this editionwassupportedbytheNationalScienceFoundationwhileoneofus(D.S.S.) wasworkingattheFoundationundertheIntergovernmentalPersonnelAct. Davis,CA,USA RobertH.Shumway Pittsburgh,PA,USA DavidS.Stoffer September2010 Contents PrefacetotheFourthEdition ........................................ v PrefacetotheThirdEdition ......................................... vii 1 CharacteristicsofTimeSeries ................................... 1 1.1 TheNatureofTimeSeriesData .............................. 2 1.2 TimeSeriesStatisticalModels................................ 8 1.3 MeasuresofDependence .................................... 15 1.4 StationaryTimeSeries ...................................... 19 1.5 EstimationofCorrelation.................................... 26 1.6 Vector-ValuedandMultidimensionalSeries..................... 33 Problems ...................................................... 38 2 TimeSeriesRegressionandExploratoryDataAnalysis............. 45 2.1 ClassicalRegressionintheTimeSeriesContext................. 45 2.2 ExploratoryDataAnalysis ................................... 54 2.3 SmoothingintheTimeSeriesContext ......................... 65 Problems ...................................................... 70 3 ARIMAModels................................................ 75 3.1 AutoregressiveMovingAverageModels ....................... 75 3.2 DifferenceEquations........................................ 88 3.3 AutocorrelationandPartialAutocorrelation..................... 94 3.4 Forecasting ............................................... 100 3.5 Estimation ................................................ 113 3.6 IntegratedModelsforNonstationaryData ...................... 131 3.7 BuildingARIMAModels ................................... 135 3.8 RegressionwithAutocorrelatedErrors ........................ 142 3.9 MultiplicativeSeasonalARIMAModels ....................... 145 Problems ...................................................... 154 ix x Contents 4 SpectralAnalysisandFiltering .................................. 165 4.1 CyclicalBehaviorandPeriodicity............................. 166 4.2 TheSpectralDensity........................................ 172 4.3 PeriodogramandDiscreteFourierTransform ................... 179 4.4 NonparametricSpectralEstimation............................ 189 4.5 ParametricSpectralEstimation ............................... 203 4.6 MultipleSeriesandCross-Spectra ............................ 206 4.7 LinearFilters .............................................. 211 4.8 LaggedRegressionModels .................................. 217 4.9 SignalExtractionandOptimumFiltering....................... 222 4.10 SpectralAnalysisofMultidimensionalSeries ................... 226 Problems ...................................................... 229 5 AdditionalTimeDomainTopics ................................. 241 5.1 LongMemoryARMAandFractionalDifferencing .............. 241 5.2 UnitRootTesting........................................... 250 5.3 GARCHModels ........................................... 253 5.4 ThresholdModels .......................................... 262 5.5 LaggedRegressionandTransferFunctionModeling ............. 266 5.6 MultivariateARMAXModels................................ 272 Problems ...................................................... 285 6 StateSpaceModels............................................. 289 6.1 LinearGaussianModel ..................................... 290 6.2 Filtering,Smoothing,andForecasting ......................... 294 6.3 MaximumLikelihoodEstimation ............................. 304 6.4 MissingDataModifications ................................. 313 6.5 StructuralModels:SignalExtractionandForecasting ............ 318 6.6 State-SpaceModelswithCorrelatedErrors ..................... 321 6.6.1 ARMAXModels .................................... 323 6.6.2 MultivariateRegressionwithAutocorrelatedErrors ....... 324 6.7 BootstrappingStateSpaceModels ............................ 328 6.8 SmoothingSplinesandtheKalmanSmoother................... 333 6.9 HiddenMarkovModelsandSwitchingAutoregression ........... 336 6.10 DynamicLinearModelswithSwitching ....................... 348 6.11 StochasticVolatility ........................................ 360 6.12 BayesianAnalysisofStateSpaceModels ...................... 367 Problems ...................................................... 378 7 StatisticalMethodsintheFrequencyDomain ..................... 385 7.1 Introduction ............................................... 385 7.2 SpectralMatricesandLikelihoodFunctions .................... 388 7.3 RegressionforJointlyStationarySeries ........................ 390 7.4 RegressionwithDeterministicInputs ......................... 399 7.5 RandomCoefficientRegression .............................. 407
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