Time Series Analysis and Its Applications: With R Examples

567 Pages·2017·16.84 MB·English

by  Robert H. Shumway

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
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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