Mathematical Foundations of Time Series Analysis

309 Pages·2017·2.53 MB·English

by  Jan Beran

Jan Beran
Mathematical
Foundations
of Time Series
Analysis
A Concise Introduction
Mathematical Foundations of Time Series Analysis
Jan Beran
Mathematical Foundations
of Time Series Analysis
A Concise Introduction
123
JanBeran
DepartmentofMathematicsandStatistics
UniversityofKonstanz
Konstanz,Germany
ISBN978-3-319-74378-3 ISBN978-3-319-74380-6 (eBook)
https://doi.org/10.1007/978-3-319-74380-6
LibraryofCongressControlNumber:2018930982
MathematicsSubjectClassification(2010):62Mxx,62M10
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Preface
The historical development of time series analysis can be traced back to many
applied sciences, including economics, meteorology, physics or communications
engineering.Theoreticaldevelopmentsofthesubjectarecloselylinkedtoprogress
in the mathematical theory of stochastic processes and mathematical statistics.
Thereareanumberofexcellentbooksontimeseriesanalysis,includingGrenander
andRosenblatt(1957),BoxandJenkins(1970),Hannan(1970),Anderson(1971),
Koopmans (1974), Fuller (1976), Priestley (1981), Brockwell and Davis (1991),
Hamilton (1994), Diggle (1996), Brillinger (2001), Chatfield (2003), Lütkepohl
(2006),DurbinandKoopmann(2012),Woodwardetal.(2016),andShumwayand
Stoffer(2017).
Timeseriesanalysisisnowawell-establishedscientificdisciplinewithrigorous
mathematicalfoundations.On the other hand,it is a verybroad subjectarea, and,
due to the diverse sciences that contributed to its development, the time series
vocabulary is permeated with terminology reflecting the diversity of applications
(cf.Priestley1981,Prefacep.vii).Thisbookisanattempttosummarizesomeofthe
mainprinciplesoftimeseriesanalysis,withthehopethattheconcisepresentationis
helpfulforteachingstudentswithamathematicalbackground.Thebookgrewout
of lectures taught to students of mathematics, mathematical finance, physics and
economicsattheUniversityofKonstanz.
I would like to thank Martin Schützner, Mark Heiler, Dieter Schell, Evgeni
Shumm, Nadja Schumm, Arno Weiershäuser, Dirk Ocker, Karim Djaidja, Haiyan
Liu,BrittaSteffens,KlausTelkmann,YuanhuaFeng,PhilippSibbertsen,Bikramjit
Das,RafalKulik,LiudasGiraitis,SucharitaGhoshandothercolleaguesforfruitful
collaboration;andto Volker Bürkelforreadingpartsof a preliminarymanuscript.
Thanks go also to the University of Konstanz for granting me a sabbatical with
the purpose of working on this book. Most importantly,I would like to thank my
family,Céline,SucharitaandSirHastings—ourCotondeTuléar—forkeepingme
motivated.
Konstanz,Germany JanBeran
November2017
v
Contents
1 Introduction................................................................. 1
1.1 WhatIsaTimeSeries?............................................... 1
1.2 TimeSeriesVersusiidData.......................................... 2
2 TypicalAssumptions ....................................................... 5
2.1 FundamentalProperties.............................................. 5
2.1.1 ErgodicPropertywithaConstantLimit.................... 5
2.1.2 StrictStationarity ............................................ 7
2.1.3 WeakStationarity............................................ 8
2.1.4 WeakStationarityandHilbertSpaces....................... 11
2.1.5 ErgodicProcesses............................................ 32
2.1.6 SufficientConditionsforthea.s.ErgodicProperty
withaConstantLimit........................................ 34
2.1.7 SufficientConditionsfortheL2-ErgodicProperty
withaConstantLimit........................................ 35
2.2 SpecificAssumptions ................................................ 39
2.2.1 GaussianProcesses .......................................... 39
2.2.2 LinearProcessesinL2.˝/................................... 40
2.2.3 LinearProcesseswithE.X2/D1.......................... 44
t
2.2.4 MultivariateLinearProcesses............................... 48
2.2.5 Invertibility................................................... 49
2.2.6 RestrictionsontheDependenceStructure.................. 63
3 DefiningProbabilityMeasuresforTimeSeries ......................... 69
3.1 FiniteDimensionalDistributions.................................... 69
3.2 TransformationsandEquations...................................... 70
3.3 ConditionsontheExpectedValue................................... 71
3.4 ConditionsontheAutocovarianceFunction........................ 73
3.4.1 PositiveSemidefiniteFunctions............................. 73
3.4.2 SpectralDistribution......................................... 77
3.4.3 CalculationandPropertiesofFandf....................... 86
vii
viii Contents
4 SpectralRepresentationofUnivariateTimeSeries..................... 101
4.1 Motivation............................................................ 101
4.2 HarmonicProcesses.................................................. 102
4.3 ExtensiontoGeneralProcesses...................................... 105
4.3.1 StochasticIntegralswithRespecttoZ...................... 105
4.3.2 ExistenceandDefinitionofZ ............................... 112
4.3.3 InterpretationoftheSpectralRepresentation............... 122
4.4 FurtherProperties .................................................... 122
4.4.1 RelationshipBetweenRe ZandIm Z...................... 122
4.4.2 Frequency .................................................... 123
4.4.3 Overtones..................................................... 124
4.4.4 WhyAreFrequenciesRestrictedtotheRangeŒ(cid:2)(cid:2);(cid:2)(cid:3)?... 125
4.5 LinearFiltersandtheSpectralRepresentation...................... 129
4.5.1 EffectontheSpectralRepresentation....................... 129
4.5.2 EliminationofFrequencyBands............................ 134
5 SpectralRepresentationofRealValuedVectorTimeSeries........... 137
5.1 Cross-SpectrumandSpectralRepresentation....................... 137
5.2 CoherenceandPhase................................................. 146
6 UnivariateARMAProcesses .............................................. 161
6.1 Definition............................................................. 161
6.2 StationarySolution................................................... 161
6.3 CausalStationarySolution........................................... 166
6.4 CausalInvertibleStationarySolution ............................... 169
6.5 AutocovariancesofARMAProcesses .............................. 170
6.5.1 CalculationbyIntegration................................... 170
6.5.2 CalculationUsingtheAutocovarianceGenerating
Function...................................................... 170
6.5.3 CalculationUsingtheWoldRepresentation................ 175
6.5.4 RecursiveCalculation........................................ 176
6.5.5 AsymptoticDecay ........................................... 177
6.6 Integrated,SeasonalandFractionalARMAandARIMA
Processes.............................................................. 185
6.6.1 IntegratedProcesses ......................................... 185
6.6.2 SeasonalARMAProcesses.................................. 186
6.6.3 FractionalARIMAProcesses ............................... 187
6.7 UnitRoots,SpuriousCorrelation,Cointegration................... 200
7 GeneralizedAutoregressiveProcesses.................................... 203
7.1 DefinitionofGeneralizedAutoregressiveProcesses ............... 203
7.2 StationarySolutionofGeneralizedAutoregressiveEquations..... 204
7.3 DefinitionofVARMAProcesses.................................... 209
7.4 StationarySolutionofVARMAEquations ......................... 211
7.5 DefinitionofGARCHProcesses .................................... 213
7.6 StationarySolutionofGARCHEquations.......................... 214
Contents ix
7.7 DefinitionofARCH(1)Processes.................................. 219
7.8 StationarySolutionofARCH(1)Equations....................... 220
8 Prediction.................................................................... 223
8.1 BestLinearPredictionGivenanInfinitePast....................... 223
8.2 Predictability ......................................................... 225
8.3 ConstructionoftheWoldDecompositionfromf................... 230
8.4 BestLinearPredictionGivenaFinitePast.......................... 235
9 Inferencefor(cid:2),(cid:3) andF................................................... 241
9.1 LocationEstimation.................................................. 241
9.2 LinearRegression.................................................... 244
9.3 NonparametricEstimationof(cid:4)...................................... 253
9.4 NonparametricEstimationoff ...................................... 262
10 ParametricEstimation..................................................... 281
10.1 GaussianandQuasiMaximumLikelihoodEstimation............. 281
10.2 WhittleApproximation .............................................. 284
10.3 AutoregressiveApproximation...................................... 287
10.4 ModelChoice......................................................... 289
References......................................................................... 293
AuthorIndex...................................................................... 299
SubjectIndex..................................................................... 303
Chapter 1
Introduction
1.1 WhatIs a TimeSeries?
Definition1.1 Letk 2N,T (cid:3)R.Afunction
xWT !Rk,t!x
t
or,equivalently,asetofindexedelementsofRk,
˚ (cid:2)
xjx 2Rk;t2T
t t
iscalledanobservedtimeseries.Wealsowrite
x (t2T)or .x/ :
t t t2T
Definition1.2 Letk 2N,T (cid:3)R,
(cid:3) (cid:4)
˝ D Rk T DspaceoffunctionsX WT !Rk;
F D(cid:5)-algebraon˝;
PDprobabilitymeasureon .˝;F/:
The probability space .˝;F;P/, or equivalently the set of indexed random
variables
˚ (cid:2)
XjX 2Rk;t2T , .X/ (cid:4)P
t t t t2T
©SpringerInternationalPublishingAG,partofSpringerNature2017 1
J.Beran,MathematicalFoundationsofTimeSeriesAnalysis,
https://doi.org/10.1007/978-3-319-74380-6_1
2 1 Introduction
Table1.1 TypesoftimeseriesX 2Rk(t2T)
t
Property Terminology
kD1 Univariatetimeseries
k(cid:2)2 Multivariatetimeseries
Tcountable,8a<b2RW T\Œa;b(cid:3)finite Discretetime
Tdiscrete,9u2RCs.t.tjC1(cid:3)tjDu Equidistanttime
T DŒa;b(cid:3)(a<b2R),TDRCorTDR Continuoustime
iscalledatimeseries,ortimeseriesmodel.Insteadof.˝;F;P/wealsowrite
X (t2T)or .X/ :
t t t2T
Moreover,foraspecificrealization! 2˝,wewriteX.!/and
t
.x/ D.X .!// Dsamplepathof .X/ ;
t t2T t t2T t t2T
.x / D.X .!// DfinitesamplepathofX:
ti iD1;:::;n ti iD1;:::;n t
Remark1.1 ˝ maybemoregeneralthaninDefinition1.2.Similarly,theindexset
T maybemoregeneralthanasubsetofR,butitmustbeorderedandmetric.Thus,
.X/ isastochasticprocesswithanorderedmetricindexsetT:
t t2T
Remark1.2 AnoverviewofthemostcommontypesoftimeseriesX 2Rk (t2T,
t
T ¤;)isgiveninTable1.1.
Remark1.3 IfX Dequidistanttimeseries,thenwemaysetw.l.o.g.T (cid:3)Z.
t
1.2 TimeSeries Versus iidData
What distinguishes statistical analysis of iid data from time series analysis? We
illustratethequestionbyconsideringthecaseofequidistantunivariaterealvalued
timeseriesX 2R(t2Z).
t
Problem1.1 IsconsistentestimationofPpossible?
Solution1.1 The answer depends on available a priori information and assump-
tionsoneiswillingtomake.Thisisillustratedinthefollowing.