Time Series Analysis. Nonstationary and Noninvertible Distribution Theory

886 Pages·2017·7.755 MB·English

by  Katsuto Tanaka

Time Series Analysis
NonstationaryandNoninvertibleDistributionTheory
KatsutoTanaka
ProfessorofStatisticsandEconometrics
GakushuinUniversity,Tokyo
SecondEdition
Thiseditionfirstpublished2017
©2017JohnWiley&Sons,Inc
LibraryofCongressCataloging-in-PublicationData
Names:Tanaka,Katsuto,1950-
Title:Timeseriesanalysis:nonstationaryandnoninvertibledistribution
theory/KatsutoTanaka,professorofstatisticsandeconometrics,
GakushuinUniversity,Tokyo.
Description:Secondedition.|Hoboken,NJ:JohnWiley&Sons,Inc.,[2017]
|Series:Wileyseriesinprobabilityandstatistics|Includes
bibliographicalreferencesandindexes.
Identifiers:LCCN2016052139(print)|LCCN2016052636(ebook)|ISBN
9781119132097(cloth)|ISBN9781119132110(pdf)|ISBN9781119132134
(epub)
Subjects:LCSH:Time-seriesanalysis.
Classification:LCCQA280.T352017(print)|LCCQA280(ebook)|DDC
519.5/5–dc23
LCrecordavailableathttps://lccn.loc.gov/2016052139
9781119132097
Setin10/12ptWarnockbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
(cid:2)
Contents
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
PartI AnalysisofNonFractionalTimeSeries 1
1 ModelsforNonstationarityandNoninvertibility 3
1.1 StatisticsfromtheOne-DimensionalRandomWalk 3
1.1.1 EigenvalueApproach 4
(cid:2) (cid:2)
1.1.2 StochasticProcessApproach 11
1.1.3 TheFredholmApproach 12
1.1.4 AnOverviewoftheThreeApproaches 14
1.2 ATestStatisticfromaNoninvertibleMovingAverageModel 16
1.3 TheARUnitRootDistribution 23
1.4 VariousStatisticsfromtheTwo-DimensionalRandomWalk 29
1.5 StatisticsfromtheCointegratedProcess 41
1.6 PanelUnitRootTests 47
2 BrownianMotionandFunctionalCentralLimit
Theorems 51
2.1 TheSpaceL ofStochasticProcesses 51
2
2.2 TheBrownianMotion 55
2.3 MeanSquareIntegration 58
2.3.1 TheMeanSquareRiemannIntegral 59
2.3.2 TheMeanSquareRiemann–StieltjesIntegral 62
2.3.3 TheMeanSquareItoIntegral 66
2.4 TheItoCalculus 72
2.5 WeakConvergenceofStochasticProcesses 77
2.6 TheFunctionalCentralLimitTheorem 81
2.7 FCLTforLinearProcesses 87
2.8 FCLTforMartingaleDifferences 91
2.9 WeakConvergencetotheIntegratedBrownianMotion 99
(cid:2)
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vi Contents
2.10 WeakConvergencetotheOrnstein–UhlenbeckProcess 103
2.11 WeakConvergenceofVector-ValuedStochasticProcesses 109
2.11.1 SpaceCq 109
2.11.2 BasicFCLTforVectorProcesses 110
2.11.3 FCLTforMartingaleDifferences 112
2.11.4 FCLTfortheVector-ValuedIntegratedBrownianMotion 115
2.12 WeakConvergencetotheItoIntegral 118
3 TheStochasticProcessApproach 127
3.1 Girsanov’sTheorem:O-UProcesses 127
3.2 Girsanov’sTheorem:IntegratedBrownianMotion 137
3.3 Girsanov’sTheorem:Vector-ValuedBrownianMotion 142
3.4 TheCameron–MartinFormula 145
3.5 AdvantagesandDisadvantagesofthePresentApproach 147
4 TheFredholmApproach 149
4.1 MotivatingExamples 149
4.2 TheFredholmTheory:TheHomogeneousCase 155
4.3 Thec.f.oftheQuadraticBrownianFunctional 161
4.4 VariousFredholmDeterminants 171
4.5 TheFredholmTheory:TheNonhomogeneousCase 190
(cid:2) 4.5.1 ComputationoftheResolvent–Case1 192 (cid:2)
4.5.2 ComputationoftheResolvent–Case2 199
4.6 WeakConvergenceofQuadraticForms 203
5 NumericalIntegration 213
5.1 Introduction 213
5.2 NumericalIntegration:TheNonnegativeCase 214
5.3 NumericalIntegration:TheOscillatingCase 220
5.4 NumericalIntegration:TheGeneralCase 228
5.5 ComputationofPercentPoints 236
5.6 TheSaddlepointApproximation 240
6 EstimationProblemsinNonstationaryAutoregressive
Models 245
6.1 NonstationaryAutoregressiveModels 245
6.2 ConvergenceinDistributionofLSEs 250
6.2.1 ModelA 251
6.2.2 ModelB 253
6.2.3 ModelC 255
6.2.4 ModelD 257
6.3 Thec.f.sfortheLimitingDistributionsofLSEs 260
6.3.1 TheFixedInitialValueCase 261
6.3.2 TheStationaryCase 265
(cid:2)
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Contents vii
6.4 TablesandFiguresofLimitingDistributions 267
6.5 ApproximationstotheDistributionsoftheLSEs 276
6.6 NearlyNonstationarySeasonalARModels 281
6.7 ContinuousRecordAsymptotics 289
6.8 ComplexRootsontheUnitCircle 292
6.9 AutoregressiveModelswithMultipleUnitRoots 300
7 EstimationProblemsinNoninvertibleMovingAverage
Models 311
7.1 NoninvertibleMovingAverageModels 311
7.2 TheLocalMLEintheStationaryCase 314
7.3 TheLocalMLEintheConditionalCase 325
7.4 NoninvertibleSeasonalModels 330
7.4.1 TheStationaryCase 331
7.4.2 TheConditionalCase 333
7.4.3 ContinuousRecordAsymptotics 335
7.5 ThePseudolocalMLE 337
7.5.1 TheStationaryCase 337
7.5.2 TheConditionalCase 339
7.6 ProbabilityoftheLocalMLEatUnity 341
7.7 TheRelationshipwiththeStateSpaceModel 343
(cid:2) (cid:2)
8 UnitRootTestsinAutoregressiveModels 349
8.1 Introduction 349
8.2 OptimalTests 350
8.2.1 TheLBITest 352
8.2.2 TheLBIUTest 353
8.3 EquivalenceoftheLMTestwiththeLBIorLBIUTest 356
8.3.1 EquivalencewiththeLBITest 356
8.3.2 EquivalencewiththeLBIUTest 358
8.4 VariousUnitRootTests 360
8.5 IntegralExpressionsfortheLimitingPowers 362
8.5.1 ModelA 363
8.5.2 ModelB 364
8.5.3 ModelC 365
8.5.4 ModelD 367
8.6 LimitingPowerEnvelopesandPointOptimalTests 369
8.7 ComputationoftheLimitingPowers 372
8.8 SeasonalUnitRootTests 382
8.9 UnitRootTestsintheDependentCase 389
8.10 TheUnitRootTestingProblemRevisited 395
8.11 UnitRootTestswithStructuralBreaks 398
8.12 StochasticTrendsVersusDeterministicTrends 402
8.12.1 CaseofIntegratedProcesses 403
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viii Contents
8.12.2 CaseofNear-IntegratedProcesses 406
8.12.3 SomeSimulations 409
9 UnitRootTestsinMovingAverageModels 415
9.1 Introduction 415
9.2 TheLBIandLBIUTests 416
9.2.1 TheConditionalCase 417
9.2.2 TheStationaryCase 419
9.3 TheRelationshipwiththeTestStatisticsinDifferencedForm 424
9.4 PerformanceoftheLBIandLBIUTests 427
9.4.1 TheConditionalCase 427
9.4.2 TheStationaryCase 430
9.5 SeasonalUnitRootTests 434
9.5.1 TheConditionalCase 434
9.5.2 TheStationaryCase 436
9.5.3 PowerProperties 438
9.6 UnitRootTestsintheDependentCase 444
9.6.1 TheConditionalCase 444
9.6.2 TheStationaryCase 446
9.7 TheRelationshipwithTestingintheStateSpaceModel 447
9.7.1 Case(I) 449
(cid:2) 9.7.2 Case(II) 450 (cid:2)
9.7.3 Case(III) 452
9.7.4 TheCaseoftheInitialValueKnown 454
10 AsymptoticPropertiesofNonstationaryPanelUnitRoot
Tests 459
10.1 Introduction 459
10.2 PanelAutoregressiveModels 461
10.2.1 TestsBasedontheOLSE 463
10.2.2 TestsBasedontheGLSE 471
10.2.3 SomeOtherTests 475
10.2.4 LimitingPowerEnvelopes 480
10.2.5 GraphicalComparison 485
10.3 PanelMovingAverageModels 488
10.3.1 ConditionalCase 490
10.3.2 StationaryCase 494
10.3.3 PowerEnvelope 499
10.3.4 GraphicalComparison 502
10.4 PanelStationarityTests 507
10.4.1 LimitingLocalPowers 508
10.4.2 PowerEnvelope 512
10.4.3 GraphicalComparison 514
10.5 ConcludingRemarks 515
(cid:2)
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Contents ix
11 StatisticalAnalysisofCointegration 517
11.1 Introduction 517
11.2 CaseofNoCointegration 519
11.3 CointegrationDistributions:TheIndependentCase 524
11.4 CointegrationDistributions:TheDependentCase 532
11.5 TheSamplingBehaviorofCointegrationDistributions 537
11.6 TestingforCointegration 544
11.6.1 TestsfortheNullofNoCointegration 544
11.6.2 TestsfortheNullofCointegration 547
11.7 DeterminationoftheCointegrationRank 552
11.8 HigherOrderCointegration 556
11.8.1 CointegrationintheI(d)Case 556
11.8.2 SeasonalCointegration 559
PartII AnalysisofFractionalTimeSeries 567
12 ARFIMAModelsandtheFractionalBrownianMotion 569
12.1 NonstationaryFractionalTimeSeries 569
12.1.1 Caseofd= 1 570
(cid:2) (cid:2)
2
12.1.2 Caseofd> 1 572
2
12.2 TestingfortheFractionalIntegrationOrder 575
12.2.1 i.i.d.Case 575
12.2.2 DependentCase 581
12.3 EstimationfortheFractionalIntegrationOrder 584
12.3.1 i.i.d.Case 584
12.3.2 DependentCase 586
12.4 StationaryLong-MemoryProcesses 591
12.5 TheFractionalBrownianMotion 597
12.6 FCLTforLong-MemoryProcesses 603
12.7 FractionalCointegration 608
12.7.1 SpuriousRegressionintheFractionalCase 609
12.7.2 CointegratingRegressionintheFractionalCase 610
12.7.3 TestingforFractionalCointegration 614
12.8 TheWaveletMethodforARFIMAModelsandthefBm 614
12.8.1 BasicTheoryoftheWaveletTransform 615
12.8.2 SomeAdvantagesoftheWaveletTransform 618
12.8.3 SomeApplicationsoftheWaveletAnalysis 625
12.8.3.1 TestingfordinARFIMAModels 625
12.8.3.2 TestingfortheExistenceofNoise 626
12.8.3.3 TestingforFractionalCointegration 627
12.8.3.4 UnitRootTests 627
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13 StatisticalInferenceAssociatedwiththeFractionalBrownian
Motion 629
13.1 Introduction 629
13.2 ASimpleContinuous-TimeModelDrivenbythefBm 632
13.3 QuadraticFunctionalsoftheBrownianMotion 641
13.4 Derivationofthec.f. 645
13.4.1 StochasticProcessApproachviaGirsanov’sTheorem 645
13.4.1.1 CaseofH =1∕2 645
13.4.1.2 CaseofH >1∕2 646
13.4.2 FredholmApproachviatheFredholmDeterminant 647
13.4.2.1 CaseofH =1∕2 649
13.4.2.2 CaseofH >1∕2 650
13.5 MartingaleApproximationtothefBm 651
13.6 TheFractionalUnitRootDistribution 659
13.6.1 TheFDAssociatedwiththeApproximateDistribution 659
13.6.2 AnInterestingMomentProperty 664
13.7 TheUnitRootTestUnderthefBmError 669
14 MaximumLikelihoodEstimationfortheFractional
Ornstein–UhlenbeckProcess 673
14.1 Introduction 673
(cid:2) 14.2 EstimationoftheDrift:ErgodicCase 677 (cid:2)
14.2.1 AsymptoticPropertiesoftheOLSEs 677
14.2.2 TheMLEandMCE 679
14.3 EstimationoftheDrift:Non-ergodicCase 687
14.3.1 AsymptoticPropertiesoftheOLSE 687
14.3.2 TheMLE 687
14.4 EstimationoftheDrift:BoundaryCase 692
14.4.1 AsymptoticPropertiesoftheOLSEs 692
14.4.2 TheMLEandMCE 693
14.5 ComputationofDistributionsandMomentsoftheMLEand
MCE 695
14.6 TheMLE-basedUnitRootTestUnderthefBmError 703
14.7 ConcludingRemarks 707
15 SolutionstoProblems 709
References 865
AuthorIndex 879
SubjectIndex 883
(cid:2)
(cid:2)
xi
PrefacetotheSecondEdition
Thefirsteditionofthisbookwaspublishedin1996.Thebookwaswrittenfrom
atheoreticalviewpointoftimeserieseconometrics,wherethemainthemewas
todescribenonstandardtheoryforlineartimeseriesmodelsthatarenonsta-
tionaryand/ornoninvertible.Ialsoproposedmethodsforcomputingnumeri-
callythedistributionsofnonstandardstatisticsarisingfromsuchprocesses.
The main theme of the present edition remains the same and reflects
the developments and new directions in the field since the publication of
the first edition. In particular, the discussion on nonstationary panel data
analysishasbeenaddedandnewchaptersonlong-memorydiscrete-timeand
(cid:2) continuous-time processes have been created, whereas some chapters have (cid:2)
beenmergedandsomesectionsdeleted.
Thiseditionisdividedintotwoparts:PartI:AnalysisofNonFractionalTime
SeriesandPartII:AnalysisofFractionalTimeSeries,wherePartIconsistsof
Chapters 1 through 11 while Part II consists of Chapters 12 through 14. The
distinctionbetweennonfractionalandfractionaltimeseriesisconcernedwith
theintegrationorderofnonstationarytimeseries.PartIassumestheintegra-
tionordertobeapositiveinteger,whereasPartIIrelaxesthatassumptionto
allowtheintegrationordertobeanypositiverealnumber.
Chapter1isessentiallythesameasthefirstedition,exceptfortheadditionof
anintroductorydescriptiononnonstationarypanels,andisapreludetosub-
sequentchapters.Thethreeapproaches,whichIcalltheeigenvalue,stochastic
process,andFredholmapproaches,totheanalysisofnonfractionaltimeseries
are introduced through simple examples. Chapter 2 merged Chapters 2 and
3ofthefirsteditionanddiscussestheBrownianmotion,theItointegral,the
functionalcentrallimittheorem,andsoon.
Chapters 3 and 4 discuss fully the stochastic process approach and the
Fredholmapproach,respectively.Theseapproachesareusedtoderivelimiting
characteristicfunctionsofnonstandardstatisticsthatarequadraticfunctionals
of the Brownian motion or its ratio. Chapter 5 is concerned with numerical
integrationforcomputingdistributionfunctionsviainversionofcharacteristic
functions derived from the stochastic process approach or the Fredholm
approach. Chapters 6 through 11 deal with unit root and cointegration
(cid:2)
(cid:2)
xii PrefacetotheSecondEdition
problems. Chapters 1 through 11 except Chapter 10 were main chapters of
the first edition. New topics such as unit root tests under structural breaks,
differences between stochastic and deterministic trends, have been added.
Chapter10hasbeenaddedtodiscussnonstationarypaneldatamodels,where
our main concern is to compute limiting local powers of various panel unit
roottests.Forthatpurposethemovingaveragemodelsarealsoconsideredin
additiontoautoregressivemodels.
Chapters12through14havebeenwrittennewlyunderPartII:Analysisof
FractionalTimeSeries.Chapter12discussesthebasictheoryoflong-memory
processesbyintroducingARFIMAmodelsandthefractionalBrownianmotion
(fBm). The wavelet method is also introduced to deal with ARFIMA models
and the fBm. Chapter 13 is concerned with the computation of distributions
ofquadraticfunctionalsofthefBmanditsratio,wherethecomputationofthe
fractional unit root distribution remains to be done, whereas an approxima-
tiontothetruedistributionisproposedandcomputed.Chapter14introduces
thefractionalOrnstein–Uhlenbeckprocess,onwhichthestatisticalinferenceis
discussed.Inparticular,themaximumlikelihoodestimatorofthedriftparame-
terisconsidered,andasymptoticsasthesamplingspanincreasesarediscussed.
Chapter 15, the last chapter, gives a complete set of solutions to problems
posedattheendofmostsections.
There are about 140 figures and 60 tables. Most of these are of limiting
(cid:2) distributionsofnonstandardstatistics.Theyareallproducedbythemethods (cid:2)
described in this edition and include many distributions, which have never
appearedintheliterature.
Thepresenteditionisdedicatedtomywife,Yoshiko,whodiedin1999.
November2016 KatsutoTanaka
Tokyo,Japan
(cid:2)