Unit Root Tests in Time Series: Extensions and Developments

586 Pages·2012·3.727 MB·English

by  Kerry Patterson (auth.)

Unit Root Tests in Time Series Volume 2
PalgraveTextsinEconometrics
GeneralEditor:KerryPatterson
Titlesinclude:
SimonP.BurkeandJohnHunter
MODELLINGNON-STATIONARYTIMESERIES
MichaelP.Clements
EVALUATINGECONOMETRICFORECASTSOFECONOMICANDFINANCIALVARIABLES
LesleyGodfrey
BOOTSTRAPTESTSFORREGRESSIONMODELS
TerenceC.Mills
MODELLINGTRENDSANDCYCLESINECONOMICTIMESERIES
KerryPatterson
APRIMERFORUNITROOTTESTING
KerryPatterson
UNITROOTTESTSINTIMESERIESVOLUME1
KeyConceptsandProblems
KerryPatterson
UNITROOTTESTSINTIMESERIESVOLUME2
ExtensionsandDevelopments
PalgraveTextsinEconometrics
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Unit Root Tests in Time Series
Volume 2
Extensions and Developments
Kerry Patterson
©KerryPatterson2012
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Contents
DetailedContents vi
ListofTables xx
ListofFigures xxiii
SymbolsandAbbreviations xxvii
Preface xxix
1 SomeCommonThemes 1
2 FunctionalFormandNonparametricTestsforaUnitRoot 28
3 FractionalIntegration 76
4 Semi-ParametricEstimationoftheLong-MemoryParameter 154
5 SmoothTransitionNonlinearModels 240
6 ThresholdAutoregressions 325
7 StructuralBreaksinARModels 381
8 StructuralBreakswithUnknownBreakDates 436
9 ConditionalHeteroscedasticityandUnitRootTests 497
References 528
AuthorIndex 542
SubjectIndex 546
v
Detailed Contents
ListofTables xx
ListofFigures xxiii
SymbolsandAbbreviations xxvii
Preface xxix
1 SomeCommonThemes 1
Introduction 1
1.1 Stochasticprocesses 1
1.2 Stationarityandsomeofitsimplications 3
1.2.1 Astrictlystationaryprocess 4
1.2.2 Stationarityuptoorderm 4
1.3 ARMA(p,q)models 6
1.4 Thelong-runvariance 7
1.5 Theproblemofanuisanceparameteronlyidentifiedunder
thealternativehypothesis 9
1.5.1 Structuralstability 9
1.5.2 Self-excitingthresholdautoregression(SETAR) 14
1.5.3 Bootstraptheteststatistics 16
1.5.4 Illustration:Hog–cornprice 18
1.5.4.i Chowtestfortemporalstructuralstability 19
1.5.4.ii Thresholdautoregression 20
1.6 Thewayahead 21
Questions 22
2 FunctionalFormandNonparametricTestsforaUnitRoot 28
Introduction 28
2.1 Functionalform:linearversuslogs 30
2.1.1 Arandomwalkandanexponentialrandomwalk 30
2.1.2 Simulationresults 32
2.2 Aparametrictesttodiscriminatebetweenlevelsandlogs 34
2.2.1 Linearandlog-linearintegratedprocesses 34
2.2.2 Teststatistics 35
2.2.2.i TheKMteststatistics 35
2.2.2.ii Criticalvalues 37
2.2.2.iii Motivation 39
vi
Contents vii
2.2.2.iv Powerconsiderations 40
2.2.3 Illustrations 40
2.2.3.i Ratioofgoldtosilverprices 41
2.2.3.ii Worldoilproduction 41
2.3 Monotonictransformationsandunitroottestsbased
onranks 43
2.3.1 DFrank-basedtest 45
2.3.2 Rank-scoretest,BreitungandGouriéroux(1997) 47
2.3.2.i Seriallycorrelatederrors 49
2.3.2.ii Simulationresults 51
2.4 Therangeunitroottest 51
2.4.1 Therangeandnew‘records’ 52
2.4.1.i Theforwardrangeunitroottest 53
2.4.1.ii RobustnessofRF 54
UR
2.4.2 Theforward–backwardrangeunitroottest 55
2.4.3 RobustnessofRF andRFB 56
UR UR
2.4.4 Therangeunitroottestsfortrendingalternatives 59
2.5 Varianceratiotests 60
2.5.1 Abasicvarianceratiotest 61
2.5.2 Breitungvarianceratiotest 64
2.6 Comparisonandillustrationsofthetests 66
2.6.1 Comparisonofsizeandpower 67
2.6.2 Linearorexponentialrandomwalks? 69
2.6.3 Empiricalillustrations 70
2.6.3.i Ratioofgold–silverprice(revisited) 70
2.6.3.ii Airrevenuepassengermiles(US) 71
2.7 Concludingremarks 73
Questions 74
3 FractionalIntegration 76
Introduction 76
3.1 Afractionallyintegratedprocess 78
3.1.1 Aunitrootprocesswithfractionallyintegratednoise 78
3.1.2 Binomialexpansionof(1−L)d 79
3.1.2.i ARcoefficients 79
3.1.2.ii MAcoefficients 80
3.1.2.iii Thefractionallyintegratedmodelinterms
oftheGammafunction 80
3.1.3 TwodefinitionsofanI(d)process,dfractional 82
3.1.3.i Partialsummation 82
3.1.3.ii Directdefinition 83
3.1.4 ThedifferencebetweentypeIandtypeIIprocesses 84
viii Contents
3.1.5 TypeIandtypeIIfBM 84
3.1.6 Deterministiccomponentsandstartingvalue(s) 86
3.1.6.i Deterministiccomponents 86
3.1.6.ii Startingvalue(s) 86
3.2 TheARFIMA(p,d,q)model 87
3.2.1 Autocovariancesandautocorrelationsofthe
ARFIMA(0,d,0)process,d≤0.5 88
3.2.1.i Autocovariances 88
3.2.1.ii Autocorrelations 88
3.2.1.iii Inverseautocorrelations 89
3.2.2 GraphicalpropertiesofsomesimpleARFIMA
models 89
3.3 Whatkindofmodelsgeneratefractionald? 93
3.3.1 Theerrordurationmodel(Parke,1999) 93
3.3.1.i Themodel 93
3.3.1.ii Motivation:thesurvivalrateoffirms 94
3.3.1.iii Autocovariancesandsurvivalprobabilities 94
3.3.1.iv Thesurvivalprobabilitiesinalong-memory
process 96
3.3.1.v Thesurvivalprobabilitiesinashort-memory,
AR(1),process 97
3.3.2 Anexample:thesurvivalrateforUSfirms 97
3.3.3 Errordurationandmicropulses 99
3.3.4 Aggregation 99
3.3.4.i Aggregationof‘micro’relationships 100
3.3.4.ii TheAR(1)coefficientsare‘draws’fromthe
betadistribution 101
3.3.4.iii Qualifications 101
3.4 Dickey-Fullertestswhentheprocessisfractionally
integrated 102
3.5 AfractionalDickey-Fuller(FDF)testforunitroots 105
3.5.1 FDFtestforfractionalintegration 106
3.5.2 AfeasibleFDFtest 107
3.5.3 Limitingnulldistributions 107
3.5.4 Seriallycorrelatederrors:anaugmentedFDF,AFDF 108
3.5.5 AnefficientFDFtest 110
3.5.6 EFDF:limitingnulldistribution 112
3.5.7 Seriallycorrelatederrors:anaugmentedEFDF,AEFDF 113
3.5.8 Limitingnulldistributionoftη(dˆT) 114
3.6 FDFandEFDFtests:deterministiccomponents 114
3.7 Locallybestinvariant(LBI)tests 118
3.7.1 AnLM-typetest 118
Contents ix
3.7.1.i Noshort-rundynamics 118
3.7.1.ii Deterministiccomponents 121
3.7.1.iii Short-rundynamics 122
3.7.2 LMtest:aregressionapproach 124
3.7.2.i Noshort-rundynamics 124
3.7.2.ii Short-rundynamics 125
3.7.2.iii Deterministicterms 126
3.7.3 AWaldtestbasedonMLestimation 126
3.8 Power 127
3.8.1 Poweragainstfixedalternatives 127
3.8.2 Poweragainstlocalalternatives 128
3.8.3 Theoptimal(power-maximising)choiceofd
1
intheFDFtest(s) 129
3.8.4 Illustrativesimulationresults 131
3.9 Example:USwheatproduction 135
3.10 Concludingremarks 141
Questions 142
Appendix3.1Factorialexpansionsforintegerandnon-integerd 149
A3.1Whatisthemeaningof(1−L)d forfractionald? 149
A3.2Applyingthebinomialexpansiontothefractionaldifference
operator 150
A3.3TheMAcoefficientsintermsofthegammafunction 151
Appendix3.2FDFtest:assumingknownd 152
1
4 Semi-ParametricEstimationoftheLong-memoryParameter 154
Introduction 154
4.1 Linearfilters 156
4.1.1 Ageneralresult 156
4.1.2 ThespectraldensityofanARMAprocess 157
4.1.3 Examplesofspectraldensities 158
4.1.4 Thedifferencefilter 159
4.1.5 Processeswithfractionallong-memoryparameter 160
4.1.5.i ARFIMAprocesses 160
4.1.5.ii Fractionalnoise(FN)processes 161
4.2 Estimatingd 162
4.2.1 Priortransformationofthedata 162
4.2.2 ThediscreteFouriertransform,theperiodogramand
thelog-periodogram‘regression’,poolingandtapering 163
4.2.2.i Reminders 163
4.2.2.ii Deterministiccomponents 163
4.3 Estimationmethods 164
4.3.1 Alog-periodogramestimator(GPH) 164