The Foundations of Modern Time Series Analysis

476 Pages·2011·2.948 MB·English

by  Terence C. Mills (auth.)

The Foundations of Modern Time Series Analysis
PalgraveAdvancedTextsinEconometricsseries.
SeriesEditors:
TerenceC.Mills,UniversityofLoughborough,UK
KerryPatterson,UniversityofReading,UK
Editorialboard:
InChoi,SogangUniversity,SouthKorea
WilliamGreene,LeonardN.SternSchoolofBusiness,USA
NielsHaldrup,UniversityofAarhus,Denmark
TommassoProietti,UniversityofRome,ItalyandUniversityofSydney,Australia
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The Foundations of Modern
Time Series Analysis
Terence C. Mills
ProfessorofAppliedStatisticsandEconometrics,DepartmentofEconomics,
LoughboroughUniversity,UK
Palgrave
macmillan
©TerenceC.Mills2011
Softcover reprint of the hardcover 1st edition 2011 978-0-230-29018-1
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Firstpublished2011by
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10 9 8 7 6 5 4 3 2 1
20 19 18 17 16 15 14 13 12 11
Contents
ListofTables vi
ListofFigures ix
1 Prolegomenon:APersonalPerspectiveandanExplanation
oftheStructureoftheBook 1
2 YuleandHookerandtheConceptsofCorrelationandTrend 6
3 Schuster,BeveridgeandPeriodogramAnalysis 18
4 DetrendingandtheVariateDifferencingMethod:Student,
PearsonandTheirCritics 30
5 NonsenseCorrelations,RandomShocksandInducedCycles:
Yule,SlutzkyandWorking 64
6 PeriodicitiesinSunspotsandAirPressure:Yule,Walkerand
theModellingofSuperposedFluctuationsandDisturbances 116
7 TheFormalModellingofStationaryTimeSeries:Wold
andtheRussians 142
8 GeneralizationsandExtensionsofStationaryAutoregressive
Models:FromKendalltoBoxandJenkins 183
9 StatisticalInference,EstimationandModel
BuildingforStationaryTimeSeries 207
10 DealingwithNonstationarity:Detrending,
SmoothingandDifferencing 261
11 ForecastingNonstationaryTimeSeries 289
12 ModellingDynamicRelationshipsBetweenTimeSeries 317
13 SpectralAnalysisofTimeSeries:ThePeriodogram
RevisitedandReclaimed 357
14 TacklingSeasonalPatternsinTimeSeries 375
15 EmergingThemes 396
16 TheSceneisSet 403
Notes 419
References 431
Index 453
v
List of Tables
2.1 MarriagerateandtradepercapitaintheUK,1857–1899 14
2.2 Correlationsbetweenthedetrendedmarriagerateandlagsof
detrendedtradepercapita 16
4.1 CorrelationcoefficientsforItalianeconomicindices.
Probableerrorsarealllessthan0.03ascalculatedusing
equation(2.10) 35
4.2 Valuesoftheratioσ2 /σ2 andtheirapproachto4−(2/d) 36
(cid:3)dy (cid:3)d−1y
4.3 Cross-correlationcoefficientsbetweentworandomseries
andtheirdifferences 47
4.4 CoefficientsofcorrelationbetweenSauerbeck’spriceindices
andLondonclearings,1868–1913 57
4.5 CoefficientsofcorrelationbetweenSauerbeck’spriceindex
andLondonclearingsfromtheirrespectivelinearsecular
trendsforthetwoperiods1868–1896and1897–1913
togetherwithcoefficientsforlagdifferences 58
5.1 Deviationsfromthemeanofthesampleinsamplesof
10termsfromarandomseries,averagingseparatelysamples
inwhichthefirstdeviationispositiveandsamplesinwhich
thefirstdeviationisnegative:averageoffirstdeviations
takenas+1,000 76
5.2 Coefficientsofthetermsinthedeviationsfromthemean
ofthesample,inasampleof10termsfromaserieswith
randomdifferencesa,b,c,…,l 78
5.3 Coefficientsbetweendeviationsfromthemeanofthesample,
inasampleof10termsfromaseriesofrandomdifferences 78
5.4 Deviationsfromthemeanofthesampleinsamplesof
10termsfromaserieswithrandomdifferences,averaging
separatelysamplesinwhich(a)firstdeviationispositive,
(b)firstdeviationis–,(c)lastdeviationis+,(d)last
deviationis–.Theaverageoffirstorlastdeviations,
respectively,called+1,000 79
5.5 Coefficientsofthetermsinthedeviationsfromthe
meanofthesample,inasampleof10termsfroma
seriesofwhichtheseconddifferencesarerandom 80
5.6 Coefficientsbetweendeviationsfromthemeanofthe
sample,inasampleof10termsfromaseriesofwhichthe
seconddifferencesarerandom 80
vi
ListofTables vii
5.7 Deviationsfromthemeanofthesample,insamplesof
10termsfromaseriesofwhichtheseconddifferencesare
random,averagingseparatelysamplesinwhich(a)first
deviationispositive,(b)firstdeviationis–,(c)lastdeviation
is+,(d)lastdeviationis–.Theaverageoffirstorlast
deviations,respectively,called+1,000 81
5.8 Comparisonofserialcorrelationsforthreeserieswith
randomdifferences,withfittedarithmeticalprogressions 85
5.9 Comparisonofserialcorrelationsforthreeserieswith
correlateddifferences,withfittedcubicseries 88
5.10 SerialcorrelationcoefficientsforModelsI–III 105
6.1 Decompositionofthefirst30termsofthesimulatedseries
usedinFigure6.2intocomplementaryfunction(simple
harmonicfunction)andparticularintegral(functionofthe
disturbancesalone) 121
6.2 Meansandstandarddeviationsofdisturbancesin
successiveperiodsof42years.(Y)correspondstoperiods
investigatedbyYule(1927,TableII) 133
6.3 Serialcorrelationsofthesunspotnumbersandthededuced
partialcorrelationsfortheextendedsampleperiod
1700–2007.Intheserialcorrelations,1denotesthe
correlationbetweenxt andxt−1,i.e.,r(1),andsoon.Inthe
partialcorrelations,2.1denotesthecorrelationbetweenx
t
andxt−2 withxt−1constant,i.e.,r(2·1),andsoon 135
8.1 Distributionofintervalsfrompeaktopeakandupcrossto
upcrossforSeriesI 187
8.2 Partialcorrelationsofthesheeppopulationandwheat
priceseries 191
9.1 GoodnessoffitstatisticsobtainedfromfittinganAR(2)
schemetoKendall’sSeriesI 214
9.2 GoodnessoffitstatisticsobtainedfromfittinganAR(2)
schemetothesunspotindex 214
9.3 TwentysimulationsoflengthT=100fromafirst-order
movingaveragewithβ=0.5 236
9.4 TwentysimulationsoflengthT=100fromafirst-order
movingaveragewithβ=0.5 237
9.5 TwentysimulationsoflengthT=100fromafirst-order
autoregressive-movingaveragemodelwithφ=0.8
andθ=0.5 240
9.6 Calculationofthe[a]sfrom12valuesofaseries
t
assumedtobegeneratedbytheprocess(1−0.3B)x =
t
(1−0.7B)a 245
t
viii ListofTables
9.7 Behaviouroftheautocorrelationandpartial
autocorrelationfunctionsofvariousARMA(p,q)processes.
φ isthekthpartialautocorrelation,beingthe
kk
coefficientonthekthlagofanAR(k)process 253
9.8 Alternativemodelestimatesforthesunspotindex.
Standarderrorsareshowninparentheses.AR(9)*
denotesanAR(9)modelwiththerestrictionsφ =···=
3
φ =0imposed 257
8
12.1 Correlationmatricesofthehogseries 353
12.2 Correlationquotientsandtheirsuccessivedifferences 354
12.3 LatentrootsofR R−1 354
k k−1
12.4 Correlationmatricesofthecanonicalvariablesy ,y
1,t 2,t
andy 355
3,t
List of Figures
2.1 MarriagerateandtradepercapitaintheUK,1857–1899,
withnineyearcentredmovingaveragesuperimposed 15
2.2 DetrendedmarriagerateandtradepercapitaintheUK,
1861–1895 15
3.1 Annualmeannumberofsunspots,1700–2007 25
3.2 Periodogramofsunspotactivity,1700–2007 25
3.3 Periodogramsofsunspotactivity:A:1750–1825;B:1826–1900;
C:1901–2007 26
3.4 Beveridge’swheatpriceindexandIndexofFluctuation,
1500–1869 27
3.5 PeriodogramoftheIndexofFluctuation 28
3.6 PeriodogramofthefirstdifferenceoftheIndexofFluctuation 29
4.1 Italianeconomicindices,1885–1912 34
4.2 Idealisticrepresentationofatimeseriesasthesumoftrend,
cyclicalandirregularcomponents 41
4.3 Londonbankclearings(inmillionsofpounds),1868–1913,with
straightline(A),parabola(B)andcompoundinterestcurve(C)
fittedtodata 53
4.4 Sauerbeck’sindexnumbersofwholesaleprices,1868–1913,
withstraightline(A)andparabola(B)fittedtodata 53
4.5 Sauerbeck’spriceindices(P)andLondonclearings(C),1868–1913,
withtheirrespectivenine-yearmovingaverages,1872–1909 54
4.6 DeviationsofLondonclearings(C)andSauerbeck’sprices
(P)fromtheirrespectivenine-yearmovingaveragesecular
trends,1872–1909 55
4.7 DeviationsofLondonclearings(C)andSauerbeck’sprices(P)from
theirrespectivelinearseculartrends,1868–1913 55
4.8 DeviationsofLondonclearings(C)andSauerbeck’sprices(P)from
theirrespectiveparabolicseculartrends,1868–1913 56
4.9 DeviationsofLondonclearingsfromtrendascompoundinterest
curve(C)andSauerbeck’spricesfromlineartrend(P),1868–1913 56
4.10 Sauerbeck’spriceindex(P)andLondonclearings(C),1868–1913,
withtwostraightlinesfittedtobothseries,1868–1896and
1897–1913,respectively 59
4.11 DeviationsofSauerbeck’sprices(P)andLondonclearings(C)
fromtheirrespectivelineartrendsforthetwoperiods1868–1896
and1897–1913 59
ix