Table Of Content(cid:2)
ForecastingwiththeThetaMethod
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Forecasting with the Theta Method
TheoryandApplications
KostasI.Nikolopoulos
PrifysgolBangorUniversity
(cid:2) Gwynedd,UK (cid:2)
DimitriosD.Thomakos
UniversityofPeloponnese
Tripolis,GR
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Thiseditionfirstpublished2019
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ISBN:9781119320760
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Kostas
Iextendmydeepandsinceregratitudetoallthosewhohavecontributedto
mypost-DEngjourneyinacademiafromlate2002onwards:
First,my“father”Vasillis,andmybig“brother”Dimitrioswithhis
unbelievablecapacity.
Then,myyounger“brother”Fotioswhofortunatelyavoidsthemistakes
Imade,andmyoldestmateinthefieldArisforthereallygreatthings
wehavedonetogether.
Also,mygoodoldfriendsandfrequentcollaboratorsKostasandVicky.
AlwaysindebttomywisementorsShantiandBrian.
(cid:2) (cid:2)
Lastbutnotleast,unbelievablyproudofmyacademic“children”to
date:Nicolas,Vassilios,Sam,Vivian,Greg,Sama,Christina,Tolu,
Azzam,Waleed,Ilias,Chris,andAxilleas.
Dimitrios
Thisvolumeisjustaquickstopinajourneythatbeganmanyyearsago.
AheartfeltthankyoutoKostasforsharingtheroadwithme,andmaywe
continuetononlinearpathsintheyearstocome.Theintellectualcuriosity
andresearchsupportofmyPhDstudentsLazarosRizopoulosandFoteini
Kyriaziinpreparingthemanuscriptisgratefullyacknowledged.Thisvolume
isdedicatedtomyparents,thosewhofirstallowedmetoprosperfreelyinmy
endeavors.
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vii
Contents
AuthorBiography ix
Preface xi
PartI Theory,MethodsandModels 1
1 The𝚯-legacy 3
1.1 TheOrigins… 3
1.1.1 TheQuestforCausality 5
(cid:2) 1.2 TheOriginalConcept:THETAasinTHErmosTAt 6 (cid:2)
1.2.1 Background:ADecompositionApproachtoForecasting 7
1.2.2 TheOriginalBasicModeloftheThetaMethod 7
1.2.3 HowtoBuildandForecastwiththeBasicModel 10
1.2.4 SESwithDrift 11
1.2.5 TheExactSetupintheM3Competition 12
1.2.6 ImplementingtheBasicVersioninMicrosoftExcel 15
1.2.7 TheFutuReisWritteninR 18
1.A Appendix 18
2 Fromthe𝜽-methodtoa𝜽-model 37
2.1 StochasticandDeterministicTrendsandtheirDGPs 37
2.2 The𝜃-methodAppliedtotheUnitRootwithDriftDGP 38
2.2.1 MainResults 38
2.2.2 AlternativeTrendFunctionsandtheOriginal𝜃-lineApproach 42
2.2.3 Implementingthe𝜃-methodundertheUnitRootDGP 45
2.3 The𝜃-methodAppliedtotheTrend-stationaryDGP 45
2.3.1 Implementingthe𝜃-methodundertheTrend-stationaryDGP 49
2.3.2 IstheAR(1)-forecasta𝜃-forecast? 50
3 TheMultivariate𝜽-method 51
3.1 TheBivariate𝜃-methodfortheUnitRootDGP 51
3.2 SelectionofTrendFunctionandExtensions 54
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viii Contents
PartII ApplicationsandPerformanceinForecasting
Competitions 59
4 EmpiricalApplicationswiththe𝜽-method 61
4.1 SettinguptheAnalysis 61
4.1.1 SampleUse,EvaluationMetrics,andModels/MethodsUsed 61
4.1.2 Data 64
4.2 SeriesCREDIT 65
4.3 SeriesUNRATE 73
4.4 SeriesEXPIMP 79
4.5 SeriesTRADE 85
4.6 SeriesJOBS 91
4.7 SeriesFINANCE 103
4.8 SummaryofEmpiricalFindings 115
5 ApplicationsinHealthCare 117
5.1 ForecastingtheNumberofDispensedUnitsofBrandedandGeneric
Pharmaceuticals 117
5.2 TheData 118
5.2.1 Prescribedvs.Dispensed 119
5.2.2 TheDataset 120
(cid:2) 5.3 ResultsforBranded 121 (cid:2)
5.4 ResultsforGeneric 142
5.A Appendix 152
PartIII TheFutureofthe𝜽-method 161
6 𝜽-ReflectionsfromtheNextGenerationofForecasters 163
FotiosPetropoulos
6.1 Design 164
6.2 SeasonalAdjustment 164
6.3 OptimizingtheThetaLines 166
6.4 AddingaThirdThetaLine 168
6.5 AddingaShort-termLinearTrendLine 169
6.6 ExtrapolatingThetaLines 170
6.7 CombinationWeights 172
6.8 ARobustThetaMethod 174
6.9 ApplyingThetaMethodinRStatisticalSoftware 174
7 ConclusionsandtheWayForward 177
References 181
Index 185
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ix
AuthorBiography
KostasI.NikolopoulosisProfessorofBusinessAnalyticsatBangorBusiness
School, UK. He is also the Director of forLAB, a forecasting laboratory, and
former Director of Research for the College of Business, Law, Education,
andSocialSciences.HereceivedbothhisEngineeringDoctorateDiplomain
ElectricalandhisComputerEngineering(MEng)fromtheNationalTechnical
University of Athens, Greece. He is an expert in time series analysis and
forecasting,forecastingsupportsystems,andforecastingtheimpactofspecial
events. Professor Nikolopoulos has published in a number of prestigious
(cid:2) journals and is an Associate Editor of Oxford IMA Journal of Management (cid:2)
MathematicsandSupplyChainForum:AnInternationalJournal.
Dimitrios D. Thomakos is Professor of Applied Econometrics and Head of
the Department of Economics at the University of Peloponnese, Greece, and
SeniorFellowandMemberoftheScientificCommitteeattheRiminiCenterfor
EconomicAnalysisinItaly.DimitriosholdsanMA,MPhil,andPhDfromthe
DepartmentofEconomicsofColumbiaUniversity,USA.Hisresearchworkhas
appearedinseveralprestigiousinternationaljournalsineconomicsandfinance
suchastheReviewofEconomicsandStatistics,theCanadianJournalofEco-
nomics,theReviewofInternationalEconomics,andmanyothers.
Kostas and Dimitrios have worked together extensively coauthoring in the
Journal of Management Mathematics and the Journal of Forecasting among
manyotherjournalsandrecentlycoeditedtwovolumesforSpringer-Palgrave
(AFinancialCrisisManualandTaxationinCrisis).
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xi
Preface
The Theta method is the most successful univariate time series forecasting
techniqueofthepasttwodecades,sinceitsoriginationinthelate1990s.Jointly
withthedampedtrendexponentialsmoothing,thesearethetwobenchmarks
that any newly proposed forecasting method aspires to outperform, so as to
pass the test of time. This performance was originally demonstrated in the
M3 competition in 2000 that the Theta method was the only method that
outperformed Forecast Pro (www.forecastpro.com) and dominated a pool of
18otheracademicmethodsandfivemoresoftwarepackages.
(cid:2) The method originated in 1999 from Kostas and his supervisor Professor (cid:2)
VassilisAssimakopoulos,andwasfirstpresentedintheInternationalDSI991
whilethefirstfull-fledgedacademicarticleanddescriptioncamein2000inthe
InternationalJournalofForecasting.Inthesamejournalcamethefirstcritique
threeyearslater,ledbyProfessorRobHyndmanandhisteaminMonashforthe
similarityofthebasicmodelfromthemethodtosimpleexponentialsmoothing
(SES)withdrift.
Thejournalmadeamistakeinthattheyhadneitherusedourreviewonthe
paper(asourobjectionswereleftunanswered)norhadtheyofferedusacom-
mentarytobepublishedalongthatarticle–thatisastandardpracticeinsimilar
situations.Thiscreatedmoreconfusionasanimmediateclearanswerfromus
andVassiliscouldnotseethelightofday;andittookafewmoreyearsuntilwe
foundthemeansandmediatopostourresponse.That2003argumentisnow
deadandburied,aseveryyearanewanddifferentsuccessfulmodeliscoming
out from the Theta method; and none of them is an SES with drift. This was
alwaysourcounterargument–thatthemethodismuchmuchbroaderandwe
cannotseetheforestforthetrees…Butthisishistory;andactuallythat2003
articlehasprobablycreatedevenmoreinterestinthemethod,sowelcomeitis
evenifitdidnotdojusticetothemethod.
A few working papers from 2005 onward and an article of ours in 2011 in
SpringerLNEEhavesettherecordstraight;andfromthatpointon,jointwork
1 AssimakopoulosandNikolopoulos(1999).
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xii Preface
withDimitriosintheIMAJournalofManagementMathematics2 provedwhy
themethodworkswellforunitroots–thatwerethedominantdata-generating
processes (DGPs) in the M3 data – while more work in the Journal of Fore-
castingextendedthemethodinthebivariatespace,providingatthesametime
varioussuccessfulextensionsofit.Thistrendwaspickedupbyresearchteams
inBrazil,theUnitedStates,and,mostnotably,intheUniversityofBathandwith
Dr. Fotios Petropoulos – a scholar of the method – providing further exten-
sionsandoptimizations.ThesamestandsforthenewstudentsofVassilisinthe
ForecastingandStrategyUnit(www.fsu.gr)inGreecewhocontinuehislegacy.
Whentheselinesarewritten,theoriginalarticleof2000hasachieved209cita-
tionsinGoogleScholar.
Sonowweknowthemethodisfar,farmorethanjustSESwithdrift.Itwasnot
aneasyjourneytothispoint–buttheforecasters’journeywasnevermeantto
beaneasyoneanywayfromtheveryfirstdaythatthefieldwasfoundedfrom
Spyros Makridakis, Robert Fildes, and Scott Armstrong. From here onward,
now that we have a fair share of the focus of the academic community – in
forecastingandbeyond–itisgoingtobeafarmoreexcitingone;andtokick
starttheprocessweproposedtoWiley–andthepublishinghousegladlyand
wiselyaccepted–todeliverthefirstbookonthemethodthatwouldcapturethe
workdonetodateand,moreimportantly,laythenewtheoryandfoundations
(cid:2) fortheyearstocome. (cid:2)
The book also includes a series of practical applications in finance, eco-
nomics, and health care as well a contribution from Fotios on his honest
viewofwhathappenedinthepastandwherethingsshouldgofromnowon.
ThisistoppedupbytoolsinMSExcelandguidance–aswellasprovisional
access–fortheuseofRsourcecodeandrespectivepackages.
Whenallissaidanddone,wedobelievethatthisbookisavaluableanduseful
toolforbothacademicsandpractitionersandthatthepotentialofthemethod
is unlimited. This is a forecast from forecasters and it comes with the usual
caveat:thepastsuccessofthemethoddoesnotguaranteethefutureone…but
onlytimewilltell!
February2018 Kostas&Dimitrios
2 AnarticlethatfortwoyearswasreviewedintheInternationalJournalofForecasting(IJF)and
nevergotpublishedthere–withoutasinglemistakebeingfoundinit.Wealwaystookthestance
thatweshouldfirstsubmitinIJF–forhistoricreasons–anynewworkonthemethod,butthat
resultedinunnecessarydelaysinthepublicationprocess.
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