Análisis de la evolución de los réditos de los bonos soberanos, utilizando las técnicas de ondículas

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www.elsevier.es/cesjef

Cuadernos

de

economía

ARTICLE

Analysis

of

the

evolution

of

sovereign

bond

yields

by

wavelet

techniques

David

Chinarro

a,∗

_,

_Eduardo

_Martínez

b

_,

_Simón

_J.

_Sosvilla

c

a_San_Jorge_University,_Spain

b_La_Laguna_University,_Spain

c_Complutense_University,_Spain

Received29June2015;accepted3July2015 Availableonline5August2015

JEL CLASSIFICATION C14; C21; E43; F34; F37; G12; G15; G17 KEYWORDS Sovereignbond; Wavelet; Coherence; Entropy; Timeseries; Correlation; Comovement

Abstract Theterm‘‘wavelets’’coversasetofresourcesfromthemathematicalanalysisthat

hasproventheirefﬁciencyinsystemidentiﬁcationonareassuchashydrology,geology,

glaciol-ogy,climatologyandenergy resourcesoptimization. Themethodologyundergoneonsystems

engineeringcouldbeextrapolatedtoeverythingconceptualizedas‘‘complexsystem’’whatever

itsnaturebe.Thewavelettechniquesprovidethedescriptionofnon-stationarycomponents

andtheevolutionofmacroeconomicvariablesinthefrequencydomain.Theidentiﬁcationof

predominantfrequentialscalesandtransienteffectsintimeserieshighlightsthe

multiresolu-tionalanalysisthatwouldbemoredifﬁculttotreatwithtraditionalmethodsofeconometrics.

Areviewoftheliteraturewillshowthepotentialproblemsthatcanbesolvedwiththese

tech-niques,includingpredictionofbeneﬁtscalculated ontheevolutionoftheriskpremiumofa

country,theextractionofsymmetricmacroeconomicshocksincountryclusters,ordetectionof

transienteffectsonthemutualinﬂuenceofsovereignbondsbetweenpairsofcountries,among

others.Thedissertationwillculminateinspeciﬁcapplicationsthatshowthepowerofwavelet

techniquesinidentifyingpossibledeterminantsandcorrelationoftheevolutionofsovereign

bondyieldsintheeuroareacountries.

reserved.

CÓDIGOSJEL

C14; C21; E43;

Análisisdelaevolucióndelosréditosdelosbonossoberanos,utilizandolastécnicas deondículas

Resumen Eltérmino‘‘ondículas’’cubreunconjuntoderecursosdelanálisismatemáticoque

hademostradosueﬁcaciaenlaidentiﬁcacióndesistemasenáreastalescomolahidrología,

∗_{Corresponding}_author.

E-mailaddress:dchinarro@usj.es(D.Chinarro).

http://dx.doi.org/10.1016/j.cesjef.2015.07.001

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F34; F37; G12; G15; G17 PALABRASCLAVE Bonosoberano; Wavelet; Coherencia; Entropía; Seriestemporales; Correlación; Comovimiento

geología,glaciología,climatologíayoptimizacióndelosrecursosenergéticos.Lametodología

utilizada en la ingenieríade sistemas podría extrapolarse a todolo conceptualizado como

‘‘sistemacomplejo’’,fuerecualfueresunaturaleza.Lastécnicasde‘‘ondículas’’aportanla

descripcióndeloscomponentesnoestacionariosylaevolucióndelasvariables

macroeconómi-casenelcampodelafrecuencia.Laidentiﬁcacióndelasescalasfrecuencialespredominantesy

losefectostransitoriosenlasseriestemporalesenfatizaelanálisismulti-resolución,quepodría

sermásdifícildetratarconelusodemétodoseconométricostradicionales.Unarevisiónde

laliteraturareﬂejarálosproblemaspotencialesquepuedensolucionarseutilizandoestas

téc-nicas,talescomolaprediccióndelosbeneﬁcioscalculadossobrelaevolucióndelaprimade

riesgoenunpaís,laextraccióndelosshocksmacroeconómicossimétricosenlosclustersdel

país,oladeteccióndelosefectostransitoriossobrelainﬂuenciamutuadelosbonossoberanos

entrelosparesdepaíses,entreotros.Ladisertacióndesembocaráenaplicacionesespecíﬁcas

quereﬂejaránelpoderdelastécnicasdeondículasenlaidentiﬁcacióndeposibles

determin-antes,ylacorrelacióndelaevolucióndelosréditosdelosbonossoberanosenlospaísesdela

zonaEuro.

derechosreservados.

1. Introduction

Classically, thedebt issuedby anational governmentwas

transformedinto bondsin the currency of a countrywith

a more stable economy. It could be used as a source

of income for pension funds, university donations,

char-ities and other institutions. Recent developments in the

euro-zoneandtheincursionofotherdevelopedand

emerg-ing markets have contributed toa shift in thinkingabout

the materiality of responsible investment in this type of

credit investing, which casts doubt on the existence of

risk-free sovereign bonds. The Principles for Responsible

Investment(PRI)givenbyKohutetal.(2013)exploresthe

ways of applying to ﬁxed income environmental, social and governance(ESG) factorsin deciding where investors couldput theirmoneyin. Theenvironmental factors gen-erally consideredareclimate change, biodiversity,energy resources, biocapacity and ecosystem qualityand natural disasters. Social factors include: human rights, educa-tion,healthlevels,demographicchangeemploymentlevels, socialexclusionandpoverty,trustininstitutions,crimeand safety, and food security. Governance factors are institu-tionalstrength,corruption,regimestability,politicalrights, regulatoryeffectiveness andaccountingstandards,among others.

Intimeseriesanalysis,thereareseveralobjectiveswhich canbeclassiﬁedintothecategoriesofdescription, explana-tion,forecasting,controlandmonitoring.Asimpleobjective ofanytimeseriesanalysisistoprovideaconcise descrip-tion of the past values of a series which involves the underlyinggeneratingprocessofthetimeseries.Dataplot complementstheformalandquantitativeanalysistoshow importantfeaturesoftheseriessuchastrend,seasonality, outliersanddiscontinuities.Timeseriesobservationstaken ontwoormorevariablescanbeusedtoexplainthatthe vari-ationinonetimeseriesproducescorrespondentchangesin theothertimeseries.

Systemengineeringisdefinedbroadlyastheartand sci-enceof creatingwhole solutions to complexproblems. It providesmethods to find the right model adapted tothe peculiarities of the system in study. Identification tech-niques,developedtorepresenttypicalengineeringartificial systemsthroughlinearandnonlinearmodels,canbeapplied inthestudyofeconomysystems.Thetechniquealsocalled data-drivenmodelingisbasedonanalysisofdatafromone system,seekinginparticularconnectionsbetweenthe sys-temstatevariables(inputandoutputvariables).

Mathematical models are formal statements or equa-tions to express the relationship between system inputs, outputsandoperations,toperformsimulationsorforecast thefutureeconomystate. Mathematicalmodelstructures andcomputability power that ever have helped to ﬁnan-cial analysts are strongly required now in order to shed light on a complex problem where a lot of these factors areimmeasurable assomeof ESG.The immediateimpact is the rise of yields for the sovereign bonds of the issu-ing country and also, likely an international propagation ofthisimpact.The hypothesisisthatfromacomovement study ontime seriesevolution of sovereign bonds in sev-eralEuropean countries, and applyingwell-known system engineering methods, we can assess, or at least put in evidence, the periocity and intensivity of these impacts propagation. In this regard, special tools, as thosebased on wavelet transform, are being considered in prepar-ing time series such as smoothing of signals, spectral analysis, coherence levels and outliers detection, among others.

Theliteraturerapidlyspreadoutandwaveletanalysisis nowusedextensivelyinphysics,geophysics(Grinstedetal., 2004),epidemiology(Cazellesetal.,2007),neuroscience, signal processing (Ricker, 2003), hydroclimatology (Rivera etal.,2007),oceanography(Meyersetal.,1992), hydroge-ology(Labatetal.,1999;Chinarroetal.,2012),electricity demand(Hernndezetal.,2013),remotesensingdata(Ebadi

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etal.,2013),computingcomplexproblemasMaxwell’scurl equations(AmatandMuoz,2008),amongotherﬁelds.

Intheeconomicandﬁnancialliterature,comovementis assessedinthetimedomainthroughthewell-known corre-lationcoefﬁcient.TheworkofKydlandandPrescott(1982)

andLongandPlosser(1983),andmanyauthorshave inves-tigatedthepropertiesofbusinesscycles.

The comovement and dependencies between the eco-nomic time series analyzed in the frequency domain can be found in the pioneering paper of Croux et al. (2001)

whopropose thecohesion, asameasurein thefrequency domain of dynamic comovement between several time series.Theydynamiccorrelationstoestimatedeterminants ofoutputcomovementbetweenOECDcountriesarestudied by Fidrmucet al. (2012), where trade intensity,ﬁnancial integration,and specialization patterns have signiﬁcantly differenteffectsoncomovementsatdifferentfrequencies.

Genay et al. (2001) supplied a unified view of filter-ingtechniques withaspecialfocusonwaveletanalysisin financeandeconomicsofferingtestingissueswith descrip-tive focus on access to a wide spectrum of parametric andnonparametric filteringmethods. Cariani (2012) com-pareswaveletsbasedmethodtoanalyzethebusinesscycles in Romania between 1991 and 2011, with the standard approachesinliterature.SangbaeKima(2007)establisheda relationshipbetweenchangesinstockpricesandbondyields intheG7countriesthroughWaveletanalysis.Businesscycles inatransitioneconomyarerevealedbyCariani(2012).Ona wavelet-basedapproach,theyshowanevidenceofbusiness cyclespropertiesvaryingintime,frequencyandintensity.

Thewaveletanalysisisproposedtoassesssimultaneously thecomovementatthefrequencyscalesandalongthetime line. So,it is possible tocapture the timeand frequency varyingfeaturesofcomovementwithinauniﬁedframework whichconstitutesareﬁnementtopreviousapproaches.

Aftercollectingtimeseriesofsovereignbondfromeleven countries(Fig.1),thispaperemphasizesthemethodsand explanationsonwaveletstechniques,withspecialfocuson exactlywhatwaveletanalysiscanrevealabouteconomics timeseries.Itofferstestingissueswhichcanbeperformed withwaveletsinconjunctionwiththemulti-resolution anal-ysis to coevaluate two or more time series of sovereign bonds.Thepaperisorganizedasfollows.InSection2,the methodologyoftheproposedmethodisdescribed.In Sec-tion3the usabilityof themethod isdemonstrated inthe applicationofsovereign bondcoevolutioninseveral Euro-peancountries.ThepaperisconcludedinSection4.

2. Method

description

2.1. Dynamicalbreakdownofthetime

TheexpressionL(R)2isaHilbertspace,withinnerproduct

Rf(t)g∗(t)dt(g*isgcomplexconjugated),forallfandg functionsdefinedinthisspace.Moreover,fandgshouldbe squareintegrablefunctions,so_Rf(t)g∗(t)dt<∞. Since thisintegralisusuallyreferredtoastheenergyofthe func-tionf,thisspaceisalsoknownasthespaceoffunctionswith finiteenergy.Therefore,withinnerproductinL(R)2canbe definedasanassociatednorm||f||=f,f2.

ThePlancherelrelationisf(t),g(t)=ˆf(t),ˆg(t)where

f(t),g(t)∈L2₍_R₎_{and ˆ}_f _denotes_de_Fourier_transform_of_f₍_t_): ˆ

f(t)=_−∞∞ f(t)e−i2ωt_dt_. _The _Plancherel _identity_says _that

an energyofa functionispreserved by theFourier trans-form:||f(t)||2₌_||_ˆ_f₍_t₎_||2₍_Korner,₁₉₈₈_).

To discover properties at different frequencies, it is common to utilize Fourier analysis. However, under the Fouriertransform,thetimeinformationofatime-seriesis completely lost. Because of this loss of information it is hard todistinguishtransient relationsor toidentify when structural changes do occur. Moreover, these techniques aredeﬁnitivelynotappropriatetodealwithnon-stationary time-series. To overcome the problems of analyzing non-stationary data, Gabor (1946) introduced the Short Time FourierTransform.Thebasicideaistobreakatimeseries intosmallersamplesandapplytheFouriertransformtoeach sample. However,this approachisinefﬁcient becausethe frequency resolution is the same across all different fre-quencies(Raihanetal.,2001).

2.2. Waveletanalysis

Wavelets are well appropriated for analyzing signals that hold local nonlinearities and singularities. Applying wavelets,importantinformationappearsthrougha simulta-neousanalysisofthesignaltimeandfrequencyproperties (Mallat, 1998). The continuous wavelet transform (CWT) ˜

f(t)s, ofafunctionf(t),onthebaseofthewaveletmother s, is deﬁnedastheinner productf(t), s,inthe

mea-surableandsquareintegralspacesL(R)2(Daubechies,1990; Mallat,1989),asissetoutinEq.(1):

[W f](s,)= ∞ −∞ f(t) ∗s,(t)dt (1) s,(t)= 1 √ s t− s (2) where * is the complexconjugate. The motherwavelet

(t) is normalized || ||2₌∞

−∞| (t)|2dt=1. All wavelet

functions usedinthetransformationarederivedfromthe mother wavelet through translation (shifting) and scaling (dilationorcompression),modifyingsandin s,.

Thecrosswaveletspectrum(XWS)betweenf(t)andg(t) signals aroundtimet andscale s is deﬁnedby (3) asthe convolutionoftheCWTofbothsignals(Mallat,1989).

X [f(t),g(t)]s, =[W f](s,)⊗[W g](s,) (3)

The wavelet power spectrum (WPS) or autocorrelation functionofthewavelettransformationofasignalisdeﬁned as

P [f(t)]s,=[W f](s,)⊗[W f](s,) (4)

The representation of the continuous wavelet trans-form(Eq.(4))producesunnecessaryredundantinformation. Therefore,forpracticalpurposesthediscretewavelet trans-formtoasetof selectedaccordingtothepower seriesis usedbands.Thisdiscretizationproducesanarrayofpowers:

P [f(t)]a,b=[Ci,j] (5)

where j=20_, ₂1_, _._._._, ₂k _for _each _band _of _frequency and i≥0 are the values of discrete time considered. A

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50 Austria Belgium Finland France Germany Greece Ireland Italy Netherland Portugal Spain 45 40 35 30 25 20 Y ields on so v e reign bonds 15 10 5 0

12Feb9312Feb9412Feb9512Feb9611Feb9711Feb9811Feb9911Feb0010Feb0110Feb0210Feb0310Feb0409Feb0509Feb0609Feb0709Feb0808Feb0908Feb1008Feb1108Feb1207Feb13

Figure1 Representationofsovereignbondyieldforeachcountryinstudyfrom1993to2014.

scalogram or spectrogram is the graphic representation of the normalized wavelet power of a signal f(t) in a time-varyingfrequencydomain,withinathree-dimensional space: time (x), scale/frequency (y) and power (z). The examplegivenin Fig.2 isadiscretepowerspectrum rep-resentation.Colortonesstandforcoefﬁcientvaluesinthe spectral matrix (Eq. (5)), Y-axis is the range of selected powerbandsandXisthetimingofthesignal.

The wavelettransform appliedto a ﬁnite length time series inevitably suffers from border distortions. The affectedregionisdelimitedbytheconeofinﬂuence(COI) ofthewaveletspectrumthatcanbecalculatedaccording toGrinstedetal.(2004).

Waveletpowercoherence(WPC)isarelativemeasureof thecorrelation betweentwosignals.Sois thenormalized crossspectrum(Eq.(4))of twosignalsin both powerand

2 3 3 4 5 6 7 8 10 11 13 16 19 23 27 32 38 45 Y ields on so v ereign bonds 54 64 76 91 108 128 152 181 215 256 304 362 431 512 12Feb9327Oct9311J ul94

23Mar9505Dec9516Aug9630Apr9712Jan9824Sep9808Jun9918Feb0001No v00

16Jul0128Mar0210Dec0222Aug0305Ma y04

17Jan0529Sep0513Jun0623Feb0707No v07

21Jul0802Apr0915Dec0927Aug1011Aug1123Jan1204O ct12

18Jun13

1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 Spain-wavelet spectrum

Figure2 WaveletpowerspectrumofsovereignbondevolutioninSpain.TheY-axisisthewaveletscale(bondyieldinpercent).The

X-axisistime(days).Curvedlinesoneithersideindicatetheconeofinfluence(dashedline)whereedgeeffectsbecomeimportant. Spectralstrengthisshownbycolorsrangingfromdeepblue(weak)todeepred(strong).Thethickblackcontourdesignatesthe5% significancelevelagainstrednoise.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe webversionofthearticle.)

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12Feb9327Oct9311Jul9423Mar9505Dec9516Aug9630Apr9712Jan9824Sep9808Jun9918Feb0001No v00

21Jul0802Apr0915Dec0927Aug1011Aug1123Jan1204Oct1218Jun13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 8 16 32 64 128 256 512 1024 2048 Y ields on so v ereign bonds

Spain-Germany wavelet coherence

Figure3 CoherencespectrumobtainedthroughtheNormalizedCrossWaveletSpectrumofsovereignbondstimeseriesinSpain andGermany.The5%ofsigniﬁcancelevelsareplottedwithblackthick line.Theleft axisisthewaveletscale(h).The bottom axisistime.CurvedlinesoneithersideindicatetheCOI.Thespectralstrengthspreadsoutfromweakvalues(blue)tostrongones (darkred)forhighcoherence.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderisreferredtotheweb versionofthearticle.)

phase.WPC is the normalized measurementof the cross-powerspectrum,asexpressedin:

C [f(t),g(t)]s, =

|X [f(t),g(t)]s,|

|P [f(t)]s,P [f(g)]s,|1/2

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Thecoherencevaluesrangefrom0to1andrevealhow much input and output series are correlated in terms of powerandphaseatagivenfrequencyinatimeinstant.The WPCisusefulforelucidatingwhichofthemultipleinput sig-nalsourcesrepresentthemaincontributiontotheresponse signal. This contribution is shown in frequency and time bands.An exampleis Fig.3 for therepresentation ofthe crossspectralpowerbetweentwocountries’bonds.

2.3. Waveletentropy

Shannon (1948) put forward the concept of information entropyﬁrstly;heputtheentropyastheuncertainty ofa randomeventortheamountofinformation.Theentropyis themeasureofthedegreeofdisorderofthesignal.Alow valueofentropycanrevealusefulinformationaboutcertain underlying dynamical processes or events associatedwith thesignal.Inspectralanalysis,averyorderedprocessis rep-resentedbyaveryclearandhighlightedband.Thewavelet powerdistributionofadisorderedsignalwillbealmostzero exceptinthewaveletresolutionlevel.

Sincewaveletanalysishasgoodabilityoftime-frequency localization, the combination of multi-resolution wavelet andinformationentropyleadstowaveletentropydeﬁnition of a signal. A timeseriesgenerated by a randomprocess

18No v93

02Aug9414Apr9527Dec9509Sep9622Ma y97 03Feb9816Oct9 8 30Jun9913Ma r00 23No v00

07Aug0119Apr0201Jan0315Sep0327Ma y04 08Feb0521Oct0 5 05Jul0619Ma r07 29No v07 1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 0 10 20 30 40 50 60 70 80 90 100 2 3 3 4 5 6 7 8 10 11 13 16 19 23 27 32 38 45 54 64 Y ields on so v ereign bonds

Figure4 TheleftsideisthewaveletpowerspectruminhighfrequenciesofIrelandsovereignbonds.Rightsideistheentropy representationofthewaveletcoefﬁcientsbyfrequencybands.Peaksoflowentropyissuggestingtheemphasizedthebands.

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3 4 6 8 11 16 23 32 45 64 91 128 181 256 362 512 724 1024 1448 2048 12Feb9327O ct93

11Jul9423Mar95 05Dec95 16Aug9630Apr97 12Jan9824Sep98 08Jun9918Feb0001No v00

16Jul0128Mar02 10Dec02 22Aug0305Ma y04

17Jan0529Sep05 13Jun0623Feb0707No v07 1/256 1/258 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 Y ields on so v ereign bonds Greece-wavelet spectrum

FigureA.1 WaveletpowerspectrumofsovereignbondevolutioninGreece.

will represent a disordered behavior and will have a ﬂat powerrepresentationforallfrequencybands.Consequently theentropywilltakehighvaluesfor allfrequencies. Nev-ertheless,timeseriesofeconomicdataarenotcompletely disordered or ordered. Wavelet entropy (WE)can help in providingaproceduretodetect frequencybandswithlow entropy,i.e.an ordered behaviorandconsistent periodic-ity.Shannon(1948)givesausefulcriterionforanalyzingthe entropyandestimatingaprobabilitydistribution.Thetotal waveletentropybased onnormalizedShannon entropyat scalej,inamovingdatawindow,isestimatedbythedetail coefﬁcientsateachtime,asisshowedinEq.(7):

Ej=−

∞

k=1[Ci,j]log[Ci,j] (7) wherewisthewindowlengthofthemovingwindowinthe fundamentalfrequencyoftheoriginalsignal.

3. Data

analysis

3.1. Timeseriesanalysisinthetimedomain

Timeseriesareconstitutedbythevaluesofsovereignbonds from 11 countries of the Euro area, with daily sampling except non-working days(Fig. 1).To ensure continuityin thetimeseries,ithasbeenassumedthatthevaluesremain constantduring non-working days.The timeseriesranges coverfrom12/02/1993to27/01/2014forAustria,Belgium, Finland, France, Germany, Ireland, Italy, Netherland and Spain;from15/07/1993forPortugal;andfrom31/03/1999 forGreece.Onceobtainedgraphicallytheevolutioninthe time domain of the time series for each country, an ini-tialassessment of apossible co-evolutionof thepricesof sovereignbondscanbeobserved.

3.2. Timeseriesanalysisinthefrequencydomain

Onesingularcharacteristicof waveletanalysisis the arbi-trarychoiceofthewaveletfunction(inFourierandinother transformsthebasis functionisﬁxed). Butseveralfactors shouldbeconsideredin ordertoselectasuitablewavelet togetbestperformance.Figs.2andA.1---A.5depictthe evo-lutionofsovereignbondsinGermany,Greece,Ireland,Italy, Portugal,andSpain,respectively.

Inordertohelprevealthehighlightedfrequencybands inthespectrum,awaveletentropyis applied(Eq.(7)).A veryordered timeseriesshould present a periodic mono-frequencysignal (signal witha narrowband spectrum).In thesovereignbondsconsideredataglancethewavelet rep-resentationisresolvedina fewwaveletresolution levels, withwaveletpower valuesnear zeroin high frequencies. Sometimes the graphic observation cannot be sufficiently clear. Colored frames are extended and interact both in thehorizontaldimension(time)andverticaldimension (fre-quency),makingblurred thebands powerrepresentation. Therefore,additionaltoolsareneededtoassistthe graph-icalinterpretation,dilucidatetheperiodicitiesandconfirm theclusteringoffrequencybands.Inordertodisclose possi-blehiddeninformation,arestrictedfocusonhighfrequency areaofspectrumis performed(4).So,anewpower spec-trumvaluesinsomebandsappear,althoughyetwithblurred dispersionthat couldbe confusinga primaface observer. Inordertoprecise theexistence of frequency bands,the graphical observation is supported by analysis of wavelet entropy(WE).The leftsideof Fig.4isthewaveletpower spectrum in high frequencies of Ireland sovereign bonds. Right side is the entropy representation of the wavelet coefficientsbyfrequencybands.Peaksoflowentropyarean indicatorofahighlightband.Thisisappliedinthisrestricted areaanddepictedinFig.4.

Now,forthisexample,twobandsoflowentropyindicate ahighorderlevelsintheprobabilitydistributionforthese

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3 4 6 8 11 16 23 32 45 64 91 128 181 256 362 512 724 1024 1448 2048 12Feb9327O ct93

11Jul9423Mar9505Dec9516Aug9630Apr9712Jan9824Sep9808Jun9918Feb0001No v00

17Jan0529Sep0513Jun0623Feb0707No v07 1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 Y ields on so v ereign bonds Ireland-wavelet spectrum

18Jun13

FigureA.2 WaveletpowerspectrumofsovereignbondevolutioninIreland.

3 4 6 8 11 16 23 32 45 64 91 128 181 256 362 512 724 1024 1448 2048 12Feb9327O ct93

17Jan0529Sep0513Jun0623Feb0707No v07 1/256 1/512 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 512 Y ields on so v ereign bonds Italy-wavelet spectrum

18Jun13

FigureA.3 WaveletpowerspectrumofsovereignbondevolutioninItaly.

bands.The most notableis between14 and16 frequency bandwhich is showinga weekly period.Inthe sameway, wecannoteabout30-daysperiodtoexistandalso,another peakoflowentropycanbenotedbetween45and50band. Therefore,thesignalfollowssomepossibleimportantthree eventswiththeseperiodicities.

A detailed wavelet spectral analysis was performed on each country making a progressive focus on differ-ent areas of the spectrum, starting from the lowest

frequency band (long period) to the lowest frequency. The results are presented in Table 1 with the help of entropy analysis to clarify the peak of frequency

or central value of frequency in each band (Eq.

(7)).

Very similarfrequency componentsare observed in all countries.Itisnoteworthytoshowtheannualand semian-nualvalues.Buttheobservercanbeespeciallystruckbythe veryprecisequarterlyvaluethathappensinallcountries.

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3 4 6 8 11 16 23 32 45 64 91 128 181 256 362 512 724 1024 1448 2048 12Feb9327Oct9311J ul94

23Mar9505Dec9516Aug9630Apr9712Jan9824Sep9808Jun9918Feb0001No v00

17Jan0529Sep0513Jun0623Feb0707No v07 1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 Y ields on so v ereign bonds Portugal-wavelet spectrum

18Jun13

FigureA.4 WaveletpowerspectrumofsovereignbondevolutioninPortugal.

3 4 6 8 11 16 23 32 45 64 91 128 181 256 362 512 724 1024 1448 2048 12Feb9327O ct93

17Jan0529Sep0513Jun0623Feb0707No v07 1/256 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 256 Y ields on so v ereign bonds Germany-wavelet spectrum

18Jun13

FigureA.5 WaveletpowerspectrumofsovereignbondevolutioninGermany.

Also,athighfrequencies periodsaround45days,3weeks and2weeksusuallyappear.

3.3. Crosswaveletanalysis

Crosswaveletanalysisandwaveletcoherencearepowerful methods for testing proposed linkages between two time

series.Thereforeitwouldbesuitablefortheidentiﬁcation ofcomovementbetweentwosovereignbondstimeseries.

Figs.3,A.3,A.4,A.6,andA.7depictthecorrelationin termsofcoherencebetweenthesovereignbondinGermany andwithoneinSpain,Greece,Ireland,ItalyandPortugal, respectively.

Fig.3 shows the wavelet coherence between the Ger-man and Spanish bond yields, following Eq. (6). In each

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Germany-Greece wavelet coherence

12Feb9327Oct93 11J ul94

23Mar95 05Dec95 16Aug9630Apr97 12Jan9824Sep98 08Jun9918Feb0001No v00

16Jul0128Mar02 10Dec02 22Aug0305Ma y04

17Jan0529Sep05 13Jun0623Feb0707No v07

FigureA.6 NormalizedCrossWaveletPowerSpectrum(coherence)ofsovereignbondevolutionsinGermanyandGreece.

Germany-Ireland wavelet coherence

FigureA.7 WaveletcoherencebetweenbothsovereignbondevolutionsinGermanyandIreland.

pointof the frequency-time map,the coherence resultis a value between 0 for dark blue color, and 1 for dark red color. A value of 1 for a given frequency band, indi-cates that the response energy is 100%, due to both signal enters in clear relationship. This suggests there are two behaviors of this bivariate set acting as a sys-tem: (a) A strong coherent behavior that happens when

the system clearly reacts in a correlated mode or there is a clear relationship between both signals. The strong coherence between January 1998 and April 2009 indi-cates a co-movement of both signals. (b) A weak or non-coherent behavior that occurs before January 1998 and after April 2009 when both signals evolve indepen-dently.

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Table1 Frequency bandsdetected in allcountries

ana-lyzedbywaveletspectrum.

Country Frequencybands Observation

Austria 363,304*,185,91,46*,28,18 *Weak Belgium 362,177,91,24,14 Finland 365,176,90,48,27,21 France 310,178,91,44,26,23* *Weak Germany 390,191,91,40,24,12 Greece 370,181,91,54,30(26)*16,13 *Uncertain Ireland 390,174,91,70*,46,30,16,14 *Weak Italy 368,190,90,37,24,14* *Uncertain Netherland 363,185,91,45,23,14 Portugal *,181,91,28,13 *Blurred Spain 362,173,91,41,23,14

4. Conclusion

Theuseofwavelettransformsallowedtodiscoverthemain

periodic features of the sovereign bonds of 11 countries

inEurope. Adetailedwaveletspectralanalysisrevealson

each countrydifferent frequency bands.Entropy provides

the central value of frequency in each band. The result

isthatverysimilarfrequencycomponentsareobservedin

allcountries.Inadditionofannualandsemiannualvalues,

Germany-Italy wavelet coherence

FigureA.8 WaveletcoherencebetweenbothsovereignbondevolutionsinGermanyandItaly.

thereis averyprecise quarterlyvalue thathappensinall

countries. Also, it is remarkable the frequencies periods

around45days,3-weeksand2-weekscycles.

Inthedetailedanalysisofthepossibleco-evolution

Ger-man versus Spanish sovereign bonds, a strong coherent

behavior of signals is unveiled by the wavelet

coher-ence analysis. The strong coherence between January

1998 and April 2009, shows the co-movement of both

signals or the period when both sovereign bond yields

are impacted by the same factors. The asynchronism,

between both signals, due to low wavelet coherence,

is observed before January 1998 and after April 2009

whenlikely differenteconomic factorsare independently

involved. The methodologyused in the example for

Ger-manyandSpaincanbeappliedtoanalyzeothercouplesof

countries.

Appendix

A. Graphic

results

This appendix reﬂects the main graphic results obtained.

Dataprocessing wascarried out by programming

applica-tionsinMatlablanguage(MathWorkCo.)andsomeroutines

codedinJavaLanguage,whichhavebeendesigned

specif-ically for this purpose. Also,system identiﬁcation library

and wavelet functions provided by the Matlab package

havebeenincludedtoimprovethealgorithmperformance

(11)

Germany-Portugal wavelet coherence

FigureA.9 WaveletcoherencebetweenbothsovereignbondevolutionsinGermanyandPortugal.

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