Interest rate changes and stock returns in Spain: A wavelet analysis

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www.elsevier.es/brq

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ARTICLE

Interest

rate

changes

and

stock

returns

in

Spain:

A

wavelet

analysis

Pablo

Moya-Martínez

a

_,

_Roman

_{Ferrer-Lape˜}

_na

b

_,

_Francisco

_{Escribano-Sotos}

a,∗

a_Department_of_Economic_and_Financial_Analysis,_University_of_Castilla-La_Mancha,_Plaza_de_la_Universidad_1,₀₂₀₇₁_Albacete,

Spain

b_Department_of_Actuarial_and_Financial_Economics,_University_of_Valencia,_Avenida_Tarongers,_s/n₄₆₀₂₂_Valencia,_Spain

Received12September2013;accepted28July2014 Availableonline26September2014

JEL CLASSIFICATION C14; E43; G12 KEYWORDS Wavelets; Interestrates; Stockreturns; Industry;

Haarátrouswavelet

Abstract Thispaperinvestigatestherelationshipbetweenchangesininterestratesandthe

SpanishstockmarketattheindustrylevelovertheperiodfromJanuary1993toDecember2012 usingawavelet-basedapproach.TheempiricalresultsindicatethatSpanishindustriesexhibit, ingeneral,asigniﬁcantinterestratesensitivity,althoughthedegreeofinterestrateexposure differsconsiderablyacrossindustriesanddependingonthetimehorizonunderconsideration.In particular,regulatedindustriessuchasUtilities,highlyindebtedindustriessuchasRealEstate, UtilitiesorTechnologyandTelecommunications,andtheBankingindustryemergeasthemost vulnerabletointerestrates.Further,thelinkbetweenmovementsininterestratesandindustry equityreturnsisstrongeratthecoarsestscales.Thisﬁndingisconsistentwiththeideathat investorswithlong-termhorizonsaremorelikelytofollowmacroeconomicfundamentals,such asinterestrates,intheirinvestmentdecisions.

Introduction

Understandingthelinkagebetweeninterestratesandstock pricesisacriticalissuetomanyareasofﬁnance,including assetallocation,portfoliomanagement,riskmanagement, assetpricingandmonetarypolicytransmission,anditmay

∗_{Corresponding} _author _at: _Facultad _de _Ciencias _Económicas _y

Empresariales de Albacete, Universidad de Castilla-La Mancha, PlazadelaUniversidad1,02071Albacete,Spain.

Tel.:+34969179100x8272;fax:+34967599216.

E-mailaddress:[email protected](F.Escribano-Sotos).

be, therefore, of special interest to investors, portfolio managers,corporatemanagersandpolicymakers.Thelink betweenchangesininterestratesandstockreturnsisbased onfinancial theory. Modern financial theory assumes that anyfirm generates a streamof futurecashflows andthe stockpriceofthatfirmisequaltothepresentvalueofall expectedfuturecashflows discounted at theappropriate discountrate.Interestratesaffectstockpricesthroughtwo primarychannels.First,movementsininterestrateshavea directeffectonthediscountrateusedinequityvaluation. Second, interest rate changes affect firms’ expectations about futurecash flowsby altering the costof financing,

http://dx.doi.org/10.1016/j.brq.2014.07.004

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96 P.Moya-Martínezetal.

mainlyinthehighlyindebtedcompanies. Consequently,it isexpectedthat interestrates willbea significant deter-minantofstockprices.In thisregard,accordingtosurvey evidencebyGrahamandHarvey(2001),interestrateriskis rankedbyU.S.firmmanagers asthesecondmostrelevant financialriskfactor,onlybehindcreditrisk.

Therelationship betweeninterestratefluctuationsand themarketvalueofcompanieshasreceivedagreatdealof attentionintheliterature,althoughmuchofthisresearch hasfocused on financialinstitutions asa large proportion ofincomeandexpensesof thesefirmsdirectly depend on interest rates. Nevertheless, interest rate variations may bealsoimportantfornon-financialcorporations,principally throughtheireffectontheborrowingcostsandthevalueof financialassetsandliabilitiesheldbythesecompaniesThe classical ordinary least squares (OLS) regression has been themostcommonapproachusedintheliteraturetoassess the relation between changes in interest rates and stock returns.Morerecent studieshave utilized,however,more sophisticatedtimeseriesmethodsinthetimedomain,such ascointegration, Granger causality,vector autoregressive (VAR) or generalized autoregressive conditional het-eroscedasticity(GARCH)models.Amajorlimitationofthe existingempiricalresearchisthatitisrestrictedtooneor atmosttwotimescales,i.e.theshortrunandthelongrun. Despiteremarkableadvancesinmodelingtechniques, lit-tle attention has been paid so far to the influence of a potentiallyrelevantfeaturesuchastheinvestmenthorizon on the interest rate-stock market link. Financial secu-ritymarkets suchasbondandstock markets arecomplex systems consisting of thousands of heterogeneous agents making decisionsover different timehorizons (from min-utestoyears),whocollectivelydetermineaggregatemarket behavior.Forexample,agentswithshortinvestment hori-zons,suchasdaytraders,aretypicallylinkedtospeculative activities andmake decisions largelybased onephemeral phenomena such as sporadic events (i.e. announcements of earning surprises, mergers), market sentiment or psy-chologicalfactors.In contrast,agents withlong horizons, such as big institutional investors, are more involved in investment activities and follow more closely macroeco-nomic fundamentalssuch asthe businesscycle, inflation, monetarypolicystance,etc.

Inthisway,itisnotunreasonabletothinkthattheextent ofconnectionbetweeninterestratesandstockpricesmay vary across time scales associated with different invest-ment horizons of market participants. In such a context, wherethestrengthanddirectionofrelationshipsbetween economicandﬁnancial variablesmaydifferwiththe time scale, waveletmethods appearasa veryappealing alter-native.Waveletanalysisisacomparativelynew,atleastin ﬁnance,andpowerfultoolforsignal processingthattakes intoaccountbothtimeandfrequencydomains.The essen-tialadvantageofwaveletsistheirabilitytodecomposeany signalintoitstimescalecomponents.1_This_signal

decompo-1_As_noted_by_Schleicher₍₂₀₀₂₎_,_the_wavelet_{decomposition}_of_a signalcanbecomparedto theactivity ofacamera-lens. Zoom-ingoutthelens bringsa broadlandscape,while zoominginthe lensallowstoﬁnddetailsthatarenotobservableinthelandscape portrait.

sitionpropertyprovidesan opportunitytostudyeconomic relationshipsonascale-by-scalebasis. Thus,itis possible toseparatethosetimehorizonsatwhichtheconnectionis statisticallysigniﬁcantfromthosehorizonsatwhichitisnot and,therefore,toobtainabroaderpictureascomparedto thetimedomainapproachwhichinfactaggregatesalltime horizonstogether.

The purpose of this paper is to examine the linkage between changes in interest rates and the Spanish stock market at the industry level across different investment horizons by using a wavelet approach. More specifically, three complementary wavelet-based tools are employed, namelywaveletvariance,correlationandcross-correlation. The centralresearchquestionof thisstudy iswhetherthe relationshipbetweeninterestratefluctuationsandindustry equityreturnsvaries acrossinvestmenthorizonsandifso, howitdoes.Apriori,weexpectthatthisrelationshipwill bestrongeratlonginvestmenthorizonsbecausemarket par-ticipantswithaverylongview(years)aremorelikelytobe interestedinmarketfundamentals,suchasinterestrates, thanagentswithshorterhorizons,whoaremainlyinfluenced bytransitoryfactors.

Thisworkcontributestothecurrentliteratureinseveral aspects.First,tothebestofourknowledge,thisstudyisthe firsttoinvestigateinterrelationshipsbetweenmovementsin interestratesandstockreturnsintheSpanishcaseat multi-pleinvestmenthorizonsbyusingthewaveletmethodology. In thisrespect, theSpanish equity marketoffers an ideal settingtoassesstheinterestrateexposuregiventhegreat importance of financial, regulated and heavily indebted firms,allofthembeingparticularlysensitivetointerestrate fluctuations,in thismarket. Second,noprevious research hasexaminedthelinkagebetweeninterestratesandstock returns at the industry level throughwavelettechniques. The analysisonan industrybasis is, however,appropriate because market aggregation may mask significant differ-encesamongindustriesintermsofinterestratesensitivity. Third,thisstudyisalsouniqueinthesensethatanovel dis-cretewavelettechniqueis applied,whichiscalled Haarà trouswavelettransform(hereafterreferredtoasHTW).The HTWwasintroducedbyMurtaghetal.(2004)andovercomes thedrawbacksoftraditionalwaveletfunctions.Asisargued byJammazi(2012b),themajorreasonbehindthechoiceof thisalternativewavelettransformisitseaseindealingwith the boundary problem without the need toremove data. Therefore,theHTWtransformensurestheconservationof thewholeinformationincludedintheoriginalsignals, mak-ingit possibletobettercharacterize therealinteractions betweeninterestratechangesandstockreturns.

Our results indicate that the Spanish equity market exhibits a remarkable degree of interest rate sensitiv-ity, although sizeable differences can be observed across industriesand depending onthetimehorizonconsidered. Unsurprisingly,regulatedindustriessuchasUtilities,heavily indebted industriessuch asRealEstate, Utilitiesor Tech-nology and Telecommunications, along with the Banking industryemergeasthemostinterestratesensitive.Onthe contrary,thereisawiderangeofindustriessuchas Chem-icals and Paper, Financial Services, Construction, Health Care,andIndustrialshardlyinﬂuencedbyinterestrates.Itis alsofoundthattheinterestrate-stockmarketlinkbecomes more pronounced at longer investmenthorizons (low

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fre-quencies),suggestingthattheroleofinterestratesasakey drivingfactorofstockmarketperformancemaybeheldonly inthelongrunforsomespecificindustries.Asexpected,the interestratesensitivityispredominantlynegative, indicat-ingthatSpanishfirmsare,onaverage,adverselyimpacted byinterestraterises.Inaddition,asignificantbidirectional relationshipbetweenchangesininterestratesandindustry returnsisfoundatlongerhorizons.

The remainder of the paper is structured as follows. Literature review section 2 briefly reviews the relevant relatedliterature.Datasetsectionintroducesthewavelet methodologyandthewavelettoolsusedtocharacterizethe relationship between changes in interestrates and indus-tryreturns.Econometricmethodologysectiondescribesthe datasetemployed.Empiricalfindingsarediscussedin Empir-ical findings section. Finally, Concluding remarks section concludesthepaper.

Literature

review

The relationship between changes in interest rates and stock returns has given rise to a prolific research activ-ityover the pastfew decades.The bulk ofthis literature has concentrated on the Bankingindustry becauseof the particularly interest rate sensitive nature of the financial intermediationbusiness.Specifically,thematuritymismatch ordurationgapbetweenbanks’financialassetsand liabili-tiesresultingfromthematuritytransformationfunctionof banking firms (i.e. the financing of long-term loans with short-termdeposits)hasbeenusuallyidentifiedasthemain cause of the interest rate sensitivity of banks (Ballester etal.,2011;Czajaetal.,2009,2010).Thispositive dura-tion gap (the average duration of banks’ assets exceeds theaveragedurationof banks’liabilities)impliesthat ris-ing interest rates have an adverse effectonthe value of banks and vice versa. This occurs because when interest rates increase, the present value of banks’ assets falls more than that of banks’ liabilities and also the costs of banks’ liabilities rise faster than the yields on banks’ assets.

However,movements ininterest ratesmay alsohave a significanteffectonthevalueofnon-financialcorporations through several channels. First, within the framework of present value models hikes in interest rates increase the costof capital for firms,which translatesinto higher discount rates for future cash flow valuation, thereby adversely affecting firms’ stock prices. Second, interest rate rises increase the interest expense of leveraged companiesandcanalsoreducethedemandforproductsby heavilyindebtedconsumers,whichmeanslowercorporate profitsandhas,inturn,anegativeimpactonshareprices. Third,interestrate fluctuationsalterthe marketvalueof financial assets and liabilitiesheld by non-financial firms. Finally,movementsininterestratesaffecttheopportunity cost of equity investments. Higher interest rates make bonds more attractive as an alternative to hold stocks, whichleadsinvestorstorebalance theirportfoliosby buy-ingbondsandsellingstocks,thus depressingstock prices. All these effects suggest an inverse relationship between interestratefluctuationsandstockreturns.Nevertheless,it isalsopossibletofindapositivecorrelationbetweenthese

variablesifinterestrates andequitymarketsmove inthe samedirectionfollowingchangesinmacroeconomicfactors suchasinﬂationaryexpectationsoreconomicprospects.

A broad consensus emerges from this body of litera-tureregardingseveralrelevantissues.Firstly,theempirical researchinthisareahastraditionallyprovidedevidenceof asignificant negativerelationship between movements in interestratesandstockreturns ofbothfinancialand non-financialcompanies(DinenisandStaikouras,1998;Lyngeand Zumwalt, 1980; Prasad and Rajan, 1995). However, some morerecent studies, suchas thoseof Czaja etal. (2009)

andKorkeamäki (2011), show that the impact of interest ratefluctuationsonequityreturnshasdeclined over time primarilyduetotheincreasedavailabilityofimprovedtools formanaging interest raterisk. Inparticular, the extraor-dinarygrowthin interest ratederivative markets andthe expansionofcorporatebondmarketsmayhaveplayedakey roleinthiscontext.Secondly,stockreturnstendtobemore sensitivetomovementsinlong-terminterestratesthanto short-termrates(Bartram,2002;Czajaetal.,2009;Ferrer etal., 2010;Oertmannetal.,2000).Thirdly,nonfinancial firmsin regulated and/or highly indebted industriessuch asUtilities, RealEstate, BasicResources, and Technology andTelecommunications arecommonly recognized asthe mostinterest rate sensitive (Bartram,2002; Reilly etal., 2007;Sweeney andWarga, 1986). Two basic reasons help toexplainthisresult.First,profitsand,hence,stockprices ofheavilyindebtedcorporationsarestronglydependenton interestratedevelopmentsinsofarasthecostoftheirdebt isdirectlyrelatedtothelevelofinterestrates.Second, reg-ulatedcompaniessuchasutilitiesadjustthepricesoftheir productsandserviceswithsomelagbehindcostincreases duetotheconstraints imposedbyregulators.Thisrigidity inpricescontributestostrengtheningthenegativeimpact ofinterestraterises onsharepricesof thesefirmsasthe increasedfinancialcostsarenotoffsetbyahigherincome. Sofar,theempiricalliteratureonthelinkbetween inter-est rates and stock prices has developed primarily in the timedomainbyusingabroadrangeoftimeseriesmethods, includingOLS regression (Korkeamäki, 2011; Reilly etal., 2007;SweeneyandWarga,1986),VARtechniques(Campbell andAmmer, 1993; Laopodis,2010), cointegration analysis (Chan etal., 1997;Das, 2005;Hatemi-J andRoca, 2008), Granger causality tests (Alaganar and Bhar, 2003; Panda, 2008;Shahetal.,2012),non-linearmodels(Ballesteretal., 2011; Bartram, 2002), and GARCH-type models (Elyasiani andMansur,1998;Faffetal.,2005;Kasmanetal.,2011).

Nevertheless,thereareafewrecent papersexamining theinterest rate-equitymarket nexus through the use of wavelet techniques (C¸ifter and Ozun, 2008; Hamrita and Triﬁ, 2011; Kim and In, 2007; Tiwari, 2013). These stud-iesfocuson aggregatestock markets of variouscountries andshowthat theconnection between interestrates and sharepricesisdependentonthetimehorizon,increasingin importanceatlongerhorizons.Theseauthors,withtheonly exceptionofTiwari(2013),employavariantofthediscrete wavelettransformknownasthemaximaloverlap discrete wavelettransform(hereafterreferredtoasMODWT).This transformisthemostwidelyusedwaveletineconomic appli-cationsduetovariousattractivefeatures.In particular,it canhandle anysample size(not restricted toa power of two),itistranslationorshiftinvariantasashiftinthesignal

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doesnotchangethepatternofwaveletcoefficients,andit providesincreasedresolutionatcoarsertimescales.Despite this,asnotedbyJammazi(2012b),theMODWTalsohassome drawbacks.First,itisaffectedbyboundaryeffectsthatarise fromapplyingthewavelettransformneartheedgeoffinite signalsduetothelackofdatabeyondtheboundary.Since thenumberofboundaryelementsincreaseswithscalethere willbemanymoreboundary-affectedcoefficientsathigher scales.Theboundaryproblemsmayleadtobiasedestimates ofthe waveletvariance and, hence,tospurious and mis-leadingresults(PercivalandWalden,2000).Inordertoget anunbiasedestimatorof thewaveletvariance,the exclu-sion of all the wavelet coefficients affected by boundary conditionshasbecome a verycommon practice.This loss of edge information inherent to the MODWT may, there-fore,make it difficult thecorrect characterizationof the realinteractionsbetweenvariables,especiallyatthe coars-est scales. Second, the MODWT is typically implemented withthe Daubechies least asymmetric (LA) wavelet filter oflengthL=8, denotedby LA(8),asitallowsan accurate alignmentintimebetween waveletcoefficientsat various scalesandtheoriginalsignal.However,theLA(8)filterisnot appropriatetocapturethenon-linearandchaoticbehavior typicalofmanyeconomicandfinancialtimeseries(i.e.oil price,inflation,stockreturns,etc.).

The present study differs from the above mentioned papersintheuseoftheHTWtransform,anewdiscreteand robust wavelettechnique that overcomes the major con-straintsoftheMODWTandhasbeenrarelyappliedtodate ineconomic andﬁnanceareas.Infact,JammaziandAloui (2010)andJammazi(2012a,b)aretheonlyones whohave usedtheHTWwithinaﬁnancialframework,particularlywith thepurposeofexploringthedynamicrelationshipbetween oilpricechangesandstockreturns.

Mostoftheliteratureontheinterestrate-stockmarket linkhasfocusedonafew countrieswithhighlydeveloped financial markets, especially the U.S. and more recently Germany, the U.K. or Australia. Concerning the Spanish case,therearesomestudies (Ferreretal.,2010; Jareño, 2008;Sotoetal.,2005),basedprimarilyonmultifactor lin-earregressionmodels,whichdocumentasignificantlinkage betweeninterestratemovementsandfirms’stockreturns, confirmingthehighinterestratesensitivityoftheSpanish equitymarket.

Data

set

Thisstudydealswiththerelationshipbetweenmovements ininterestratesandstockreturnsintheSpanishcaseover theperiodfromJanuary1993toDecember2012.The start-ingdate of our analysisis January 1993 toavoid possible distortionsintheinterestrate-equitymarketnexuscaused bytheturbulencesinﬁnancialmarketsinthecontextofthe crisisof theEuropeanmonetary systemduring thesecond half of 1992. Along the lines of, amongothers, Campbell (1987), Kim and In (2007), Korkeamäki (2011) and Reilly etal.(2007),monthlydataseriesareemployed(atotalof 240observations). Monthly data(end-of-the month obser-vations)arepreferredto weeklyor daily datafor several reasons. Firstly, monthly data are less contaminated by noiseandcanthereforebettercaptureinteractionsbetween

interest rates and equity prices. Secondly, monthly data havesmallerbiasesduetonon-synchronoustradingofsome stocks.Thirdly,theresultsintermsofsmoothnessand differ-entiationamongtimehorizonsgeneratedbytheapplication of wavelet analysis on monthly data are much harder to obtain with higher frequency data. In fact, it would be necessarytouseaverylargenumberofdecomposition lev-elswhenconsidering weeklyor dailydatain ordertoﬁnd comparableresultstothoseachievedwithmonthlydatain termsofthetimerangecovered.

Inlinewithpreviousresearchontheinterestrate-stock marketlink(Bartram,2002;Ferreretal.,2010;Reillyetal., 2007;SweeneyandWarga,1986),thisanalysisiscarriedout attheindustrylevel.Variousreasonsareusuallyputforward tojustify anindustry-basedapproach.First,theformation ofindustryportfoliosprovidesanefficientwayof condens-ing asizable amount ofinformation regardingstock price behavior.Second,theuseofportfolioshelpstosmooththe noiseinthedataproduced bytransitoryshocksin individ-ualstocks,whichleadstomorepreciseestimates.Thus,all firmslistedontheSpanishStockExchangeforatleastone fullyearofthesampleperiod(atotalof249companies)are assignedtoanyoftheindustriesconsidered. Then, value-weightedindustrystockindicesareconstructedfromstock prices,adjustedforsplitsanddividends,ofindividualfirms withineachindustryportfolio.

The fourteenindustries coveredare: ConsumerGoods, Consumer Services, Technology and Telecommunications, RealEstate,Banking,FinancialServices,Utilities, Construc-tion, Chemicals and Paper, Basic Resources, Health Care, FoodandBeverages,Industrials, andEnergy.Thisindustry breakdownissimilartothewell-knownIndustry Classifica-tionBenchmark(ICB),butadaptedtothesingularfeatures of theSpanishequity market.Forexample,the Construc-tionandRealEstateindustrieshavebeenaddedsincethey wereoneofthemajorgrowthenginesoftheSpanish econ-omyovertheperiod1996---2007,representingalmost18%of SpanishGDPin2007.Theuseoftheofficialindustry classifi-cationoftheSpanishStockExchangehasbeendiscardedin thisstudyfortwomainreasons.First,theSpanishindustrial classificationhasundergoneseveralrestructuringprocesses overthelasttwodecades,withJanuary2005beingthemost recent.Thisimpliesthattherearenosufficientlylongtime series of industry stock indices for conducting a credible empiricalanalysiswithmonthly data.Second,thecurrent industry classification of the Spanish market only distin-guishes six basic industriesand,therefore, it seemsmore appropriatetousemoredisaggregatedindustryleveldata. TheIndiceGeneraldelaBolsadeMadrid(IGBM),the broad-est index of the Spanish equity market, is utilized asan indicator of the stock exchange as a whole. Equity mar-ket data are collected from the Madrid Stock Exchange database.

Interestratesusedinthisstudyaretheyieldson10-year Spanishgovernmentbonds,whichhavebeentakenfromthe BankofSpain’swebsite.Thischoicehasbecomeincreasingly popular in the literatureon the linkage between interest ratesandstockmarket(Ballesteretal.,2011;Elyasianiand Mansur,1998;Faffetal.,2005;Oertmannetal.,2000)andis justiﬁedforseveralreasons.First,long-terminterestrates containmarketexpectationsaboutfutureprospectsforthe economyanddeterminetoalarge extentthecostof

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bor-rowing funds. Thus, long-term rates are likely to have a criticalinfluenceoninvestmentdecisionsandprofitabilityof firmsand,hence,ontheirstockmarketperformance. Sec-ond,long-term governmentbondsareoften consideredas closermaturitysubstitutestostocks,whichmaypresumably increase the extent of linkage between the two financial assets.Industryreturns arecalculatedasthefirstlog dif-ferenceofindustrystockindices.Changesininterestrates arecomputedasthefirstdifferencesinthelevelofinterest ratesbetweentwoconsecutiveobservations.

Table1presentsdescriptivestatisticsfor thedata.The averagemonthlyreturnispositiveformostindustriesaswell astheoverallstockmarketinlinewiththegeneral increas-ing trend in stock prices. The average monthly change in 10-year government bondyields is, however, negative, reflecting theclear downward trend in Spanishlong-term ratesduringthesampleperiod.Basedonthestandard devi-ation,all industryand marketreturns have, asexpected, highervolatilitythantheseriesofchangesin10-year Span-ishbondyields.Themeasuresofskewnessindicatethatthe majorityofindustryreturnsarenegativelyskewed,meaning thatnegativeshocksaremorecommonthanpositiveones. Furthermore, all industry return series exhibit a kurtosis significantlylarger thanthree, therebyimplying leptokur-ticbehaviorascomparedtotheGaussiandistribution.The Jarque---Berateststatisticscorroboratethisfinding, reject-ingthe nullhypothesisof normalityin allcasesat the1% level. This result is consistent with that reportedfor the Spanish stock market by Miralles et al. (2011). A similar distributionalpictureemergesfor the10-yeargovernment bond yield change series. The Augmented Dickey---Fuller (ADF)andPhillips---Perron(PP)unitroottestsindicatethat theseriesofchangesin10-yearbondratesandmarketand industryequityreturnsareallstationary(integratedoforder zero).Thisfindingisinlinewiththatofearlierworkon finan-cialreturn data (Badillo et al., 2010; Czaja et al.,2009; Mirallesetal.,2012).

Fig.1presentsthedynamicsoftheSpanishequity mar-ket index,proxied by the IGBM,and the yieldon Spanish 10-yeargovernmentbondsovertheperiod1993---2012.The stockmarketexhibits ageneralupwardtrendduringmost ofthestudy period,only interruptedbytheInternet bub-bleburstinMarch2000andtheglobalﬁnancialcrisisfrom late2007. On the contrary, theyields on10-year govern-ment securitiesdisplay adownward trendupto thestart of the sovereign debt crisis in the euro zone during the spring of 2010. This ﬁgure shows that the interest rate-stockmarketlinkoverthefull sampleperiodis somewhat unclear.Inparticular,theSpanishmarketindexand10-year governmentbondyieldshavemovedpredominantlyin oppo-site directions until approximatelymid-1998. Since then, theconnectionbetweenbothvariablesismoreambiguous, although apositive correlation seemstoexistfor mostof the2000s.

Econometric

methodology

Waveletanalysis:basicconsiderations

Waveletanalysisisatechniqueforsignaldecompositionthat wasborninthe1980sandovercomesthemajorlimitations

oftheFouriertransform.Thewavelettransformcombines information from both time and frequency domains and, therefore, preserves time information. Moreover, it does not require the stationarity of signals. This greater ﬂex-ibility has fostered the rapid expansion of the wavelet approachindisciplinessuchasgeophysics,engineering, cli-matologyormedicine,althoughitsapplicationineconomics and ﬁnance is only a relatively novel phenomenon (i.e.

Genc¸ayetal.,2002;RamseyandLampart,1998a,b).Given itsabilitytopartitionanysignalintocomponentsassociated todifferenttime scales, waveletanalysis providesa sim-pleand intuitive wayto study interactionsbetween time serieson a scale-by-scale basis, helping to uncover rela-tionshipsthatstandardtimeseriesmethodsarenotableto detect.

Waveletsaresmallwavesthatgrowanddecayinalimited timeperiod.Thewavelettransformisbasedontwospecial functions:thefatherwavelet(orscalingfunction)˚(t)and themotherwavelet(t).Basedonthemotherwavelet,a familyofdaughterwavelets, u,s(t), canbegenerated by simplyscalingandtranslating:

u,s(t)= 1 √ s t−u s (1)

where s is a scaling or dilation parameter that controls the length of the wavelet, while u is a location parame-terwhich determinesitspositioninthetimedomain.The fatherwaveletintegratestooneandisgoodatrepresenting thesmoothandlow-frequencypartsofasignal,whereasthe motherwaveletintegratestozeroandisusefulincapturing thedetailandhigh-frequencycomponents.

Thereare twotypes ofwavelettransform: the contin-uouswavelettransform(CWT)anditsdiscretecounterpart (DWT).TheDWTisacompactrepresentationofthedataand isparticularlysuitablefornoisereductionanddata compres-sion, whereas the CWT is better for feature extraction purposes.Foralongtime,waveletapplicationsineconomics haveconcentrated onthe DWTdue toitsgreater simplic-ityandmoreparsimoniousnature.TheDWTproducesonly theminimalnumberofcoefﬁcientsnecessarytoreconstruct theoriginalsignal.This reductionisachievedby discretiz-ingtheparameters u ands, sothat u=k2−j _and_s₌₂−j_, where j and k are integers representing the set of dis-cretedilatations andtranslations.Inthe DWTthenumber ofobservationshastobedyadic,i.e. aninteger powerof two.

ByapplyingtheDWT, anytimeseriescanbeprojected ontoasequenceoffatherandmotherwaveletsfroma spe-ciﬁc family. A number of discrete wavelet families have proventheirusefulnessinﬁnancialdataanalysis,for exam-pletheHaar,Daubechies, Morlet,Mexican Hat,Symmlets, etc.Thus,atimeseriesXt canberepresentedasalinear

combinationofwaveletfunctionsasfollows:

Xt= k sJ,k˚J,k(t)+ k wJ,kJ,k(t)+···+ k w1,k1,k(t) (2)

where J is the number of multiresolutionlevels or scales andkrangesfrom1tothenumberof coefﬁcientsineach level.

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Table1 Sampledescriptivestatisticsofindustryequityreturnsand10-yearSpanishgovernmentbondyieldchanges.January1993---December2012.

Returnsandfactors Mean Median Min. Max. Std.dev. Skewness Kurtosis JBstatistic ADFstatistic PPstatistic Consumergoods 0.0094 0.0101 ₋0.2173 0.2039 0.0659 ₋0.54** _4.16*** _24.85*** ₋_15.70*** ₋_15.71***

Consumerservices 0.0027 0.0072 −0.3154 0.2395 0.0643 −0.75*** _6.05*** _115.45*** ₋_15.18*** ₋_15.22***

Technologyandtelecom 0.0070 0.0047 −0.2659 0.3121 0.0804 0.03 4.58*** _24.93*** ₋_15.88*** ₋_15.87***

RealEstate 0.0011 0.0017 −0.3147 0.4255 0.0909 0.62** _6.63*** _146.56*** ₋_7.06*** ₋_14.78***

Banking 0.0033 0.0105 −0.2860 0.2884 0.0791 −0.56** _5.26*** _63.26*** ₋_14.25*** ₋_14.26***

Financialservices −0.0004 0.0000 −0.2544 0.2811 0.0420 0.13 17.31*** _2040.95*** ₋_12.87*** ₋_12.89***

Utilities 0.0043 0.0020 −0.1607 0.1404 0.0336 −0.16 9.71*** _448.93*** ₋_14.17*** ₋_14.33***

Construction 0.0020 0.0022 ₋0.1667 0.2013 0.0371 ₋0.20 9.26*** _392.32*** ₋_6.92*** ₋_14.44***

Chemicalsandpaper −0.0031 −0.0040 −0.3279 0.1699 0.0549 −0.63** _8.34*** _299.28*** ₋_14.61*** ₋_14.60***

Basicresources ₋0.0025 0.0000 ₋0.5683 0.4427 0.0826 ₋0.89*** _20.91*** _3224.90*** ₋_13.28*** ₋_13.30***

Healthcare −0.0032 0.0000 −0.3477 0.2520 0.0559 −1.45*** _17.31*** _2122.92*** ₋_6.77*** ₋_16.44***

Foodandbeverages 0.0058 0.0029 −0.1558 0.2125 0.0405 0.68*** _8.25*** _293.37*** ₋_14.52*** ₋_14.52***

Industrials 0.0049 0.0007 −0.2560 0.3394 0.0577 0.41* _9.29*** _401.41*** ₋_15.81*** ₋_15.91***

Energy −0.0058 0.0041 −0.4924 0.2560 0.0751 −1.57*** _12.80*** _1054.36*** ₋_13.77*** ₋_14.12***

Stockmarketreturn 0.0053 0.0117 ₋0.2492 0.1964 0.0613 ₋0.58** _4.37*** _32.33*** ₋_14.42*** ₋_14.48***

10-yearbondyieldchanges −0.0003 −0.0005 −0.0121 0.0104 0.0032 −0.15 4.94*** _38.36*** ₋_14.71*** ₋_14.93***

Notes:Thetablepresentssomedescriptivestatisticsofthemonthlyindustryequityreturnsand10-yeargovernmentbondyieldchanges,includingmean,median,standarddeviation(Std. dev.),minimum(Min.)andmaximum(Max.)valuesandalsoskewnessandkurtosismeasures.JBdenotesthestatisticoftheJarque---Beratestfornormality.Thelasttwocolumnspresent theresultsoftheaugmentedDickey---Fuller(ADF)andPhillips---Perron(PP)unitroottests,respectively.

* _Statistical_{significance}_at_10%. **_Statistical_{significance}_at_5%. *** _Statistical_{significance}_at_1%_levels.

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Spanish equity market index (IGBM) 10-year spanish govemment bond 1800 1600 1400 1200 1000 800 600 400 200 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 19931994 19951996199719981999 200020012002200320042005200620072008200920102011 2012

Figure1 EvolutionoftheSpanishequitymarketindex(IGBM)andthe10-yearSpanishgovernmentbondyieldfromJanuary1993 toDecember2012.

sJ,kandwj,k arescalingor smoothanddetailorwavelet coefﬁcients,respectively,andaregivenby:

sJ,k= X(t)˚J,k(t)dt (3) wj,k= X(t)j,k(t)dt j=1,2,...,j (4)

wj,k coefficientscapture the high frequencycontentof thetimeseries,whilesJ,kcoefficientsrepresentthesmooth behavioroftheseries.Themagnitudeofthesecoefficients offersa measureofthe contributionof thecorresponding waveletfunctiontothetotalseries.

Thus,theoriginaltimeseriescanbereconstructedas:

Xt=SJ(t)+WJ(t)+WJ−1(t)+···+W1(t) (5) where SJ= ksJ,k˚J,k(t) and Wj= kwj,k j,k(t) for j=

1,...,J are the smooth (or approximation) and detail (or wavelet)componentsofthesignal,respectively.This recon-structionisknownasmultiresolutionanalysis.

In practice, the DWT is implemented via the pyramid algorithmderivedbyMallat(1989).Thistechniqueconsists ofrecursivelyapplyingahigh-passfilter,whichisbasedon themotherwavelet,anditscounterpartlow-passfilterto a given timeseries. The high-pass filtercorresponds to a differencing operation and extracts the detail (high fre-quency)informationofthesignal,whilethelow-passfilter is associated to an averaging operation and extracts the coarse(lowfrequency)information.Specifically,theoriginal XtseriesisdecomposedintoapproximationS1(t)anddetail

W1(t) componentsby convolving the series withthe low-passandhigh-passﬁlters,respectively.The approximation componentS1(t) becomestheinput for thenextiteration

step,sothattwonewapproximationS2(t)anddetailW2(t) componentsareobtained.Thisrecursiveprocedureis con-tinueduntilthedecompositionlevelJisreached.Despiteits greatpopularity,thestandardDWThasseveraldrawbacks. First,itissubjecttoboundaryeffects,whichmayproduce unreliableresultsintheedgeregions.Second,itisnotshift invariant.Third,itrequirestimeserieswithadyadiclength.

TheHaaràtrouswavelet(HTW)transform

Inthisstudy,theHTWtransformdevelopedbyMurtaghetal. (2004)is appliedasanalternativetotraditionalDWT.The HTWcombinestheàtrouswavelet,whichisatypeof redun-dant wavelettransform, with the Haar wavelet, which is oneof thesimplestwavelets.The HTWtransformoffersa number of advantages over standard wavelet transforms. First, the use of the Haar wavelet function respects the asymmetricnatureofthetime-varyingsignal,sothat scal-ingandwaveletcoefficientsarecalculatedonlyfromdata obtainedpreviouslyintime.Therefore,theHTWdoesnot sufferfromboundaryproblemscausedbytheapplicationof waveletanalysis onfinitedatasets. This absenceofedge effectsintheHTWtransformallowstheconservationofthe wholeinformationcontainedintheoriginalseries.Second, theHTWtransformisflexibleenoughtoaccuratelyreflect thenon-linearandchaoticdynamicsofmanyfinancialtime series.Third,theredundancyinherentintheàtrouswavelet functionimpliesthatallwaveletcomponentshavethesame lengthastheoriginaltimeseries,soitiseasytorelate infor-mationateachresolutionscaleforthesametimepoint.This

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propertyofredundancyofalsomeansthattheHTWisshift invariant.2

The HTW is a redundant version of the popular Haar wavelettransform thatusesanon-symmetriclow-pass ﬁl-terhequalto1₂,12

andcanbedescribedasfollows.The scalingcoefﬁcientssj+1,kofatimeseriesXtatanyscalecan beobtainedbyconvolvingthesmoothedversionofthesignal atthepreviousscalewiththelow-passﬁlterh=1

2, 1 2 : sj+1,k= 1 2(sj,k−1+sj,k) (6)

The detailcoefﬁcientscan becalculatedasthe differ-encebetweenthe smoothedversions ofthe signal at two consecutivescales:

wj+1,k=sj,k−sj+1,k (7)

HTW-basedvariance,correlationand cross-correlation

The wavelet coefﬁcients can be manipulated to obtain several statistical wavelet-based tools, such as the waveletvariance, waveletcorrelation,andwavelet cross-correlation,whichprovideanalternativerepresentationof thevariabilityandassociationstructurebetweentimeseries on a scale-by-scale basis. These three tools are deﬁned analogously to the usual variance, correlation and cross-correlationmeasuresintimeseriesanalysis.

Thewaveletvariancedecomposesthevarianceofatime seriesonascale-by-scalebasisandhelpstoidentifywhat time scales are the dominant contributors tothe overall variabilityoftheseries.Asnotedby Gallegati(2008),the waveletvarianceatscalej ofastationarystochastic pro-cessXt,2_X(j),isgivenby:

2

X(j)=

1 2jvar

(ωj,t) (8)

wherej=2j−1andwj,tdenotethewaveletcoefﬁcientsof Xtatscalej.

AccordingtoJammazi(2012b),theHTW-basedvariance estimatorcanbealsoexpressedintermsofthenormalized sumofthesquaredwaveletcoefﬁcients:

ˆ 2 X,HTW(j)= 1 2j_N j Nj−1 t=0 ˜ ω2 j,t= 1 N Nj−1 t=0 ˜ ω2 j,t (9) whereˆ2

X,HTW(j)istheestimatedwaveletvarianceatscale

j,Nj=N/2jthenumberofwaveletcoefﬁcientsatthe reso-lutionleveljandNthesamplesize.SincetheHTWtransform isnotaffected byboundaryconditions,the HTWvariance estimatorallowstoanalyzeasignalbyusingallthewavelet coefﬁcients.

The HTW framework also enables us to derive the waveletcorrelation and cross-correlation. These wavelet-based tools make it possible to quantify the degree of associationbetweentwostochasticprocessesona scale-by-scalebasis.Analogouslytotheusualcorrelationcoefﬁcient intimeseriesanalysis,thewaveletcorrelation coefﬁcient

2_For_a_more_complete_and_detailed_discussion_of_the_Haar_à_trous wavelettransform,seeMurtaghetal.(2004)andJammazi(2012b).

at scale j, X,Y(j),can becalculated asthe ratioof the waveletcovariancebetweenXtandYt,denotedbyX,Y(j), andthesquarerootoftheirwaveletvariances:

ˆ X,Y(j)= ˆ X,Y(j) ˆ X(j)ˆY(j) (10) TheHTW-correlationatscalejcanbealsoexpressedin termsofwaveletcoefﬁcients:

ˆ HTW X,Y (j)= 1 N Nj−1 t=0 ˜ wX j,tw˜j,tY (11)

Justastheclassical cross-correlation is usedto deter-minelead/lagrelationshipsbetweentwotimeseriesinthe time-domain,thewaveletcross-correlation gives informa-tiononlead/lagrelationshipsonascale-by-scalebasis.The HTWcross-correlationcoefﬁcientestimatorforscalejand timelag,denotedbyˆHTW

X,Y,(j),canbecalculatedas:

ˆ HTW X,Y,(j)= ˆ X,Y,(j) ˆ X(j)ˆY(j) = 1 N Nj−1 t=0 ˜ wX j,tw˜j,tY+ (12)

Empirical

ﬁndings

EstimatedHTW-basedvariance,correlationand cross-correlation

This sectionexamines thelinkagebetweenchangesin 10-year Spanish governmentbond yields and industry equity returnsatdifferenttimehorizonsbyusingtheHTW trans-form with6 timescales.3 _Each _scale _is _associated_with _a particulartimeperiod.Thus,sincemonthlydataareused, thescale1representsthehighestfrequencyandis associ-atedwithatimehorizonfrom2to4months.Inturn,scales 2to6correspondto4---8,8---16,16---32, 32---64and64---128 monthperiods,respectively.

Fig.2illustratesthewaveletvariancesofindustryreturns and changesin 10-year bondrates derived fromthe HTW decomposition.The straightlinesrepresentthe estimated wavelet variances at different time horizons. Two things arenoteworthyfromthisfigure.First,thegreatest variabil-ityof alltimeseriesis observedat shortertimehorizons, suggesting that movementsin industryequity returns and 10-year interest yields are mainly caused by short-term fluctuations.Thereisanapproximatenegativelinear rela-tionship between the wavelet variance and the scale, so thattheHTW-basedvariancesof10-yearbondyieldchanges and industry returns decline as the scale increases. This decreasingtrendinvariabilityindicatesthatinvestorswith short-termhorizonsfacehigherrisksthantheirlonger hori-zon counterparts. Second, as expected, the HTW-based variancesofallindustrystockreturnsarelargerthanthose of 10-year bondratechanges over all timehorizons.This findingisconsistentwiththatreportedbyKimandIn(2007)

3_All_computations_of_the_HTW_algorithm_in_this_study_have_been performedbyusingtheExcelutilitydevelopedbyForeTrade Tech-nologiesbasedontheworkofMurtaghetal.(2004)andavailable atthewebsitehttp://www.foretrade.com/Wavelet.htm.This soft-warehasbeenemployedinanumberofpreviousstudiessuchas thoseofJammazi(2012a,b)andJammaziandAloui(2010).

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0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03 6.0E-03 7.0E-03

Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6

Wa v e le t va ri a n ce

Time scales (Investment horizons)

Interest Rate Changes 10-years (x100) Consumer Goods Consumer Services Technology & Telecom Real Estate Banking Financial Services Utilities Construction Chemicals & Paper Basic Resources Health Care Food & Beverages Industrials Energy

Figure2 HTW-basedvariance ofchanges in10-yearSpanish governmentbondyieldsandindustryequityreturns.Notes:This ﬁgurereportsthewaveletvariancesofSpanishindustryequityreturnsandchangesin10-yearSpanishgovernmentbondyieldsfor thedifferenttimehorizonsorscalesestimatedwiththeHaaràtrouswavelet(HTW)transform.

for the G7 countries,conﬁrming that thestock marketis morevolatilethanthelong-termpublicdebtmarket regard-lessoftheinvestmenthorizon.Further,allindustryreturns exhibitasimilarbehaviorintermsofwaveletvariances.

The results of the waveletcorrelation analysisfor the pairscomposedofchangesin10-yeargovernmentbondrates andeachofindustryreturnsoverthesixtimehorizonsare displayed in Fig. 3 The estimated HTW-based correlation coefficientsarerepresentedbyasolidline.The95% confi-denceintervalsarealsoshownthroughdashedlinesandthey permitreliablestatisticalinferenceastowhetherthe corre-lationsaresignificantlydifferentfromzero.4_It_is_found_that

4_Given_the_inherent_{non-normality}_of_the_correlation_coefﬁcient forsmallsamplesizes,theconﬁdenceintervalsfortheHTW-based correlationand cross-correlationhavebeenconstructed byusing

thewaveletcontemporaneouscorrelationbetween10-year bondyieldfluctuationsandindustryreturnsvariesaccording tothe timehorizonconsidered, suggesting thatthe inter-estrate-stockmarketlinkisamultiscalephenomenon.The highestlevel (inabsolute value) ofHTW-correlationtends tobedetectedatlongerhorizons,indicatingthatlongtime horizonsarethemost important contributorstothe asso-ciationbetweenyieldson10-yearbondsandstockmarket performance.Infact,thewaveletcorrelationisnot signifi-cantlydifferentfromzeroatshorterhorizons,i.e.scales1 and2,foralargenumberofindustries.Onthecontrary,the HTW-basedcorrelationissignificantatlongerhorizons,i.e.

theFisher’stransformation.Throughthistransformation,the cor-relationcoefﬁcienttendstobecomenormalquicklyasthesample sizeincreases.

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104 P.Moya-Martínezetal. Consumer goods 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 0.8 0.4 0.0 –0.4 –0.8 1 2 3 4 5 6 1 2 3 4 5 6 Consumer serv.

Technology & telecom Real estate

Banking Financial services

Utilities Construction

Chemicals & paper Basic resources

Health care Food & beverages

Industrials Time horizon W a velet correlation Energy

Figure3 HTW-basedcorrelationbetween10-yearSpanishgovernmentbondyieldchangesandindustryequityreturns.Notes: EstimatesoftheHTW-basedcorrelationcoefﬁcientsbetweenchangesin10-yeargovernmentbondyieldsandeachofindustryequity returnsatthedifferenttimehorizonsareshownwithasolidline.The95%conﬁdenceintervalsaround thewaveletcorrelation estimatesconstructedbyusingtheFisher’stransformationaredrawnwithdashedlines.

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scales5and6,formostindustries.Thisresultisinlinewith thatofC¸ifterandOzun(2008),HamritaandTriﬁ(2011)and

KimandIn(2007)fordifferentcountriesalsousingwavelet methodology.

Moreover, the wavelet correlation results reveal that there is considerable heterogeneity among industries in termsof thedegreeofconnectionbetween movementsin 10-yearbondratesandstockreturns.Inparticular,Utilities, FoodandBeverages,RealEstate,Technologyand Telecom-munications,andBankingemergeasthoseindustriesmost closelylinkedtochangesin 10-yearbondyields.This evi-dence supports the commonly held view that regulated, heavily indebtedand banking sectors arethe most inter-est rate sensitive. Again, this is consistent with earlier work focused on different countries and periods of time andusingavarietyofmethodologies(Bartram,2002;Reilly et al., 2007; Sweeney and Warga, 1986). In contrast, a broadgroupofindustriessuchasFinancialServices,Health Care, Consumer Goods, Chemicals and Paper, Industrials, andConstructionareidentifiedassectorshardlylinkedto 10-yearbondyield movements.Thisfindingis not surpris-ingsincefirmsintheseindustriesaretypicallyperceivedas moresensitive tootherrisk factorssuchasbusinesscycle fluctuations.

ThesignoftheHTW-basedcorrelationismostlynegative, implying that Spanish firmstend to benefit frominterest ratefalls. This is due tothe factthat reductions in long-term interest rates lead to a decrease in the borrowing costsofcompanies,withtheconsequentpositiveeffecton their profits and share prices. In addition, lowerinterest ratesincreasetheattractivenessofstocksasaninvestment comparedwithbondsandmightinduceashiftintostocks, pushing up equity prices. This finding is also in keeping withpriorresearchconductedinotherdevelopedcountries (Kim andIn,2007; Korkeamäki,2011;Reillyetal., 2007). Nevertheless,asignificantpositivecontemporaneous corre-lationbetweenchangesin10-yearbondyieldsandindustry returns is observedat longer horizonsfor some industries such asHealth Care,Basic Resources, and Chemicals and Paper.Onepossibleexplanationisrelatedtothepro-cyclical natureofthesesectors.Itiswellknownthatthelong-term stockmarketperformanceofpro-cyclicalindustriesishighly dependentoneconomicgrowth.Inturn,inenvironmentsof lowinterestrates,suchasthatinforcesincethelate1990s, increasesininterestratescanbeassociatedwithan improv-ingeconomicsituation.Therefore,itisnotsurprisingthat 10-yeargovernmentbondratesandequityreturnsofthese industriesmovein tandemoverlong timehorizons mainly drivenbytheeconomicoutlook.

Sincethewaveletcorrelationdoesnottakeintoaccount possible lagged interest rate effects on industry returns or vice versa, the wavelet cross-correlation is calculated toshed light onthe lead-lagrelationships between these variables at different investment horizons. Fig. 4 plots the estimated HTW-based cross-correlation coefﬁcients between10-yearbondyieldﬂuctuationsattimetand indus-tryreturnsattimet−andt+upto24-monthtimelags forthesixtimehorizons.5_The_{corresponding}_approximate

5_Due_to_space_restrictions_and_given_that_the_HTW-based cross-correlationanalysisshowsasimilarpatternforallindustries,Fig.4

conﬁdence intervals at 95% level are also displayed in ordertoassessthestatisticalsigniﬁcanceoftheestimated waveletcross-correlation.

SimilarlytotheHTWcontemporaneouscorrelation,the estimatedHTWcross-correlationshowsthatthemagnitude of the association between changes in yields on 10-year bonds and industry returns increases with the time hori-zon.At shorterhorizons, i.e.scales 1and 2,the lead-lag linkbetweenboth variablesis insignificantforvirtuallyall leadsandlagsregardlessoftheindustry.However,the cross-correlation dynamics becomes more apparent at longer horizons.The waveletcross-correlation coefficientsreveal asignificantbidirectionalrelationshipbetweenmovements in10-yearbondyieldsandindustryreturnsatscales5and6 formostindustries.Inparticular,thelagsofindustryreturns arestatistically significant,therebymeaningthat industry returnsleadchangesin10-yearbondrates,andtheleadsof industryreturnsarealsosignificant,whichimpliesthat 10-yearbondyieldfluctuationsleadindustryreturns.Further, theleadingaswellasthelagging periodsincreaseasdoes thetimescale.

Overall,theresultsofthewaveletcorrelationand cross-correlationindicatethattheinterestrate-stockmarketlink isnotfixedovervarioustimehorizons,butitincreaseswith thetimeframe,aresultwelldocumentedintheempirical literatureonthisissue(ÇifterandOzun,2008;Hamritaand Trifi,2011;Kim andIn, 2007).The presenceof astronger connectionbetween10-yearbondratechangesandindustry returnsastheinvestmenthorizonincreaseshasa convinc-ingexplanation.At shorterhorizons, thelinkage between thesevariablesispracticallynon-existentastheshort-term dynamicsof stock returns is mainly driven by ephemeral phenomena such as sporadic events, changes in market sentiment,andpsychologicalfactors(Zhou,2012).In con-trast,atlongerhorizonstherelationshipbecomesstronger asmacroeconomicvariablessuchasinterestrates exerta morepredictable influence onthe stock market.In addi-tion,long-termbondsandequitiesareclosesubstitutesfor investorswithlonghorizons,soit seemsnatural thatthey movetogetherandinfluenceeachotheroverlongtime hori-zons.

Robustnesschecks

Two robustnesstests are performed to checkthe validity ofthe above results. Inthe first test, the wavelet analy-sishas been repeated by replacing the yields on 10-year governmentbonds with a shorter-term interest rate such astheyieldon1-yearTreasurybills.Forthevast majority of industriesthe HTW-based correlation coefficients have lowerstatisticalsignificance,irrespectiveofthetime hori-zon considered, than those obtained from 10-year bond rates. Moreover, the HTW cross-correlation estimated by using1-yearTreasurybillyieldstendstobelowerin mag-nitudeand statistical significance over the different time

only displays the cross-correlation coefﬁcients corresponding to Utilities,Banking,andFoodandBeveragesindustries.Thefullset ofresultsfromthewaveletcross-correlationanalysisisavailable fromtheauthorsuponrequest.

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a

Utilities Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Time lag W a velet cross-correlation 20 10 0 –10 –20 20 10 0 –10 –20 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8

Figure4 HTW-based cross-correlationbetween 10-yearSpanish governmentbond yieldchangesand industryequityreturns. Notes:EstimatesoftheHTW-basedcross-correlationcoefﬁcientsbetweenchangesin10-yeargovernmentbondyieldsandindustry equityreturnsagainsttimeleadsandlagsforthesixtimehorizonsconsideredarerepresentedbyasolidline.Therangeofleads/lags is24months.Thecross-correlationiscalculatedbyshiftingthesecondtimeseriesinthepair(inthiscasetheindustryequityreturn). The95%conﬁdenceintervalsaroundthewaveletcross-correlationestimatesconstructedbyusingtheFisher’stransformationare drawnwithdashedlines.

horizonsthanthat previouslyprovidedusing10-year bond yields.Theseﬁndingssupportthewidespreadviewthatthe interestrate-stock market link is closerwhen considering long-terminterestrates.6

In the second robustness check, a multiscale Granger causalitytestisperformed inordertovalidatetheresults ofthewaveletcross-correlation.Thistestinvestigatesthe causalrelationshipsintheGrangersensebetweenchangesin yieldson10-yearbondsandindustryreturnsatvarioustime horizons.ItconsistsofapplyingaGrangercausalitytestona scale-by-scalebasisbetweenthewaveletdetailsofchanges

6_For_the_sake_of_brevity,_Figures_which_report_the_results_from theserobustnesstestsarenotpresentedhere,bytheyareavailable uponrequestfromtheauthors.

in10-yearbondratesandindustryreturns.Tothisend,the followingVARmodeloforderKisestimated:

Dj,tIR =ˇ01+ K k=1 ˇ(11j)Dj,tIR−k+ K k=1 ˇ(12j)Dj,tIND−k+u (j) 1,t (13) Dj,tIND=ˇ02+ K k=1 ˇ(21j)DIRj,t−k+ K k=1 ˇ(22j)DINDj,t−k+u (j) 2,t (14) where DIR

j,t is the HTW-based detail component of the seriesofchangesin10-yearbondyieldsatscalej,DINDj,t isthe HTW-baseddetailcomponentoftheseriesofreturnsofthe industryconsideredat scalej.The numberoflags ofthe model, K, ischosen according tothe Schwarzinformation criterionandAkaikeinformationcriterion.

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b

Banking Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Time lag W a velet cross-correlation 20 10 0 –10 –20 20 10 0 –10 –20 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 Figure4 (Continued)

In ordertoinvestigatewhether industryequityreturns Granger cause movements in 10-year government bond yields,wetestwhetherthenullhypothesisthatallof the

ˇ(12j) parametersarezerocanbestatisticallyrejected.And viceversa,inordertoascertainwhetherchangesin10-year bondrates Granger cause industryequity returns we test whetherthenullhypothesisthatalloftheˇ(21j) parameters arezerocanbestatisticallyrejected.Bothnullhypotheses aretestedbyusingaF-test.

The findings ofthe Grangercausality testsat different timehorizons,reportedin Table2,arebroadlyconsistent withthoseofthewaveletanalysisoutlinedaboveintermsof themultiscalenatureoftheinterestrate-stockmarketlink andtheindustrieswithhigherandlowerlevelsofconnection with10-yeargovernmentbondrates.Thus,theevidenceof Grangercausalityis ingeneralquiteweakat shorter hori-zons.However,thecausalityisdramaticallystrengthenedat longhorizons.Significant bidirectionalcausalrelationships between movements in 10-year bond yields and industry returns are found at longer horizons for most industries, confirmingthattheassociationbetween10-yearbondyield

ﬂuctuations and Spanish stock market increases with the investmenthorizon.

Concluding

remarks

This paper investigates the interrelationships between changesin10-year governmentbondyieldsandthe Span-ishstock marketfromanindustryperspectiveacross time andfrequencybyusingawaveletapproach.Because ﬁnan-cialmarketsarecomposedofavarietyofagentswhohave differentinvestmenthorizons,itispostulatedthatthe inter-estrate-stockmarketlinkmaydifferacrosstimehorizons or scales.Wavelet analysis is basedon decomposing time seriesintodifferenttimescalesandprovidesanatural plat-formforcharacterizinginteractionsbetweentimeserieson ascale-by-scalebasis.Speciﬁcally,theHaaràtrouswavelet transform, a novel technique proposed by Murtaghet al. (2004) which overcomes the main limitations of conven-tional wavelet transforms, is applied. In order to gain a deeperunderstandingoftherelationshipbetweeninterest

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c

Food & beverages

Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Time lag W a velet cross-correlation 20 10 0 –10 –20 20 10 0 –10 –20 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 Figure4 (Continued).

rate changes and industry equity returns, three wavelet-based tools are implemented, namely wavelet variance, correlationandcross-correlation.

The empirical results show that the Spanish stock marketistightlyconnectedwiththedevelopmentof long-terminterest rates, although theextent of linkage varies markedly across industries and depending on the time horizonconsidered. On the one hand, Utilities,Food and Beverages,RealEstate, and Bankingareidentified asthe industriesmost closely linked tochanges in 10-year bond rates.Incontrast,long-termbondyieldsseemtohavevery littleinfluenceonthestockmarketperformanceofabroad rangeofindustriessuchasHealthCare,FinancialServices, ChemicalsandPaper,Construction,andIndustrials.This het-erogeneouspattern of connection is in line with previous researchin this field (Bartram, 2002; Reilly et al., 2007; SweeneyandWarga,1986).

Moreover, the relationship between movements in 10-year bondrates and industryreturns is a function of the measurementhorizon.Thenexus betweenthesevariables isweakat shortertimehorizons(high frequencies),butit

becomesstronger at longerhorizons(low frequencies). In this regard, significant feedback effects between 10-year bond yield fluctuations andindustry returns are detected onlyatlongerhorizons.Thisfindingagreeswiththenotion thatinvestorswithlong-termhorizons(i.e.pensionfunds, mutual fundsor insurance companies) aremore likelyto followmacroeconomicfundamentals,suchasinterestrates, intheirinvestmentdecisionsthaninvestorswithashorter time frame such as speculative traders. Therefore, the widelyheldbeliefthatinterestratesrepresentamajor driv-ingforceofstockmarketperformanceappearstobetruein theSpanishcaseonlyoverlongtimehorizonsandforsome industries.Asexpected,thesignoftheinterestrate-stock market link is primarily negative, suggesting that Spanish companies are,on average, benefited by falls in interest rates.

Thehorizon-dependenceofthelinkagebetween move-mentsin10-yearbondyieldsandindustryreturnsindicates that time horizon-conscious investorsshould consider the variationsinthedegreeofconnectionbetweenthese varia-blesforportfolioselectionandriskmanagementpurposes.

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Table2 Granger causalitytestsbetweenchangesin10-yeargovernmentbondyieldsandindustryequityreturnsatvarious timehorizons.

Directionofcausality Timehorizons

Scale1 Scale2 Scale3 Scale4 Scale5 Scale6 Changesin10-yearbondrates→consumergoods 0.74 0.82 0.51 0.09 5.96*** _8.03***

Consumergoods→changesin10-yearbondrates 0.88 1.54 0.548 1.81 3.21** _4.65**

Changesin10-yearbondrates_→consumerservices 2.19* _0.93 _1.20 _0.27 _1.20 _3.41**

Consumerservices→changesin10-yearbondrates 1.69 2.40** _2.96** _1.90 _3.91*** _4.60***

Changesin10-yearbondrates_→Technologyandtelecom. 1.99* _0.83 _1.21 _3.26*** _1.92 _8.55***

Technologyandtelecom→changesin10-yearbondrates 1.05 0.65 0.81 3.71*** _1.43 _3.66***

Changesin10-yearbondrates_→RealEstate 3.01** _0.95 _3.54*** _1.43 _3.01** _2.60**

RealEstate→changesin10-yearbondrates 2.37** _1.68 _1.56 _3.35** _3.99** _0.66

Changesin10-yearbondrates→Banking 0.13 1.30 0.98 1.69 0.66 4.49***

Banking→changesin10-yearbondrates 0.93 0.79 0.32 1.92* _0.28 _3.38***

Changesin10-yearbondrates→ﬁnancialservices 1.00 0.56 0.63 0.76 0.59 2.70**

FinancialServices_→changesin10-yearbondrates 1.35 0.90 0.59 0.03 0.58 3.46***

Changesin10-yearbondrates→Utilities 1.49 1.45 2.48** _2.31* _2.92** _2.97**

Utilities_→changesin10-yearbondrates 2.51** _2.01* _1.97* _2.64** _3.40** _2.57**

Changesin10-yearbondrates→construction 0.46 0.57 1.52 1.14 5.98*** _3.64**

Construction→changesin10-yearbondrates 1.43 0.95 1.46 0.72 5.01*** _3.84***

Changesin10-yearbondrates→chemicalsandpaper 0.89 1.79 0.78 2.28** _4.94*** _2.70**

Chemicalsandpaper→changesin10-yearbondrates 1.63 1.60 1.13 0.94 4.44*** _3.79***

Changesin10-yearbondrates→basicresources 0.17 0.88 0.65 1.23 2.74** _0.60

Basicresources→changesin10-yearbondrates 2.98** _2.59** _2.80** _1.55 _3.43*** _4.28***

Changesin10-yearbondrates_→healthcare 0.18 0.27 1.26 1.50 1.26 1.32

Healthcare→changesin10-yearbondrates 0.60 1.20 1.73 0.71 0.86 0.78

Changesin10-yearbondrates_→foodandbeverages 0.42 1.29 3.09*** _3.10*** _2.88** _4.14***

Foodandbeverages→changesin10-yearbondrates 4.05*** _3.22*** _3.95*** _4.81*** _0.98 _0.43

Changesin10-yearbondrates→industrials 0.49 0.10 1.79 2.05 3.24** _4.22**

Industrials→changesin10-yearbondrates 2.22* _1.29 _1.17 _0.62 _1.65 _4.84***

Changesin10-yearbondrates→energy 1.98 0.70 2.79** _4.32** _4.54** _9.58***

Energy_→changesin10-yearbondrates 1.50 2.82** _1.64 _4.93*** _1.78 _7.53***

Notes:ThistablereportstheresultsofthemultiscaleGrangercausalitytestsbetweenchangesin10-yearSpanishgovernmentbond yieldsandindustryequityreturnsatdifferenttimehorizons.Scale1representstheﬁnesttimehorizonandcapturesdynamicswitha periodlengthof2---4months.Scales2,3,4,5and6capturedynamicswithaperiodof4---8,8---16,16---32,32---64and64---128months, respectively.

* _Statistical_{significance}_at_10%. ** _Statistical_{significance}_at_5%. *** _Statistical_{significance}_at_1%_levels.

In fact, the effectiveness of portfolio selection and risk managementstrategiescouldbesigniﬁcantlyenhancedby incorporatingthehorizon-dependentlinkbetweeninterest rates and equity market. In addition, the evidence pre-sented here clearlyshows that the traditionalcorrelation coefﬁcientinthetimedomaincannotbeusedasa univer-sal measure of thedegree of linkage between changesin interestratesandstockreturns.

Acknowledgements

TheauthorswishtoexpresstheirsinceregratitudetoRania Jammazi for their valuable comments and suggestions to improve thispaper. The authorswish toacknowledge the commentsof twoeditorsof the BRQjournal,Lucio Fuen-telsaz and Susana Menéndez, as well as two anonymous

referees.TheauthorsarealsoindebtedtoJoseMiguelFerrer forhisdetailedandcriticalreadingofthepaper.

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