Stock return distribution in the BRICS

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This is the published version of a paper published in Review of Development Finance.

Citation for the original published paper (version of record):

Adu, G., Alagidede, P., Karimu, A. (2015)

Stock return distribution in the BRICS.

Review of Development Finance, 5(2): 98-109

http://dx.doi.org/10.1016/j.rdf.2015.09.002

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ScienceDirect

ReviewofDevelopmentFinance5(2015)98–109

Stock

return

distribution

in

the

BRICS

George

Adu

a

,

Paul

Alagidede

b

,

Amin

Karimu

c,∗

a_Department_of_Economics,_Kwame_Nkrumah_University_of_Science_and_Technology,_Kumasi,_Ghana b_Wits_Business_School,_University_of_the_{Witwatersrand,}_{Johannesburg,}_South_Africa c_Department_of_Economics,_Umeå_School_of_Business_and_Economics,_Umeå_University,_Sweden

Abstract

Stockreturnsinemergingmarketeconomiesexhibitpatternsthatare distinctivelydifferentfromdevelopedcountries:returnsarenotedto

behighlyvolatileandautocorrelated,andlonghorizonreturnsarepredictable.Whilethesestylizedfactsarewellestablished,theassumption

underlyingthedistributionofreturnsislessunderstood.Inparticular,theempiricalliteraturecontinuestorelyonthenormalityassumptionasa

startingpoint,andmostassetpricingmodelstendtooverstretchthispoint.Thispaperquestionstherationalebehindthissuppositionandproceedsto

testmoreformallyfornormalityusingmultivariatejointtestforskewnessandkurtosis.Additionally,thepaperextendstheliteraturebyexamininga

numberofempiricalregularitiesforBrazil,Russia,India,ChinaandSouthAfrica(theBRICSforshort).Ourmainfindingsarethatthedistribution

ofstockreturnsfortheBRICSexhibitspeakednesswithfatterandlongertails,andthisisinvarianttoboththeunitofmeasurementandthetime

horizonofreturns.Volatilityclusteringisprevalentinallmarkets,andthisdecaysexponentiallyforallbutBrazil.Therelationshipbetweenrisk

andreturnisfoundtobesignificantandriskpremiumsareprevalentinoursample.

JELclassiﬁcation: F23;F36;G10;G30

Keywords: Normality;Returnpredictability;Leverageeffect;Volatilityclustering;Efficiency;Emergingmarkets

1. Introduction

SinceGoldmanSachseconomist,JimO’neilcoinedtheterm BRICintheearly2000s,theeconomiesofBrazil,Russia,India andChinahavetaken centrestageinboththe global politics and economics. In 2010, South Africa joined the club, offi-cially spreading the tentacles of the largestemerging market economiesoverfourcontinents.By2013,theBRICSaccounted for almost 3 billion of the world’s population, with a com-binednominalGDPofUS$16.039trillion.AboutUS$4trillion of foreignreservesare heldbythe BRICS,withChinaalone accountingformorethanaquarter.

∗_{Corresponding}_author._Tel.:₊₄₆₉₀₇₈₆₈₀_14.

E-mailaddresses:gadu.cass@knust.edu.gh,gyadu2011@gmail.com

(G.Adu),paul.alagidede@wits.ac.za,alagidede@gmail.com(P.Alagidede),

amin.karimu@econ.umu.se(A.Karimu).

PeerreviewunderresponsibilityofAfricagrowthInstitute.

Overthe past fewyears, theperformanceof BRICS stock marketshasbeensterling.DatafromReuters(2012)showsthat viewedina10yearhorizon,theMorganStanleyCapital Interna-tional(MSCI)BRICindexreturnedastriking450%,compared to the 320% and98% returns on other emerging and devel-opedmarketsrespectively.Between2001and2007theMSCI’s BRIC index returnedover 500%,significantly outperforming otheremergingmarkets.However,recentevidenceshowsthat thehaydaysmaybeoversoon.Therehavebeenslumpsinthe mostrecentperiodwithlossesof8.6%inthepastfiveyearsin dollarterms.TherearealsoindicationsthatChina’simpressive doubledigitgrowthspurtisfading.BrazilandSouthAfrica’s growthhasbeenanaemic,andRussiafacesproblemsintheoil andgassectorwhilereformsinIndiahavebeensluggish.The volatilityingrowthratesandstockmarketperformanceraises importantquestionspertinenttoinvestments,portfolio diversi-ficationandtheoverallroleoftheBRICSinglobaleconomic growth.WilltheBRICSassetscontinuetoreceivetheattention theyhaveenjoyedoverthepastdecade?Whatisthenatureofthe riskreturnrelationshipinthesemarkets?Questionssuchasthese

http://dx.doi.org/10.1016/j.rdf.2015.09.002

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botheronthedistributionalpatterns,volatilityandpredictability ofstockreturnsaswellastheefficiencyoftheBRICS.This arti-cleconcernsitselfwithreturndistribution,andonthetimeseries propertiesofstockreturns.Thepaperextendstheliteraturein twodirections.

Thefirst ismethodological.Standard assetpricing models suchas themeanvariancemodel takesthe normalityofasset returnsasgiven.Althoughthisassumptionhasbeenpointedout tobehighlyunrealistic(seeMandelbroit,1963;Rachev,2003) evenfor developedmarkets,a significantamount of research continuetofocusonthenormalityofreturnsasastartingpoint. Thisisnotsurprisingsincethecomputationalintensity under-lyingalternativedistributionsisbothtimeconsumingandmore daunting.Moreso,thepropertiesofthenormaldistributionare sufficientlywellknownandstudiedintheliterature.However, theconsequenceofrelyingonmodelsofnormalreturnsmaylead tosignificantunderestimationoftheriskofinvestinginemerging markets,particularlyifthedistributionisskewedandfattailed. Thispaper thus questions the over reliance on the normality assumptionthatexistintheextantliteratureonthedistribution ofreturnsinemergingmarkets.Departingfromextantliterature weemployamultivariateskewnessandkurtosistestofMardia

(1970)andthe jointskewness andkurtosis testofHenze and

Zirkler(1990).

The second contribution is to extend the literature on the peculiaritiesofassetreturnsintheBRICS.Anumberofresearch effortshavebeenexpendedinunderstandingthereturn distribu-tionofemergingmarketsgenerally,however,alotremainstobe learnedabouttheBRICSstockmarketsefficiencyinallocating scarceresources.Moreover,themostcomprehensivestudy of thereturndistributionofemergingmarketsappearednearlytwo decadesago(Bekaertetal.,1998).Ontheempiricalregularities emergingmarketsarenotedtohavelow/ornegativecorrelations withthemoredevelopedworld(seeHarvey,1995;Alagidede, 2010); emerging market economies offer returns that exceed industrial-marketreturns(Buckberg,1995;Reuters,2012).Both ofthesefactssuggestthatunexploitedprofitopportunitiesmay exist.Atthesametime,emergingmarketreturnstendtoexhibit highvolatilityandautocorrelation,longrunpredictabilityand generallylow levels of liquidity (Bekaert and Harvey, 1997;

Aggarwaletal.,1999;Kasmanetal.,2009;Blitzetal.,2013;

HullandMcGroarty,2014).Thesestylizedfeaturesmaysignal

market inefficiencyandopportunities for profitablearbitrage. Understandingthe dynamic behaviour of stockreturns inthe BRICS iscrucial for portfolio managers, policy makers, and researchers. We contribute to this strand of the literature by accountingforreturndynamicsindifferenttimehorizons and currencies.

1.1. StylizedfactsofBRICSstockmarkets

Thekeyfacts aboutBRICSstockmarketsare indicatedin

Table1.Forthesakeofbrevity,andinlinewithdataavailability,

theWorldDevelopmentIndicatorsforthestockmarketvariables areonlyreportedfor2012.Themarketcapitalization,turnover ratioandtradingvalueareallexpressedasapercentageofGDP. Marketcapitalizationisthesharepricemultipliedbythenumber

ofsharesoutstanding,anditisaroughbenchmarkforjudginga company’snetworth.Theturnoverratioisderivedbydividing thevalueoftotalsharesbythemarketcapitalization.Whilethe totalvaluetradedratiocapturestradingrelativetothesizeofthe economy,andtheturnoverratiomeasurestradingrelativetothe sizeofthestockmarket.Inpractice,theturnoverratioproxies theliquidityofthemarket:highturnoverisanindicatoroflow transactioncosts.

FromTable1,nosingleBRICScountrydominatesintermsof

allindicators,unsurprisinglyconfirmingthediversityofdepth, performance and influence of the national stock exchanges. Judgedbymarketcapitalization,theChinesemarketstandsout. The Shenzen, ShanghaiandHongKong stockmarketshada marketcapitalisationof$3.7trilliondollarsattheendof2012. TheShenzenstockexchangeisoverwhelminglydominatedby state owned enterpriseswhichare the back bone of the Chi-neseeconomy,whiletheShanghaiisnotfullyopenedtoforeign investors.BrazilandIndiahavemarketcapitalisationofabout $1.2and$1.3 trillionas of 2012,respectively. Inthe BRICS, RussiaandSouthAfricaarethesmallestmarketsusingthis indi-catorat$874billionand$612billion,respectively.Inrelationto thesizeofthedomesticeconomy,however,SouthAfrica dom-inatesasseenfromTable1.Thesizeofthestockmarketasa proportionofGDPisawhopping159%.Thisgivesahighvalue of sharestradedas proportionofGDPinSouthAfrica(81%) thananycountryinTable1.Interestingly,China’sstockmarket is44%ofGDP,slightlybiggerthantheRussian43%butless thanBrazils54%andIndia’s68%.WiththeexceptionofSouth AfricaandChina,totalvaluetradedasashareofGDPisless than40%asof2012.

Thenumberoflisteddomesticcompaniesamountedto5191 inIndiain2012.Thisisabout15timesthenumberof compa-niesinBrazilandSouthAfricaandabout19timesthenumber of domesticcompanieslistedinRussia’sstockmarket.China comes second with 2494 companies.The most liquid of the BRICS stock markets is China (164%), followed by Russia (87%)andBrazil(67%).IndiaandSouthAfricahaveaturnover ratioofabout54%.

2. Empiricalstrategyanddata

The analysisof thedatafor thisstudyfollows threesteps. First we examine thenature of theprobabilitydistributionof theindexreturnseriesfortheBRICSmeasuredinbothUS dol-larsandlocalcurrencyandfordifferentholdingperiods:daily, weeklyandmonthly.Whilethisanalysisisanendinitself,italso offers important informationrelevant forselecting the appro-priatestatistical modelforperforminginferenceonthereturn generatingprocess.Toachievethisaim,weemploytheMardia

(1970)skewnessandkurtosis,andHenzeandZirkler(1990)test

forjointskewnessandkurtosis.

Testsandestimatesbasedonthesamplemeanvectorand sam-plecovariancematrixhavebeenshowntohavepoorefficiency propertieswhenheavytailednoisedistributions arepresentin

adataset.Mardia(1970,1974and1980)pioneeredmeasures

of skewnessandkurtosis, anddemonstratedthatfunctionsof thethirdandfourthmomentsareasymptoticallydistributedas

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Table1

FactsaboutBRICSstockmarkets(2012). Marketcapitalization (CurrentUS$) Marketcapitalization (%ofGDP) Listeddomestic companies

Totalvaluetraded (%ofGDP) Turnoverratio(%) Brazil $1.229trn 54.60 353 37.05 67.88 Russia $874bn 43.41 276 36.34 87.64 India $1.26trn 68.60 5191 33.80 54.63 China $3.69trn 44.94 2494 70.82 164.44 SouthAfrica $612bn 159.33 348 81.13 54.93

Source:CompiledfromWorldDevelopmentIndicatorsoftheWorldBank(Onlineversion).

chi-squareandstandardnormal.Bygeneralizingtheunivariate skewnessandkurtosistoamultivariatesetting,Mardia(1970)

demonstratedthatthepowerofthetestissufficientlyimproved. Intheunivariatecase,standardizedthirdandfourthmomentsb1,

b2areoftenusedtoindicatetheskewnessandkurtosis.Fora ran-domsamplex1,...,xnfromap-variatedistributionwithsample

meanvector¯xandsamplecovariancematrixS,Mardiadefined the p-variatestatistic as b1,p=avei,j[(xi−¯x)TS−1(xj−¯x)]

3

and b2,p =avei[(xi−¯x)TS−1(xi−¯x)]

2

respectively. The statistics b1,p andb2,p arefunctionsofthe standardizedthird and fourth moments respectively. Thus b1,p and b2,p are invariant under affine transformation. In the univariate case these reduce to the usual univariate skewness and kurtosis statistics b1 andb2.Mardiaadvocateusing the skewnessand kurtosistotestformultinormalityastheyaredistributionfree undernormality.Theskewnessstatisticisthusdecomposedas

b1,p =6 j<k<1 [avei{zijzikzil}]2+3 j=/k [avei{z2ijzik}] 2 + j [avei{z3ij}]

andthekurtosisissimilarly

b2,p =

j=/k

avei{z2ijz2ik}+

avei{z4ij}

Under multinormality, b1,p and b2,p are asymptotically

normal and asymptotically independent and consequently the limiting distributions of n(b1,p/6) and √n((b2,p− p(p+2))/√8p(p+2)) are a chi-square distribution with

p(p+1)(p+2)/6andaN(0,1)distributionrespectively. However,despite its widespread use thestatistic hasbeen found wanting in distinguishing well between ‘skewed’ and ‘non-skewed’distributions(seeMecklinandMundfrom,2005). Thusitispossibletocombinethisstatisticintoanomnibustest toimprovethepowerofthetest.Consistentandinvarianttests proposedbyEppsandPulley(1983)areanexample.TheEpps

andPulley(1983)statisticisbasedon

T =

_∞

−∞|Φn(t)− ˆΦ0(t)| 2

dG(t)

whereΦn(t)istheempiricalcharacteristicfunction, ˆΦ0(t)isan

estimateof the characteristicfunctionof thenormal distribu-tion,and,G(t)isaweightfunction.HenzeandZirkler (1990)

proposed a multivariate extension to the test statistic above, namely Dnβ= d|Φn(t)− ˆΦ0| 2 ϕβ(t)dt

whereΦn(t)istheempiricalcharacteristicfunctionofthe

stan-dardized observations, ˆΦ0(t)isthecharacteristicfunctionofa multivariate standardnormal distribution, and ϕβ(t) is a

ker-nelfunction.HenzeandZirkler(1990)usethedensityfunction ofaNp(0,β2Ip)randomvector(β ∈ )inderivingtheirtest

statistic andtheyshow that the teststatistic hasa lognormal asymptoticdistributionandderiveaclosedformexpressionfor

Dnβ.UsingvariousvaluesofβHenzeandZirkler(1990)

con-ductedasimulationstudytocomparetheirstatisticwithothers, including Mardia’s (1970)multivariate measuresof skewness andkurtosis.Thechoiceofβ=0.5inasimulationexercisecan produce apowerful testagainst alternative distributions with heavytails,andincomparisonwithotherkindsofdistributions: independent marginals, mixtures of normal distributions,and sphericallysymmetric distributions,the multivariate jointtest for skewnessandkurtosis isnotedtohavegoodpower prop-erties, and morereliable than thoseobtained from univariate descriptivestatistics(seeHenzeandZirkler,1990).

Thesecondstepinvolvestestingforthepresenceof autore-gressiveconditionalheteroscedasticity(ARCH)inthemeanof theindexreturnsfortheBRICS.Again,whilesthepresenceof ARCHeffectinthemean of thereturnseriesofferimportant informationaboutthereturnbehaviourandefficiencyof mar-kets,italsooffersvitalinformationinselectingtheappropriate modelforthereturngeneratingprocess.

Inthethirdstep,weestimateastatisticalmodelthat appro-priatelycapturestheessentialfeaturesofthereturngenerating process.

The rests of this section is devoted to the description of the econometric or estimationtechnique adopted, andabrief descriptionofourdata.Tobeginwith,weassumethatthelogof thestockpriceindexfollowsarandomwalkwithadrift.That is:

lnPt =μ+lnPt−1+εt (1)

Eq.(1)impliesthefollowingexpressionforthereturn gen-eratingprocess(RGP):

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Inmanyapplications,itturnsoutthattheeffectof positive newstendstodifferfromnegativenewsofequalsize,onaverage. Toaccountforthis,weadopttheexponentialgeneralized autore-gressiveconditionalheteroscedasticityinthemean (EGARCH-M) of the return series. We also augment the standard EGARCH-MspecificationwithARMA(p,q)model.Thespecific functionalformofthemodeladoptedforthisstudyis:

Rt =μ+ p i=1 βiRt−i+ q j=1 θjεt−j+ϕht+εt (3) ε |ηt−1 ≈t·d(0,ht,vt) (4)

Withtheabovespecification,Rt isthemarketreturnandμ

istheexpectedreturn,ht istheriskpremium(volatility),εt is

theerrortermthat weassumetohaveastudent-Tdistribution duetotheskewnessandexcesskurtosis ofeachofthe series, whilepandqaretheoptimallagordersfortheautoregressive (AR)andmovingaverage(MA)termsrespectively.The inclu-sionoftheriskpremiumterminthemeanEq.(4)istomodel therelationshipbetweenexpectedreturnandrisk.Asignificant andpositiveϕimpliesthatexpectedreturnispositivelyrelated toriskandanegativerelationifthereverseistrue.

Additionally, to account for possible asymmetry of news onreturn-riskrelationship, we applythe exponential general-izedautoregressiveconditional heteroscedasticity(EGARCH) modelproposedbyNelson(1991),inwhichthevolatility(ht)

aspresentedinEq.(3)isexpressas;

ln(ht)=ω+α1zt−1+γ1(|zt−1|−E(|zt−1|)+δ1ln(ht−1) (5)

Thisspecification is much more flexiblesince we do not need to explicitly impose non-negativity restrictions on the parametersintheARCHandGARCHtermsinthemodel.The reasonisthatthelogtransformationensuresthatthevarianceis non-negativeandthereforefreefromtheproblemsofpossible negativevarianceasinthecaseofthestandardGARCHmodels. Theα1istheparameterthatcaptureleverageeffect,implying thatasignificantpositiveα1isanindicationofpositive innova-tionshavingmoreeffectsonreturnsthannegativeinnovations,

γ1isthesymmetrytermandδ1theGARCHterm,whileωisthe constantterminthevolatilityequation.

2.1. Datasourcesanddescriptions

ThedataemployedinthispaperistheMorganStanleyCapital International(MSCI)indices foremergingmarkets,focussing mainlyonBrazil,Russia,India, ChinaandSouthAfrica. We usetheMCSIWorldIndexasaproxyfortheWorldportfolio. TheMSCIdataforemergingmarketscaptureslargeand mid-capitalisationstocksanditcoversroughly85%ofthefree float-adjustedmarketcapitalizationineachcountry.Thedataisfrom theperiodJanuary1995toMay2014.Forcompletenessweuse daily,weeklyandmonthlydatareportedinbothUSdollarsand localcurrencyvalues.Thisallowsustocomparethebehaviour of returns across different time horizons and to examine the impactofexchangeratevariationsonstockreturns.Alltheseries underconsiderationare obtained fromThomson DataStream.

FollowingPukthuanthongandRoll(2009),wefilterreturnsto purgeholidaysandnontradingdays.

3. Emergingmarketreturndistribution

Inthissectionweexaminethedistributionofthereturnseries. Firstwe lookatthesummarystatistics fromthereturnseries. Nextwetestforthe presenceofskewnessandkurtosisinthe returns(bothinUSdollarsandlocalcurrency)ondaily,weekly andmonthlyfrequencies.Summarystatisticsonallthe series that we examined are reported in the Appendix. Judging by theJarque–Berastatistic,thereportedsummarystatisticsonthe returnseriesindicatenon-normaldistributionforalltheseries both in US dollars andlocal currency for daily, weekly and monthlyholdingperiods.Inalmostallthecasesexaminedhere, theskewnesscoefficientisnegativewiththeexceptionsbeing daily dollar returns for Chinaand daily returns in local cur-rencyunitsforBrazilandChina.Thustheimplicationfromthe summarystatisticssuggestthattheprobabilitydistributionsof BRICSreturnsfordaily,weeklyandmonthlyholdingperiodsare negativelyskewed,indicatingthatthelefttailsofthedistribution areeitherlongerorfatter(orboth)thantherighttail.

Followingthecluefromthesummarystatistics,weundertake aformalstatisticaltesttoproduceevidenceontheskewnessand excesskurtosisofthereturnseriesfortheBRICS.Theresults arereportedinTable2(fordailyreturns),Table3(forweekly returns)andTable4(formonthlyreturns).

Threedifferenttestsareemployed– Mardiaskewnesstest, MardiakurtosistestandHenze–Zirklerjointskewnessand kur-tosistest.AscanbeseenfromtheupperpanelofTable2,the nullhypothesisthattheskewnesscoefficientisnotstatistically differentfromzeroforthedailyreturnseriesisflatlyrejected by theMardia skewnesstest forBrazil, Russia,South Africa and the World.In the case of India andChina, however, the nullhypothesisisnotrejectedattheconventionallevelof sta-tisticalsignificance.Thereisstrongevidenceofexcesskurtosis (leptokurtosis) inthe dailystockreturnseries for the BRICS market indexesas wellas theWorld stockmarket index.The Mardiakurtosistestforexcesskurtosisinthereturnseriesflatly rejectedthenullhypothesisinallthecasesat1%levelof statis-ticalsignificance.Thestrongevidenceofexcesskurtosisinthe dailydollarreturnseries isconfirmedbyHenze–Zirklerjoint testfortheevidenceofskewnessandexcesskurtosis.Thenull hypothesisisrejectedatthe5%levelofstatisticalsignificance orlower.Theevidencereportedheregivesstrongindicationthat theprobabilitydistributionofthedailydollarstockreturnsfor theBRICSexhibitspeakednesswithfatterandlongtails.Jointly, theseindicatethattheprobabilitydistributionisnon-normal,and specialattentionneedstobetakenwhenperformingstatistical inferenceonthem.

Thisconclusionisinvarianttotheunitinwhichthereturnsare measured.Thisisconfirmedbytheresultsreportedinthelower panel of Table 2 where the null hypotheses of no skewness, nokurtosis andnojoint skewnessandkurtosisindailystock returns inlocal currencyunitsareallrejected atconventional levelofsignificance.Ithasbeenpointedintheextantliterature thatthenormalityassumptionontheprobabilitydistributionof

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Table2

SkewnessandKurtosistestondailyreturns.

Brazil Russia India China SouthAfrica World

DailyUSDollarReturns

Skewness 0.0094 [7.946] (0.0048) 0.9947 [168.3] (0.0000) 0.0029 [2.408] (0.1207) 0.0008 [0.697] (0.4038) 0.1759 [148.5] (0.0000) 0.1376 [116.16] (0.0000) Kurtosis 10.5850 [12,127.2] (0.0000) 14.8363 [29,531.3] (0.0000) 9.6360 [9282.4] (0.0000) 8.7725 [7023.9] (0.0000) 8.4823 [6335.3] (0.0000) 10.8188 [12,886.4] (0.0000) Joint 81.462 [96.73] (0.0000) 166.91 [126.78] (0.0000) 71.39 [91.64] (0.0000) 114.81 [110.59] (0.0000) 61.2102 [85.881] (0.0000) 91.1044 [101.15] (0.0000)

DailyLocalCurrencyReturns

Skewness 0.7469 [29.637] (0.0000) 1.2042 [47.779] (0.0000) 0.2847 [11.296] (0.0008) 0.2967 [11.771] (0.0006) 0.5797 [23.0000] (0.0000) 2.2989 [91.214] (0.0000) Kurtosis 5.7088 [70.931] (0.0000) 7.4317 [189.854] (0.0000) 4.8019 [31.386] (0.0000) 4.1349 12.451 (0.0004) 5.0858 [42.054] (0.0000) 8.4045 [282.352] (0.0000) Joint 2.5986 [10.005] (0.0016) 3.912 [14.423] (0.0001) 1.6422 [6.008] (0.0142) 3.0689 [11.680] (0.0006) 1.7083 [6.313] (0.0120) 5.6917 [19.178] (0.0000)

Note:In[]arethechisquaredstatisticsand()aretheprobabilityvalues.

stockreturns aremorelikelytobeviolatedinhighfrequency datathanalowone. Wethereforerepeattheabove skewness andkurtosis testonweeklyandmonthlystockreturnsforthe BRICSaswellastheWorld.Again,weconsiderboththereturns measuredinlocalcurrencyunitsandtheUSdollar.Thefindings ontheweeklyreturnsseriesconcerningskewnessandkurtosis oftheirunderlyingprobabilitydistributionsarenotatvariance withthosereportedondailyreturnseries.

WiththeexceptionofRussiawherethenullhypothesisofno skewnessisnotrejectedfor bothreturnsinthelocalcurrency

units and US dollarweekly series, the null hypothesis of no skewnessintheprobabilitydistributionofreturnsisrejectedfor theremainingBRICSweeklyreturnseries.Intermsofkurtosis, the nullhypothesisisrejectedfor alltheweeklyreturnseries measured interms of both the US dollar and local currency units.

TheHenze–Zirklerjointtestforskewnessandkurtosisinthe probability distribution of weekly BRICS returns (bothlocal currencyandUSdollars)rejectedthenullhypothesisflatlyfor alltheBRICSmarketaswellastheWorldstockmarketreturns.

Table3

SkewnessandKurtosistestonweeklystockreturns.

WeeklyUSDollarReturns

Skewness 0.4231 [71.723] (0.0000) 0.0156 [2.654] (0.1033) 0.1361 [23.073] (0.0000) 0.0722 [12.236] (0.0000) 0.0372 [6.302] (0.0121) 1.2880 [218.326] (0.0000) Kurtosis 7.0750 [699.5] (0.0000) 8.5842 [1313.6] (0.0000) 5.2467 [212.639] (0.0000) 5.9561 [368.1] (0.0000) 7.6655 [916.942] (0.0000) 12.8531 [4089.623] (0.0000) Joint 11.6893 [31.589] (0.0000) 28.2706 [51.143] (0.0000) 3.9810 [14.085] (0.0002) 10.9712 [30.365] (0.0000) 13.2744 [34.115] (0.0000) 13.4321 [34.355] (0.0000)

WeeklyLocalCurrencyReturns

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Table4

SkewnessandKurtosistestonmonthlystockreturns.

MonthlyUSDollarReturns

MonthlyLocalCurrencyReturns

Skewness 1.2993 [51.553] (0.0000) 1.1155 [44.258] (0.0000) 0.4244 [16.837] (0.0000) 0.2909 [11.543] (0.0007) 2.1596 [85.688] (0.0000) 2.8078 [111.408] (0.0000) Kurtosis 6.9145 [148.126] (0.0000) 7.0115 [155.554] (0.0000) 5.1774 [45.829] (0.0000) 4.1089 [11.887] (0.0006) 10.0213 [476.548] (0.0000) 9.7901 [445.684] (0.0000) Joint 2.4569 [9.462] (0.0021) 3.5946 [13.443] (0.0002) 1.2176 [3.949] (0.0469) 3.0417 [11.610] (0.0007) 2.2856 [8.785] (0.0030 5.1941 [17.956] (0.0000)

Note:In[]arethechisquaredstatisticsand()aretheprobabilityvalues.

The implicationhere isthat the weeklyreturn seriesinherits thefatterandlongertailsandpeakednessthatcharacterizedthe probabilitydistributionof dailystockreturns, atleast for the BRICSandtheWorldstockmarketasawhole.

Wenowconsiderthetailandpeakbehaviourofthe probabil-itydistributionofmonthlyindexreturnsfortheBRICSandthe WorldStockmarkets.Intheprecedingparagraph,wearrivedat theconclusionthatweeklyreturnseriesinheritstheproperties ofthe distributionof dailyreturns.Itisthusimportant toask whethermonthlyreturnsalsoinheritsthepropertiesofthe prob-abilitydistributionofweeklyreturnsandhencethoseofdaily returns.Wethussearchforevidenceofskewnessandkurtosis inmonthlyreturnsfortheBRICSinbothdollarseriesandlocal currencyseries.TheresultsarereportedinTable4.Consistent withourresultsondailyandweeklyreturns,theMardiatestsfor skewnessandkurtosisrejectedthenullhypothesisofno skew-nessandnokurtosisintheprobabilitydistributionofmonthly stockreturns.Henze–Zirklerjointtestforskewnessandkurtosis intheprobabilitydistributionof monthlyreturnsalsorejected thenull hypothesisinall cases.Thisconclusion isnot sensi-tivetotheunitinwhichmonthlyreturnsaremeasured(eitherin USdollarsorlocalcurrencyunits).Thustheevidenceof non-normalityinthedistributionofBRICSindexandWorldmarket indexreturns is robust.In line withBekaert etal.(1998) we areabletoestablishthattheBRICSstockreturnsexhibittime varyingskewnessandkurtosis.Theunderlyingstructureofthe BRICS:their potentialfor largegrowthswings, susceptibility toregulatoryandpoliticalchanges,andcontinuedintegration intothe global economic andfinancial systemmaythuslead their stockreturns to deviatesignificantly from the Gaussian assumption.Asaresult,anystatisticalinferencewhichrelieson thenormalityofthedistributionofreturnshasthepotentialof makingerroneousconclusions.

4. Volatility,predictabilityandrisk-returnrelationship

Inthissectionweestimateandreportevidenceonthe volatil-ity,predictabilityandrisk-return relationships for the BRICS indexreturns.Duetothesimilaritiesinthedistributionof the returnseriesforthedaily,weeklyandmonthlyholdingperiods, we onlyfocus onmodellingdailyreturnbehaviour. To begin with,wetestforthepresenceofARCHeffectsinthemeanof thedailyreturnsusingtheLagrangemultiplier(LM)test.The resultsarereportedinTable5.

TheresultsoftheLagrangemultiplier(LM)testforthe pres-enceofARCHeffectsinthemeanofBRICSindexreturnseries flatlyrejectedthenullhypothesisofnoARCHeffectsat1%level ofstatisticalsignificanceforallthecountries.Thisconclusion isinvarianttowhethertheBRICSindexreturnsare measured inthelocalcurrenciesor inUnitedStatesdollars. The invari-anceofthepresenceofARCHeffectsinthemeanofthereturns fortheBRICSsuggeststhatexchangeratemovementshavenot affectedBRICSindexreturnsvolatility;anempiricallytestable proposition.

Thevarianceistimedependentandthusneedtobeaccounted forinanystatisticalmodelaimedatcapturingthereturn gener-atingprocessofthesecountries.

Giventheevidenceofthepresenceofautoregressive condi-tionalheteroscedasticityinthemeanof BRICSindexreturns, we proceed to model the mean and the conditional variance of each of the BRICS market indexes using ARMA(p,q)− EGARCH(m,n)−M.Theresultsofourestimatesarereported

inTable6(fordailyreturnsmeasuredinUSdollars)andTable7

(forreturns measuredinlocalcurrency.Themodeldiagnostic statisticsreportedatthelowerpartsofTables6and7indicatethat theestimatedmeanandconditionalvarianceequationswere cor-rectlyspecified.Inparticular,theLjung-BoxQ-statisticatboth

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Table5

LagrangemultipliertestforARCHeffectsinBRICSdailystockreturns.

Brazil Russia India China SouthAfrica

LagrangeMultiplierTestforARCHeffectintheMeanofDailyUSdollarReturns

Lags=1 222.26(0.000) 549.76(0.000) 136.23(0.000) 361.35(0.000) 289.60(0.000)

Lags=5 1079.97(0.000) 641.59(0.000) 317.96(0.000) 736.58(0.000) 638.81(0.000)

Lags=10 1167.28(0.000) 732.62(0.000) 413.28(0.000) 802.37(0.000) 827.59(0.000)

Lags=20 1235.45(0.000) 871.65(0.000) 449.68(0.000) 910.10(0.000) 988.58(0.000)

LagrangeMultiplierTestforARCHeffectintheMeanofDailyLocalCurrencyReturns

Lags=1 189.52(0.000) 571.66(0.000) 183.34(0.000) 361.33(0.000) 330.97(0.000)

Lags=5 563.23(0.000) 662.92(0.000) 385.87(0.000) 739.24(0.000) 547.31(0.000)

Lags=10 621.29(0.000) 753.76(0.000) 463.03(0.000) 804.67(0.000) 589.44(0.000)

Lags=20 658.83(0.000) 893.91(0.000) 498.62(0.000) 912.78(0.000) 639.78(0.000)

Note:TheteststatisticfortheLMtestfollowsachisquaredistribution.Thenumbersreportedintheabovetablearethechisquaredstatisticswith()housingthe probabilityvalues.

lags11and25didnotrejectthenullhypothesisthatthemean equationiscorrectlyspecifiedforalltheBRICSindexreturns. Also,theLjung-BoxQ-squaredstatisticfailedtorejectthenull hypothesis that the conditional variance equation iscorrectly

specified. The estimated ARMA(p,q)−EGARCH(m,n)− M adequatelycaptures theconditional mean and variance of thereturngeneratingprocessfortheBRICS.Wethereforeturn totheinterpretationoftheestimatedparameters.

Table6

EstimatedARMA-EGARCH-MmodelofDailyReturnsinDollars.

Brazil Russia India China SouthA World

μ 0.125 0.248*** _0.0125 _−0.0727 _−0.332 _0.0871*** (1.49) (4.27) (0.09) (−1.32) (−0.98) (8.26) ARCHM ϕ 0.0237 −0.00848* _0.0722 _0.0220* _0.126 _−0.0251** (1.90) (−2.57) (1.85) (2.11) (1.21) (−2.74) ARMA AR(1) 0.999*** _−0.0126 _0.990*** _0.998*** _−0.00115 _0.0573*** (2710.22) (−1.10) (4905.07) (633.92) (−0.08) (9.54) AR(2) 0.000151 0.00981*** _0.00808 _0.0272** (0.02) (48.78) (0.68) (2.79) AR(3) −0.0111 (−0.98) MA(1) −0.997*** −0.997*** −0.997*** −0.0646*** (−1617.12) (−1370.94) (−573.32) (−10.99) ARCH α 0.0221 0.0520 0.0226 0.0147 0.0229 −0.0135 (1.10) (1.26) (0.67) (0.44) (0.77) (−0.48) γ 0.129*** _0.362*** _0.117** _0.235*** _0.100*** _0.156*** (4.52) (5.40) (2.75) (4.68) (3.35) (4.06) δ1 0.815*** 0.135 0.0166 0.189 −0.00775 −0.267 (28.66) (0.77) (0.05) (1.50) (−0.01) (−1.65) ω 0.352*** _2.631*** _1.209** _1.416*** _1.187 _0.176 (6.85) (4.84) (3.20) (6.25) (0.87) (1.86) N 5059 5059 5059 5059 5059 5059 AIC 21,709.7 23,600.2 19,077.8 20,080.7 19,084.4 13,097.5 BIC 21,768.4 23,658.9 19,143.1 20,139.5 19,149.7 13,162.8 LBQ(11) 15.702 [0.152] 6.412 [0.844] 16.463 [0.124] 9.342 [0.590] 17.199 [0.102] 12.288 [0.342] LBQ(25) 27.412 [0.335] 32.808 [0.135] 33.033 [0.130] 30.928 [0.191] 29.527 [0.242] 28.546 [0.283] LBQ2₍₁₁₎ _14.460 [0.208] 6.424 [0.843] 16.267 [0.131] 9.204 [0.603] 16.692 [0.117] 11.300 [0.418] LBQ2₍₂₅₎ _26.749 [0.368] 32.161 [0.153] 32.592 [0.141] 29.291 [0.251] 28.983 [0.264] 27.795 [0.317]

tstatisticsinparenthesesandthenumbersinsquarebracketsarethep-values,LBQistheLjung-BoxstatisticsforserialcorrelationandLBQ2_is_the_{Ljung-statistics}

forARCHeffect,withthelagsusedintheparentheses.

* _p_<_0.05. **_p_<_0.01. ***_p_<_0.001.

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Table7

EstimatedARMA-EGARCH-MmodelforDailyReturnsinLocalCurrency.

Brazil Russia India China SouthA World

μ −0.121*** 0.197** 0.0537 −0.0837*** 0.138 0.103*** (−10.43) (2.75) (0.51) (−18.55) (0.94) (6.53) ARCHM ϕ 0.0460*** −0.00429 0.0603 0.0184*** 0.0462 −0.0391** (11.57) (−1.36) (1.66) (16.58) (0.56) (−2.77) ARMA AR(1) 0.00760 −0.0105 0.993*** _−0.987*** _1.013*** _−0.00778 (0.63) (−0.85) (4511.98) (−835.39) (7265.27) (−0.60) AR(2) 0.00416 0.00611*** _−0.00813*** _0.0230* (0.45) (26.74) (−64.04) (2.57) AR(3) −0.00542*** (−52.21) MA(1) −0.997*** _0.985*** _−0.997*** (−1324.60) (1164.69) (−877.71) ARCH α 0.00950 0.0863 0.0345 0.0165 0.0297 −0.000248 (0.52) (1.77) (0.90) (0.50) (1.19) (−0.01) γ 0.145*** 0.501*** 0.141** 0.240*** 0.0661 0.170*** (4.47) (5.09) (3.11) (4.36) (1.72) (4.42) δ1 0.831*** 0.246 0.0135 0.198*** −0.356 −0.413** (107.91) (1.95) (0.04) (8.29) (−0.70) (−2.87) ω 0.270*** 2.618*** 1.045** 1.406*** 0.699** 0.204 (28.62) (5.60) (2.97) (37.66) (2.68) (1.71) N 5059 5059 5059 5059 5059 5059 AIC 19,662.8 23,178.2 18,127.4 20,065.9 15,924.0 12,748.7 BIC 19,715.0 23,236.9 18,192.7 20,124.7 15,995.8 12,807.5 LBQ(11) 17.130 [0.104] 7.277 [0.776] 18.724 [0.066] 10.798 [0.460] 8.589 [0.659] 11.499 [0.402] LBQ(25) 32.443[0.145] 30.882 [0.192] 33.914 [0.109] 35.870 [0.073] 30.052 [0.222] 28.580 [0.281] LBQ2₍₁₁₎ _14.984 [0.183] 7.333 [0.771] 18.807 [0.064] 10.279 [0.505] 8.402 [0.676] 10.581 [0.479] LBQ2₍₂₅₎ _30.157 [0.218] 30.687 [0.199] 34.308 [0.101] 33.378 [0.121] 29.985 [0.224] 27.879 [0.313]

tstatisticsinparenthesesandthenumbersinsquarebracketsarethep-values,LBQistheLjung-BoxstatisticsforserialcorrelationandLBQ2istheLjung-statistics forARCHeffect,withthelagsusedintheparentheses.

* _p_<_0.05. ** _p_<_0.01. ***_p_<_0.001.

Accordingtothe results reportedinTable 6, theexpected daily index return is statistically not different from zero for Brazil,India,ChinaandSouthAfrica.However,theindexfor RussiaandtheWorldindexpromiseapositivereturn.The esti-mated mean return for Russia is 0.248 and it is statistically significant at 1%level. Forthe World index return, the esti-matedmeanis0.078whichisalsostatisticallysignificantat1% level.Whenreturnsaremeasuredinlocalcurrencyhowever,the expecteddailyindexreturnsforBrazilandChinaarenegative andstatisticallysignificantat1%level.Theestimatedexpected return for Brazil and China are −0.121 and −0.084 respec-tively.Onthecontrary,RussiaandtheWorldindexreturnsare bothpositiveandstatisticallysignificantontheaverage,when returns are measured inlocal currency units. ForRussia and theWorldindex,theestimatedexpecteddailyindexreturnsare 0.197(whichisstatisticallysignificantat5%level)and0.103 (whichisstatisticallysignificantat1%level)respectively.The estimatedexpecteddailyindexreturnsinlocalcurrencyunits are,however,notstatisticallydifferentfromzeroforIndiaand

SouthAfrica.Thus,forIndiaandSouthAfricathedailyindex returniszeroandpositiveforRussia,irrespectiveoftheunitin whichreturnsaremeasured.

Oneimplicationofthecapitalassetpricingmodel(CAPM) isthatexpectedreturnandriskarerelatedpositively.Fora ratio-nalrisk-averseinvestortotakeonanadditionalrisk,hemustbe compensatedwithadditionalreturn.Weconfrontthisprediction ofCAPMwithBRICSindexreturnsdata.Therisk-return rela-tionshipparameteriscapturedbythecoefficientonvarphi(ϕ)of theARCHMpartoftheresultsreportedinTables6and7.From

Table 6 the estimated coefficientonthe risk variable

(appro-priatelyinterpretedasthe riskpremium)ispositiveforallthe BRICS except Russia and the World index, for daily dollar returns. In sharp contrast with the predictions of the CAPM, thecoefficientontheconditionalvarianceinthemeanequation whichcapturestherelationshipbetweenriskandreturnare nega-tiveandstatisticallysignificantforRussiaandtheWorld.Thisis aviolationofrationalbehaviourofrisk-averseinvestorsinequity markets.However,theestimatedrisk-returnrelationshipisonly

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statisticallysignificantforRussia,China(10%levelof signif-icance)andtheWorld(5%levelofsignificance).Withreturns measuredinlocalcurrencyhowever,the estimatedcoefficient ontheconditionalvarianceinthemeanequationispositiveand statisticallysignificantforBrazilandChinabutnegativeand sig-nificantfortheWorld.However,giventhenegativemeanlocal currencyreturnforBrazilandChina,theestimatedpositive rel-ative riskaversioncoefficientsimpliesanegativerelationship betweenriskandreturn,afindingwhichisinconsistentwiththe CAPM.TheWorldlocalcurrencyindexreturnalsoviolatesthe predictedpositiverisk-returnrelationshipbythe CAPMsince theestimated meanreturnispositivewhile theestimatedrisk aversionparameterisnegativeandstatisticallysignificant.

Wenowturntothequestionofpredictabilityofdailyindex returns both in US dollars and local currency units and the implications thereof for the efficiency of BRICS markets. In well-functioning equity markets the prices of the securities tradedactasthoughtheyfullyreflectallavailableinformation andreact instantaneously and inan unbiased fashion tonew information(Fama, 1970,1991).If thiswere notthe case, it wouldbepossibletoobtainrisklessreturn,castingdoubtsabout theabilityofthestockmarkettoefficientlyallocatecapital.The weakformof the efficientmarkets hypothesisthusprecludes anypredictabilityonthebasisofpastinformation(seeFama, 1991).AreBRICSstockreturnspredictable?

ThisquestioncanbeansweredfromtheARMA(p,q) com-ponentoftheestimatedARMA(p,q)−EGARCH(m,n)−M

reportedinTables6 and7.According totheresults reported

inTable6,thecoefficientonthe AR(1)ispositiveand

statis-ticallysignificant at1%level forBrazil, India,Chinaandthe Worldindexdailyreturns.Notonlyarethesecoefficients statis-ticallysignificant,butalsoeconomicallygiventhelargesizeof thecoefficients.Forinstance,thecoefficientAR(1)are0.999, 0.990,0.998and0.057forBrazil,India,ChinaandtheWorld indexrespectively.Interestingly,thissubgroupfortheBRICS marketsalsoshowednegativeandstatisticallysignificant coef-ficientonthefirstlagofthemovingaverage(MA)component of the estimated ARMA(p,q)−EGARCH(m,n)−M. The estimatedcoefficientof thefirst lagof theshocktothemean processis−0.997forBrazil,−0.997forIndiaand−0.997for China.Theimplicationfromthisisthatpastinformationabout returns (prices) have predictive power on current and future returns(prices), a sharp contrast to the weak-form efficiency hypothesis.InthecaseofRussiaandSouthAfrica,thereisno evidencethatpastreturnsandshockstoreturnproveany infor-mationin predicting current andfuture returns(prices). Does predictabilitynecessarilyimplyinefficiency,anddostockprices intheBRICSreflectarationalassessmentoffundamental val-ues?Ouransweristhatwithoutanotionofthemodelofreturn generation onecannot draw definite conclusionson the time seriespatterns alone.Tothisendwesidewiththetheoretical conclusions of LeRoy (1973)and Lucas (1978)that rational expectations equilibrium prices need not form a martingale sequence.

Measuringreturns using localcurrency series changesthe resultsfor BrazilandSouthAfrica.In thecaseof Brazil nei-therpastreturnsnorshockstoitofferanypredictivepoweron

current andfuture returns. Thisis in sharp contrast with the finding obtained whenwe measurereturns inUSdollars. For South Africa thereispositive andstatisticallysignificant AR (1) andanegativeandstatisticallysignificantMA(1)effectin thereturngeneratingprocess.Thisimpliesthatpastreturnsand pastshockstoreturnscanbeusedtopredictcurrentandfuture dailyRandreturnsintheSouthAfricanstockmarket.This find-ing may suggest the existence of profitable arbitrage. China, IndiaandtheWorldindexarepredictable,butthisisinvariant towhether returnsare measuredinlocalcurrencyor US dol-lars.ThepredictabilityofBRICSreturns seemtoconcurwith similarstudiesonemergingmarkets(seeforexampleCooper,

1982;Darrat,1990;ErrunzaandLosq,1985),whichexamined

theweakformversionoftheefficientmarketshypothesis.The presenceoffirstorderfirst-orderserialcorrelationinstockprices showsthatinformationmaynotbefullyincorporatedinsecurity prices.DailamiandAtkin(1990)arguethatpositiveserial cor-relationmayresultinslowincorporationofnewinformation, insider trading, or infrequent trading. There may be barriers tothedisseminationof information,andcompaniesappearto divulgelessinformationwithagreatertimelagthanisthenorm. Ontheotherhand,negativeserialcorrelationsmaysignalthin tradingandsubjecttospeculativeinfluences.

4.1. Volatilitypersistence

Volatility persistence in the return series for each of the countriesisdescribedbybothα1andδtermsinTables6and7. Theparameterα1representthelaggedsquaredresidualsfromthe EGARCH-Mmodel,whileδisthelaggedconditionalvariance termintheEGARCHmodel.Volatilityissaidtobepersistent ifthesumofthetwovolatilitytermsisclosetounity,less per-sistentiflessthanunityandexplosiveifgreaterthanunity.In bothreturnseries(indollarsandlocalcurrency)asreportedin

Tables 6and7,we findstrong indicationof volatility

persis-tenceforBrazil,themagnitudeofthepersistencetermssum-up toclosetounity(0.944and0.976forreturnsindollarsandlocal currency,respectively).Wefindlessvolatilitypersistenceinthe returnseriesforalltheothercountriesandalsotheWorldindex return,irrespectiveoftheunitofcurrency(dollarsorlocal cur-rency).Theimplicationoftheseresultsisthat,alltheBRICS countries (withtheexception of Brazil)show noevidence of long-memoryintheirrespectivereturnseries.Thismeansthat shockstovolatilitytend todecayveryquickly,implying that previousvolatilitydonothaveastrongpredictivepoweron cur-rent volatility. Theseresults are, however,conditional on the model specification andthe distribution assumption made in theestimation.Sincealldescriptivestatisticsandnormalitytest along withtestingfor theeffectof autoregressive conditional heteroscedasticitypointedtowardsthemodelusedinestimating theparameterspresentedinTables6and7,theseresultsarevery satisfactory.

4.2. Asymmetryinvolatility

An avalanche of empirical research in emerging market returns distributionpoint tosignificantleverage effect, where

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highervolatilitytendtofollownegativereturns(seeAlagidede, 2011andreferences therein).Asymmetriesinthe distribution ofreturnsmayariseeitherbecauseofshockstosystematicrisk factorsthat affect the cross section of returns, or because of country-specificshocks.Thisistakenupintheresultsreported

inTables6and7.Theevidencerevealsapositiveleverageeffect

forallreturnseries(bothindollarsandlocalcurrency),except SouthAfricawhichisinsignificant.Theseresultsareconfirmed bythe misspecificationtestof Ling andMcAleer(2000) and

McAleeretal.(2007)(resultsreportedintheAppendix)thatthe

asymmetricEGARCH-Mmodelbestfit theBRICS data,and accountforthevolatilityprocesswell.

Theseresults thus contravene the findings of Bekaert and

Harvey(1995)andBekaertandHarvey(1997)whodonotfind

supportforleverageeffectsinemergingmarkets.

5. Conclusion

Thispaperexaminedstockreturnsdistributioninthebiggest emergingmarketeconomies.Previousresearchhasestablished thatstandardassetpricingmodelsconsistentlyfailtoaccountfor allthepeculiaritiesofemergingmarketsreturns.Forinstance, returns are notedtobe highly volatileandnon-normally dis-tributed. Long horizon returns are predictable while there is significant autocorrelation inreturns. Afterover two decades ofresearchinemergingmarketeconomies,theissueofreturn distributionisfar from settled.Newdataand empirical tech-niques,coupled withfastergrowthandincreasingimportance

ofemergingmarketsallowustore-examinethedistributionof stockreturnsforBrazil,Russia,India,ChinaandSouthAfrica fortheperiod1995–2014.

Usingamultivariatejointtestforskewnessandkurtosis,and accountingfor risk premia andconditional heteroscedasticity wearrivedatthefollowingfindings:(a)thedistributionofstock returnsfortheBRICSexhibitspeakednesswithfatterandlonger tails.Thisisinvarianttoboththeunitofmeasurementandthe timehorizon inwhichreturns are studied. Wearguethat the underlyingstructureoftheBRICS,particularlytheirpotential forlargegrowthswings,susceptibilitytoregulatoryand politi-calchangesmayleadtheirstockreturnstodeviatesignificantly fromtheGaussianassumptionandthisoughttobeincorporated inanyinferencesonreturndistribution.(b)Whilethestock mar-ketsof ChinaandIndiaarepredictableirrespective ofunitof measurement,returnpredictabilityforBrazilandSouthAfrica areconditionaluponwhetherwearelookingatlocalcurrency or US dollar returns. Without an explicit model of the price generating process it is difficultto judge the weakform (in) efficiencyofthesemarketsonthebasisofthetimeseries prop-erties alone. (c) Allmarkets exhibit volatility clustering,and whilethisdecaysformostmarkets,ittendstobepersistentfor Brazil.Thusalthoughshockstocurrentvolatilitymayperpetuate throughtime,thereisnoevidenceoflongmemory.(d)The so-calledleverageeffectisconfirmedforallbutSouthAfrica,while therisk-returnrelationshipisdynamicallyrelatedtoindividual countryandmodelspecification.

AppendixA. SummarystatisticsofBRICSreturns

DailyUSDollarReturns

Mean 0.023 0.039 0.024 −0.003 0.020 0.020 Median 0.058 0.060 0.021 0.003 0.076 0.068 Maximum 17.335 24.220 19.486 14.044 12.353 9.097 Minimum −18.323 −31.013 −12.041 −14.442 −13.566 −7.325 Std.Dev. 2.318 2.992 1.738 1.971 1.726 0.993 Skewness −0.097 −0.447 −0.053 0.029 −0.419 −0.371 Kurtosis 10.585 14.836 9.636 8.772 8.482 10.819 Jarque–Bera 12,135.07[0.000] 29,699.48[0.000] 9284.83[0.000] 7024.63[0.000] 6483.65[0.000] 13,002.46[0.000]

DailyLocalCurrencyReturns

Mean 0.042 0.040 0.036 −0.003 0.038 0.019 Median 0.000 0.061 0.000 0.000 0.013 0.068 Maximum 24.734 24.220 16.423 14.036 6.750 8.720 Minimum −14.217 −31.013 −12.050 −14.457 −12.208 −7.156 Std.Dev. 1.923 2.928 1.588 1.969 1.253 0.960 Skewness 0.351 −0.462 −0.133 0.025 −0.406 −0.337 Kurtosis 15.365 15.834 9.392 8.811 8.249 10.360 Jarque–Bera 32,334.44[0.000] 34,900.44[0.000] 8626.06[0.000] 7118.92[0.000] 5947.81[0.000] 11,515.55[0.000]

WeeklyUSDollarReturns

Mean 0.125 0.196 0.124 −0.011 0.099 0.101 Median 0.422 0.447 0.364 0.234 0.356 0.321 Maximum 25.617 44.899 18.366 22.536 27.601 11.636 Minimum −33.056 −31.698 −21.879 −24.336 −18.855 −22.381 Std.Dev. 5.322 7.100 3.950 4.619 3.907 2.375 Skewness −0.650 −0.125 −0.369 −0.269 −0.193 −1.135 Kurtosis 7.075 8.584 5.247 5.956 7.666 12.853 Jarque–Bera 770.805[0.000] 1316.244 [0.000] 235.577[0.000] 380.276[0.000] 923.208[0.000] 4306.687 [0.000]

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AppendixA(Continued)

WeeklyLocalCurrencyReturns

Mean 0.221 0.200 0.185 −0.011 0.191 0.096 Median 0.440 0.448 0.434 0.230 0.296 0.323 Maximum 19.189 44.916 13.660 22.422 16.264 10.736 Minimum −22.529 −31.698 −19.000 −24.341 −13.454 −21.318 Std.Dev. 4.120 6.946 3.557 4.609 2.840 2.287 Skewness −0.547 −0.104 −0.332 −0.276 −0.142 −1.040 Kurtosis 6.872 9.157 5.276 5.989 6.107 12.208 Jarque–Bera 681.831 [0.000] 1598.525 [0.000] 236.676 [0.000] 389.233 [0.000] 409.969 [0.000] 3753.445 [0.000]

MonthlyUSDollarReturns

MonthlyLocalCurrencyReturns

AppendixB. Testfornon-nestedmodels

The Ling and McAleer (2000)and McAleer etal. (2007)

testisatestingprocedurefornon-nestedmodels,for instance betweenEGARCHandGARCHmodels.Supposeourproposed modelistheEGARCHandwewanttocomparethatwiththe GJR-GARCHmodel,theproposedtestingprocedureistocheck for significant coefficient of the log variance from the GJR-GARCHmodelinthefollowingspecification:

ln(h_t)=ω+α|zt−1|+γzt−1+βln(ht−1)+δln(σ2t)

WheretheestimatedvariancefromtheEGARCHmodelis

ht,σ2_t istheestimatedvariancefromtheGJR-GARCHmodel

andzt−1isthestandardizedresidualsfromtheEGARCHmodel.

TheNullhypothesisisthat;δ=0,whichisineffecttestingfor significanceoftheδintheaboveequationandnon-significance impliesδ=0andthereforetheEGARCHinthiscasewillbe thepreferredmodelrelativetotheGJR-GARCHmodel.

RejectionpercentageoftheNullhypothesisofδ=0baseon

LingandMcAleer(2000)test

Testat5%sig. level

NullModel Alternative LocalCurrency (%)

Dollars (%)

Ling–McAleer EGARCH GJR-GARCH 0 0

Ling–McAleer EGARCH GARCH 0 33

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