T estsofthe R and om W alkHypothesisand Long-R unDepend ence onthe J ohannesb urg Stoc kE xchange Asst.P rof .O w enB eeld ers¤ Departm ent ofE c onom ic s E m ory U niversity Atlanta G a 30 32 2 -2 2 4 0 J une 1,2 0 0 0 Ab strac t W e test f or pred ic tab ilityand long-rund epend ence inf our b road South Af ric anStoc kInd exesusingthe variance ratio test,tw o testsf or f rac tional d i®erencing and the m od i¯ed resc aled range statistic .For the AllShare Ind ex and the G old Ind ex w e f ailto rejec t the nullofthe rand om w alk m od el. For the Ind ustrialInd ex, w e c anrejec t the nullhypothesisofa rand om w alkusingthe variance ratio test at the 4 and 8-w eekhoriz on,b ut w e ¯nd no evid ence oflong-rund epend ence using the test f or f rac tional d i®erencing or the resc aled range test.For the F inancialInd ex, w e ¯nd evid ence oflong-rund epend ence using a test d eveloped b y New b old and Agiakloglou (19 9 4 ), b ut this m ay b e d ue to the thintrad ing ofthese stoc ks.Inc onclusion,there islittle evid ence to suggest that there islongrund epend ence onthe J ohannesb urgStoc k E xc hange. J E L Classi¯c ationNumb er: G 14 ,G 15 K ey w ord s: R and om W alk,Variance R atio,P red ic tab ility ¤Iw oul d l ike to thankM ic helle Francisofthe J SE f or her generousand invaluab le assistance and my researc h assistant, Xiw ang G ao, f or her c apab l e assistanc e.Al lrem aining errorsare m ine.Allc orrespond anc e should b e ad d ressed to O w enB eeld ers, Departm en t ofE c onom ic s, E m ory U niversity, Atl anta G a 30 32 2 -2 2 4 0 ; tel ephone n umb er (4 0 4 ) 72 7-6650 ; f ax n umb er (4 0 4 ) 72 7-4 639 ;e-m ail , ob eel d e@em ory. ed u. 1 1 In trod uc tion T wocompetingschools ofthoughthave arisen in the debate aboutmarketef¯ciency. O ne posits thatmarkets are e±cientand returns are unpredictable. T he otherdraws on work by Fama and French (1 988)and P oterba and Summers (1 988) who¯nd thatreturns arepredictableatlongertime horizons.In fact,the evidenceseems topointtothe factthatin short-run there is alotof noise orvolatility in stock returns, butatlongerhorizons returns this washes 1 outandreturns arepredictable. T wocompetingstatisticalmodels areusedto promotethesetwoschools ofthought.T herandom walkmodel,as popularized byFama(1 965)andM alkiel(1 973),is themainworkhorseofthesupporters of e±cientmarkets.T herandom walkmodelofstockprices implies thatthebest forecastofany future price is the currentprice,price changes and returns are unpredictable atany horizon and forecaststandard errors grow linearly with the horizon.O ne drawbackofthe random walk modelis the assumption that pricechanges areindependent;this is untestable.Sincethepublication ofH arrison and Kreps (1 979), the random walk hypotheses has been superceded by themartingalehypothesiswhichmaintainsthethreeimplicationsabove,butreplacestheindependenceofpricechangeswiththatofuncorrelatedpricechanges. T he martingale hypothesis introduces an additionaltestable hypothesis ofno short-run dependence,i. e.stockpricechanges orreturns arenotseriallycorrelated,andpermits themodellingofhigherorderdependencesuchas conditional heteroskedasticity. R esearchers whosupportthe notion oflong-run dependence have adopted themodeloffractionaldi®erencingforitspurposes.FractionalA R IM A models displaylong-rundependencethatis consistentwithpredictabilityatthelonger horizons thatFamaandFrench (1 988)and others havefound.T hebasicmotivation forfractionalintegrationis aggregation within theerror-duration model where the observed process is an aggregate ofshocks with a stochastic magnitude and stochastic duration (P arke (1 999)). T his motivation is consistent with the notion thatthe longhorizon assetreturns are an aggregate ofinformation shocks ofdi®erent magnitudes and duration. T he magnitude can be determinedbytherelativeimportanceoftheshocktothecompanyoreconomy andthedurationcanbedeterminedbytheprotractednatureoftheinformation revelation.Forexample,thelitigation againstM icrosofthas amaterialimpact on the how the company will operate in the future and the duration of the shockis determinedbythelengthoftheproceedings andnegotiation.R ecently, D ieboldand Inoue(1 999)haveshownthatasmallamountofregimeswitching may be observationally equivalenttothe observation offractionaldi®erencing in economicdata.T hus weare likely to¯nd evidence offractionalintegration instockmarkets thathaveexperienced regimeswitchingorstructuralchanges. B oth short- and long- run dependencehaveconsequences in ¯nancialmodelling.First,theunderlyingmodels forvaluingderivatives excludethepossibil1 For exam pl e,Harrisonand Zhang(199 9 )¯nd that the risk-returnrelationship onlyappears at tw o-year hold ingintervalsb ec ause noise tend sto d om inate returnsat shorter horiz ons. 2 2 ity ofany form ofdependence in returns. Second, statisticaltests ofmarket e±ciency and otherhypotheses derived from ¯nancialtheory are constructed underthe assumption thatthe nullis correct.Ifthese assumptions are incorrect, the usualforms ofstatisticalinference are no longervalid and the joint hypothesis problem rears its heard. B ecause ofthe noise in assetreturns and the presence ofserialcorrelation due tothin trading, greaterfocus has been placed on testingforlong-run dependence.T he objectiveofthis paperis totestthetwocompetingtheories in thecontextofbroadstockindexes ontheJohannesburgStockExchange(JSE). Section 2 contains a briefliterature survey ofresults forothercountries. In section 3 wedescribeourdataanddiscuss thedescriptivestatistics.In Section 4,5 and6 wetestthetwocompetinghypotheses byapplyingthevarianceratio testofL oand M acKinley,twotests forfractionaldi®erencingdeveloped independently byG eweke and P orter-H udak(1 983)and N ewbold and A kiakloglou (1 994)and themodi¯ed rescaled-rangestatisticofL o(1 991 ),respectively.W e concludewith section 7. 2 Literature R eview T odatetherehas been littleresearchon markete±ciencyon theJSE.A ²eckG raves and M oney (1 975)tested the random walk hypothesis by considering the autocorrelations ofstock prices, butthere is nosubsequentresearch using morepowerfulstatisticaltests.W e expectto¯nd deviations from thee±cient marketshypothesisatshorthorizonsbecausetheJSEishighlyconcentratedand manystocks arethinlytraded,soweturn totests oflong-run predictability. InthecaseoftheU S,Campbell,L oand M cKinlay(1 997)reportthatlarge stocks followarandom walk,butsmallstocks donot.In particular,they ¯nd thattherandom walkhypothesiscanberejectedfortheequallyweightedCR SP index,butnotforthevalue-weighted CR SP index.U pon furtherinvestigation of¯ve size-ranked portfolios, the random walk hypothesis can be rejected for the smallestand middle quintile, butnotthe largestquintile. Forindividual stocks, the random walk hypothesis cannotbe rejected. O ne explanation for this resultis thatstocks havealotofidiosyncraticnoisethatis averagedoutin theportfolios.Incontrasttotheresults fortheportfolios wherevarianceratios tend to be greaterthan one and reject at the uppertail, variance ratios for individualstocks areless than oneaverage.T his suggests thatthereis alarge degreeofpositivecorrelationacross stocks thatis evidentinthevarianceratios ofthe portfolios.O ne ofthe caveats to theiranalysis is the failure toadjust the levelofsigni¯cance forthe multiple comparisons ofthe variance ratios at di®erenthorizons,consequentlythelevelofsigni¯canceatwhichtheyaretesting may be fargreaterthan the nominallevelforeach individualtest.Chowand D enning(1 993)providethe necessary statisticaltheory for¯ndingthelevelof signi¯canceofmultiplevarianceratiotests. 2 Anexc eptioninthe l itertature isLo and W ang (19 95) w ho c onsid er the im plem en tation ofoptionpric ingm od elsw henreturnsare pred ic tab l e. 3 U rrutia (1 995)examined the South and CentralA merican markets ofA rgentina,B razil,Chileand M exicoandfoundevidenceagainsttherandom walk model.H owever, he did notuse the Chowand D enning(1 993)criticalvalues thatcontrolthelevelofsigni¯cancewhentherearemultiplecomparisons.U sing thecorrectcriticalvalues,O jah and Karemera(1 999)failtorejecttherandom walkhypothesisforthesemarketsand¯ndnoevidenceoffractionaldi®erencing. A runs tests detects alackofweakform e±ciency fortheChilean market,but thisisonlyatshorthorizons. FrennbergandH annson(1 993)rejecttherandom walkmodelforSwedishstockprices becausethevarianceratios arestatistically greaterthan 1 atlags ofless than 24 months,however,acaveattotheiranalysis is thatthey donotuse the Chowand D enningcriticalvalues.A yadi and P yun (1 994)testthe random walk modelusingvariance ratios in the Korean stockmarket.A lthough this marketis alsocharacterized by thin tradingthey donot¯nd evidence torejectthe random walk model.In conclusion, there is not much evidence againstthe random walk hypothesis when the Chow and D enning(1 993)criticalvalues areused. 3 Data and Desc riptive Statistic s W efocus on thebroad,marketindex and threebroad sectorindexes thatrepresentthelargestpercentage ofmarketcapitalization,i. e.theA llShareIndex (A L SI),theIndustrialIndex(IN D I),theG oldIndex(G L D I)andtheFinancial 3 Index (FIN I). T he weekly data forthe A L SI, IN D I, G L D I and FIN I indexes wereobtainedfrom D atastream fortheperiodJanuary1 988 toD ecember1 999. T he indexes are value-weighted and are adjusted formergers, de-listings, but 4 notdividends. R eturns are computed as the naturallogoftheprice relative, i. e.R t ´1 0 0 ¢ ln(P t=P t¡1 )whereP t is thepriceindexattimet. T he purpose ofthis section is to compare the descriptive statistics ofthe indexes,tolookfordeviations from normalityin theunconditionaldistribution and todeterminetheamountofdependenceinthe¯rstandsecondconditional moments.T hedescriptivestatistics arereportedinT able1 andincludetests of skewness and excess kurtosis based on theG M M estimatorand thetraditional estimatorthatis derivedundertheassumption ofiid returns.W eincludetests basedontheG M M estimatorbecausethetraditionalestimatorsu®ers from the drawbackthatits varianceis underestimatedwhenreturns arenon-normaland conditionallyheteroskedastic(P agan (1 996)). Exceptforthe G old Index, the mean returns ofallthe broad indexes are statistically di®erent from zero.T here are a number of possible reasons for this:¯rst,weareanalyzingreturns de¯nedas logprice-relatives andnotexcess returns so the non-zero mean may re° ectthe non-zero riskless rate ofreturn underano-arbitrageequilibrium.Second,duringthesampleperiod,therateof in° ationandtherisk-freeratewas inexcess of1 0 % perannum andthis maybe 3Detail ed inf orm ationab out the ind exc onstruc tionisavailab le at the w eb site ofthe J SE , http://w w w . jse. c o. za/. 4 Som eon e said that d ivid end ad justm en t d oesnot m atter - w hat isthe ref erence? 4 re° ected in the rawreturns ofthe indexes.Finally, the statisticalsigni¯cance maybeas aresultofthelargesamplesizeof2993 observations. T urningtothesecond moment,theG old Indexis approximatelyfourtimes more volatile than the otherthree indexes. T he di®erence in volatility is re° ected in ¯gure 1 where we plotthe time series ofreturns forthe fourbroad indexes on the same axes.T he G old Index alsodisplays its uniqueness in the thirdmoment:incontrasttotheotherindexes,theG oldIndexhasapositivecoe±cientofskewness although itis notstatisticallysigni¯cant.T heimportance oftestingforskewness andkurtosis usingtheG M M estimatoris apparentfrom the huge di®erence in magnitude ofthe teststatistics.Forexample,the G old Index is negativelyskewed and has at-statisticof4. 283 undertheassumption ofiidreturns whereas theteststatisticis only0 . 0 0 2 whentheiidassumptionis relaxed.O nly the A llShareand IndustrialIndexes display statistically significantnegativeskewness,butallfourindexes display excess kurtosis relativeto thenormaldistribution thatis signi¯cantatthe1 % level. T he ¯rst¯ve autocorrelations are reported in T able 1 .T he JSE is known forits lowturnoverratioand thin tradingsoweexpecttosee some short-run dependence.ExceptfortheG oldIndex,the¯rstautocorrelationis statistically signi¯cantalthough this maybeduetoconditionalheteroskedasticity(D iebold (1 986)). T he ¯rstautocorrelation ofthe FinancialIndex is the largestofall theindexes becauseitcontains stocks thatsu®ermostfrom thin trading.T he G oldIndexis leastlikelytosu®erfrom thintradingbecausethegoldstocks are highly traded by domestic and foreign investors. T he L jung-B ox Q -statistics con¯rm thepresenceofserialcorrelation in alltheindexes andarestatistically signi¯cantatthe1 % levelatboth5 and1 0 lags.T heQ -statistics ofthesquared returnsarealsostatisticallysigni¯cantatthe1 % levelatboth5 and1 0 lags.T he presenceofconditionalheteroskedasticityis consistentwith theexcess kurtosis in returns. In conclusion, the gold sectoris di®erentto the othersectors in thatitis morevolatileand has asymmetricdistribution.A lltheindexes displayexcess kurtosisthatmaybeduetothepresenceofconditionalheteroskedasticityorleptokurtosis in theunderlyingdistribution.T hereis signi¯canttimedependence inthereturns oftheindexes especiallyatthe¯rstlagandtheserialcorrelation 5 appears toberelated tothin trading. 4 T he Variance R atio T est U nderthenullhypothesisthatastockprice,P t,followsarandom walk,thevarianceofthek-horizonreturnis proportionaltok,i. e.ifwede¯nethek¡horizon returnas R t(k)= 1 0 0 ¢ ln(P t+ k=P t),then its variancesatis¯es therelationship, V ar(R t(k))= k¢V ar(R t(1 )): (1 ) 5T hese resul ts are b road ly c onsistent w ith the d esc riptive statistic s ofd aily returns in B eeld ers (2 0 0 0 ) although the d eviations f rom norm al ity f or the w eekly returns are not as extrem e.T he id iosync rac y ofthe gold ind ex isstillpresent inthe w eekly returns. 5 W ede¯nethek-horizon varianceratioas V R (k)= V ar(R t(k)) : k¢V ar(R t(1 )) (2) U nderthenullhypothesis,thevarianceratioequals 1 atallhorizons fork> 1 . Severalstudies have found variance ratios greaterthan one athorizons ofless than one yearand variance ratios less than one athorizons greaterthan one year.T his is consistentwith investorover-reaction (positiveserialcorrelation) intheshort-runandmean-reversion(negativeserialcorrelation)inthelongrun. FollowingCampbell,L oand M cKinlay(1 997),wede¯netheq-horizon varianceratioas V R (q)´ ¾ 2c(q) ¾ 2a (3) wheretheunbiased estimators ¾ 2c(q)and ¾ 2a arede¯ned as ¾ 2a = nq X 1 (R t(1 )¡b ¹ )2 nq¡1 (4) k=1 nq 1 X (R t(q)¡q¹ b)2 lk=1 µ ¶ q l = q(nq¡q+ 1 ) 1 ¡ nq n q 1 X ¹ b = R t(1 ): nq ¾ 2c(q) = k=1 where n is the numberofnon-overlappingsub-samples oflength q.U nderthe assumption thatthereturns areseriallyuncorrelated and homoskedastic, p (5) Z 1 (q)´ nq(V R (q)¡1 )»N (0 ;2 (2q¡1 )(q¡1 )=3q): U ndertheassumptionthatthereturnsareseriallyuncorrelatedandheteroskedastic,theheteroskedasticityconsistentvarianceestimatorofV R (q)is µ(q)´4 where b ±k = ¶2 q¡1 µ X k b 1 ¡ ±k q j=1 P nq 2 2 nq j=k+ 1 (pj ¡pj¡1 ¡b ¹ ) (pj¡k ¡pj¡k¡1 ¡b ¹) : hP i2 nq 2 ¹) j=1 (pj ¡pj¡1 ¡b T heteststatisticforanullofarandom walkis p nq(V R (q)¡1 ) p Z 2 (q)´ »N (0 ;1 ): µ(q) 6 (6) (7) (8) B y recognizingthatthe application ofthe variance ratioofL oand M cKinley (1 988)involves multiple comparisons, i. e. at each horizon we conducta hypothesis test,Chowand D enning(1 993)obtain abound fortheoverallsizeof thetest.Forasequenceofm horizons wecan obtain theteststatistics forthe varianceratios, Z (q1 );:::;Z (qm); (9) where the qi are the horizons thatsatisfy the condition q1 (= 2)< q2 < ::: < qm · N =2. W e de¯ne Z ¤(q)as the largestofthe absolute values ofthe test statistics,i. e. Z ¤(q)= maxfjZ (q1 )j;:::;jZ (qm)jg: (1 0 ) T he(1 ¡®)£1 0 0 % con¯denceintervalforZ ¤(q)is de¯ned as Z ¤(q)§S M M (®;m;1 ) (1 1 ) where S M M (®;m;1 )is the asymptotic criticalvalue ofthe ®¡pointofthe studentized maximum modulus (SM M )distribution with parameterm and degrees offreedom 1 .T hecriticalvalueis obtainedfrom thenormaldistribution by the equality, S M M (®;m;1 )= Z ® + =2 , where ® + = 1 ¡(1 ¡®)1 =m.U nderthe assumption thatthe returns are uncorrelated and homoskedastic, the asymptoticcon¯denceintervalforeach varianceratiois given by p nq(V R (qi)¡1 )§(2 (2qi ¡1 )(qi ¡1 )=3qi)1 =2 S M M (®;m;1 ) i = 1 ;:::;m (1 2) andundertheassumptionthatthereturns areuncorrelatedandheteroskedastic theasymptoticcon¯denceintervalforeach varianceratiois given by p nq(V R (qi)¡1 )§µ1 =2 S M M (®;m;1 ) i = 1 ;::;m: (1 3) T he results of the variance ratio tests for both the homoskedastic and heteroskedastic consistentstandard errors are reported in T able 2. W e focus on thetests basedontheheteroskedasticconsistentstandarderrors.Itis interestingtonote thatonly the G old Index has variance ratios thatare less than 1 . In factwe expectto¯nd mean reversion in the G old Index because the main drivingforce behind this index is a commodity price, the gold price, which is typicallymodelledas ameanrevertingprocess.W erejectthenullofarandom walkforthe A llShareIndex at4 and 8 weeks atthe1 0 % levelofsigni¯cance, butnoneofthe teststatistics aresigni¯cantwhen weuse theChowand D enningcriticalvalues.Contrary toourexpectations, we failtorejectthe nullof a random walkforthe G old Index atalllags.W erejectthenullofarandom walk forthe IndustrialIndex at4 and 8 week lags atthe 1 % levelofsigni¯cance and for2 and 1 6 weeks atthe 1 0 % levelusing the Chowand D enning criticalvalues.Finally, we rejectthe nullofa random walk forthe Industrial and Financial Indexes at a 2 week lag using the Chow and D enning critical 7 values atthe 1 % levelofsigni¯cance underthe assumption ofhomoskedastic returns,butwe cannotrejectthe nullwhen we use the heteroskedasticerrors. T his extremerejection at2 weeks is consistentwith evidence ofahigh degree ofpositiveserialcorrelationandconditionalheteroskedasticityinthereturns of thesetwoindexes and underscores theimportanceofusingtheheteroskedastic consistentstandard errors.W e rejectthe nullofa random walk ata lagof4 weeks whenusingtheheteroskedasticconsistentstandarderrors.Inconclusion, thereis littleevidencetorejectthenullhypothesis ofarandom walkexceptfor theIndustrialIndex. 5 Frac tionalDi®erencing Fractionally integrated processes display long-run dependence and are a feasiblealternativetodi®erencestationarymodels becausetheydisplaythetypical spectralshapecharacterized byG ranger(1 966).A n autoregressivefractionally integrated and movingaverageprocess (A R F IM A (p;d;q))is denotedby Á(L )(1 ¡L )dP t = µ(L )"t (1 4) where jdj< 1 ; L is the lag operator, Á(L )is A R polynomialand µ(L )is the M A polynomial. A ssetprices are weakly stationary ifd< 0 :5, invertible for d> ¡0 :5 and nonstationary ford> 0 :5. . T he intuition behind the long-run dependenceofthe process is derived from the behaviorofthe spectraldensity atfrequency zero;itbehaves like !¡d as ! (the frequency)converges tozero. T hus the typicalshape ofthe spectraldensity can be captured by 0 < d< 1 . In practice,unitroottests have very lowpoweragainstfractionalalternatives so we may be di®erencing too much and not taking into account long-range dependence. G ewekeandP orter-H udak(1 983)havedevelopedatestforfractionaldi®erencingusingthespectralregression, ¡ ¢ ln(I(¸j))= ¯ 0 ¡dln 4 sin2 (¸j=2) + ´j,forj = 1 ;2;:::;n (1 5) where¸ j = 2¼j=T forj = 1 ;2;:::;n,istheharmonicordinate,thesamplesizeor thenumberofperiodogram ordinates n= g(T )¿ T ;I(¸j)is theperiodogram ofreturns atfrequency ¸j and ´j is an errorterm.G ewekeand P orter-H udak suggestthatthebestchoice ofn is given by g(T )= T 0 :5 .T henullhypothesis is d= 0 ,i. e.theprocess is arandom walk.U nderthenull,thestandard error ofdis ¼ 2 =6 andwecancomputeat-statistictotestthenullhypothesis against thealternativeoffractionaldi®erencing. T heestimated values ofdforthe fourindexes are reported in T able3afor ® = 0 :45;0 :5;0 :55 and n = T ® . W e fail to reject the null hypothesis of no long-rundependenceforeachoftheindexes ateveryvalueof®,i. e.wehaveno evidencetorejectthenullhypothesis ofarandom walk. T otesttherobustness oftheG ewekeandP orter-H udaktest,wealsoimplementa testforfractionaldi®erencingdeveloped by A kiakloglou and N ewbold 8 b L )R t = b (1 994).Forthe¯ttedA R IM A (p;q)modelofreturns,Á( µ(L )b "t,A kiakloglouandN ewboldhaveadevelopedanL M testforthehypothesis thatd= 0 as thet-testof± in theregression, where b "t = b µ(L )W Xp i=1 ¯ iW t¡i + Xq ° jZ j=1 t¡j + ±Kt(m)+ ut b Z t =b "t; Kt(m)= t = R t, µ(L ) m X j=1 j¡1 b "t¡j: (1 6) (1 7) N ewbold andA kiakloglou¯ndthatthetesthas empiricalsizeclosetonominal size,butlowpowerwhen p and qare notzeroand the process has anon-zero mean.T heyalso¯ndthatthesampleautocorrelationfunctionisseverelybiased in smallsamples soitis very di±culttodetectfractionaldi®erencingin small samples. W e estimateA R (5)models forthe returns ofthe fourindexes thus setting p = 5 and q = 0 and saving the residuals as b "t. W e construct Kt(m)for m = 3;4;5 andreporttheresults of(1 6)inT able3b.W ecannotrejectthenull ofno long-run dependence forthe A llShare and Industrialindexes although the estimates of± ° uctuate wildly across m.T he estimates of± forthe G old Index are quite uniform across m and range between 0 . 31 1 and 0 . 40 0 , butwe alsofailtorejectthenullhypothesis ofnolong-run dependence.W erejectthe nullofnolong-run dependencefortheFinancialIndexforallvalues ofm.T he rejectionofthenullhypothesis maybeduetothethintradingofstocks inthis indexand itis consistentwith thehigh levelofserialcorrelationofthereturns oftheindex. T he overwhelming support for the random walk hypothesis is surprising fortworeasons:¯rst, D iebold and Inoue (1 999)¯nd thatevidence in support offractionaldi®erencing may be due to a smallamountofregime changes in a series. Second, South A frica was immersed in economic and socioeconomic turmoilduringthe sample period and there were anumberofregime changes during the latter part of the sample period (B eelders (20 0 0 )). D espite the economicmilieu,weonly¯ndevidenceoffractionaldi®erencingintheFinancial Indexand this maybeduetothethin tradingofthesestocks. 6 T he M od i¯ed R esc al ed R ange Statistic T he rescaled range (R /S)statistic was developed forthe purpose ofdetecting long-rangedependenceandhascertainrobustnesspropertiesthatmakeitbetter suited than autocorrelations, variance ratios orspectralanalysis.M andelbrot andW allis (1 969)showthattheR /S statisticcandetectlong-rangedependence in time series that are non-G aussian and have skewness and excess kurtosis. T his is clearly and advantage for¯nancialdata where the distributions tend 9 be leptokurtic.Second,M andelbrot(1 972, 1 975)shows thatthe R /S statistic converges almostsurely even ifthe process has in¯nite variance;this does not hold for the autocorrelations or variance ratios. Finally, M andelbrot (1 972) arguesthattheR /S statisticcandetectnon-periodiccycleswithperiodsgreater than thesamplesize;this is adistinctadvantageoverspectralanalysis. L o (1 991 )develops the modi¯ed R /S statistic thatis robust to short-run dependence by using an estimate ofthe long-run standard deviation to standardizethestatistic.L o¯nds thatthemodi¯ed R /S statistichas less bias and has bettersize. Forexample, forthe alternative ofa fractionally di®erenced process, the probability thatthe modi¯ed R /S statistic rejects the nullofno long-range dependence converges to 1 . U sing the CR SP equally- and valueweightedindexes L o¯nds thatthereis noevidenceoflong-rundependence,but thedataareconsistentwith ashort-memoryprocess. T hemodi¯edR /S statisticis calculated as follows: 2 3 Xk ¡ Xk ¡ ¢ ¢ 1 4 max Q n´p (1 8) R j ¡R n ¡ min R j ¡R n 5 1 ·k·n n¢ ¾ bn(q) 1 ·k·n j=1 j=1 where ¾ b2n(q) ´ ¾ b2x + 2 ¾ b2R ´ °bj ´ Xq j=1 !j(q)b ° j; !j(q)´1 ¡ n ¢2 1 X ¡ R j ¡R n n j=1 n ¢¡ ¢ 1 X ¡ R j ¡R n R i¡j ¡R n n j=1 j ; q< n q+ 1 (1 9) (20 ) (21 ) and n is the sample size. T he key modi¯cation of the R /S is the use of a consistentestimatorofthe variance ofthe partialsums underthe assumption thatthere is short-run dependence in the data, i. e. use ¾ bn(q)instead of ¾ bR in computing the R =S test statistic. T he estimated values ofthe R =S and modi¯ed R =S test statistics are reported in T able 4. W e fail to reject the nullhypothesis ofno long-run dependence foreach ofthe fourindexes. T his resultis consistentwith thevarianceratiotests wherewefound littlelong-run dependence.W eacceptourresults withanoteofcautionbecauseP agan(1 996) has notedthatthechoiceofqis anunresolvedissue,i. e.howdowedistinguish between the short- and long-run (M ills (1 999), p. 1 1 8).T here is alsoevidence thatthedistribution oftheteststatisticis sensitivetofat-tailed distributions, thus L osuggestthatthe R =S testshould be regarded as a portmanteau test before conductinga more thorough analysis oflong-run dependence.Keeping these caveats in mind, we conclude that there is no evidence to suggest the presenceoflong-run dependencein theJSE Indexes. 10 7 Concl usion W ehaveusedthevarianceratiotest,twotestsforfractionaldi®erencingandthe modi¯ed rescaled rangestatistictotestforlong-run dependencein thereturns offourbroad South A frican Indexes. Forthe A llShare Index and the G old Indexwefailtorejectthenulloftherandom walkmodel.T his is particularly surprisingfortheG oldIndexwhereweexpectto¯ndevidenceofmeanreversion. Forthe IndustrialIndex, we can rejectthe nullhypothesis ofa random walk usingthevarianceratiotestatthe4 and8-weekhorizon,butwe¯ndnoevidence oflong-rundependenceusingthetestforfractionaldi®erencingortherescaled range test.Forthe FinancialIndex, we ¯nd evidence oflong-run dependence using the N ewbold and A giakloglou (1 994)test, butthis may be due to the thintradingofthestocks inthis index.Inconclusion,thereis littleevidenceto suggests thattheJSE is note±cientdespitetheeconomicturmoilsurrounding South A fricaduringthesampleperiodandtheonlyrejectionofthehypothesis ofe±ciencymaybeduetothin trading. R ef erences [1 ]A ²eck-G raves,J. P and A . H .M oney(1 975)\ A N oteontheR andom W alk M odel and South A frican Share P rices," South A frican Journalof Economics,43,3,382-388. [2]A giakloglou,C.andP.N ewbold(1 994)\ L agrangeM ultiplierT estsforFractionalD i®erence,"JournalofT ime Series A nalysis,1 5,253-262. [3]B eelders,O wen (20 0 0 )\ T heU nconditionalD istribution ofSouth A frican StockR eturns,"workingpaper,D epartmentofEconomics,EmoryU niversity. [4]Campbell, John Y . , A ndrew W .L o and Craig M acKinlay (1 997) T he Econometrics ofFinancialM arkets,P rincetonU niversityP ress,P rinceton, N ewJersey. [5]Chow,V . K.and K. D .D enning(1 993)\ A SimpleM ultipleV arianceR atio T est,"JournalofEconometrics,5,385-40 1 . [6]D iebold,Francis X .(1 986),\ T estingforSerialCorrelation in theP resence ofA R CH ,"P roceedings oftheA merican StatisticalA ssociation,B usiness andEconomics Statistics Section,p. 323-328. [7]D iebold, Francis X .and A tsushi Inoue (1 999)\ L ongM emory and StructuralChange,"workingpaper,SternSchoolofB usiness,N ewY orkU niversity. [8]Fama, E. F.(1 965)\ T he B ehavior of Stock M arket P rices," Journal of B usiness,38,34-1 0 5. 11 [9]Fama,E. F.andK.French(1 988)\ P ermanentandT emporaryComponents in StockP rices,"JournalofP oliticalEconomy,96,246-273. [1 0 ]G eweke,J.and S.P orter-H udak(1 983)\ T heEstimation and A pplication ofL ongM emoryT imeSeries M odels,"JournalofT ime Series A nalysis,4, 221 -238. [1 1 ]G ranger,CliveW .J.(1 966)\ T heT ypicalSpectralShapeofan Economic V ariable,"Econometrica,34,1 50 -1 61 . [1 2]H arrison, J.M ichaeland D avid M .Kreps (1 979)\ M artingales and A rbitragein M ultiperiod Securities M arkets,"JournalofEconomicT heory,2, 3,381 - 40 8. [1 3]H arrison,P auland H arold H .Z hang(1 999)\ A n Investigation oftheR isk and R eturn R elation atL ongH orizons,"R eviewofEconomics and Statistics,81 ,3,399-40 8. [1 4]L o,A ndrewW .(1 991 )\ L ong-T erm M emory in StockP rices,"Econometrica,59,5,1 279-1 31 3. [1 5]L o,A ndrewW .and A .CraigM acKinley (1 988)\ Stock M arketP rices do not follow R andom W alks:Evidence from a Simple Speci¯cation T est," R eviewofFinancialStudies,1 ,41 -66. [1 6]L o, A ndrew W .and Jiang W ang (1 995)\ Implementing O ption P ricing M odels W henA ssetR eturns A reP redictable," JournalofFinance,50 ,1 , 87-1 29. [1 7]M alkiel,B urtonG .(1 973)A R andom W alkD ownW allStreet,W . W .N orton,N ewY ork. [1 8]M andelbrot, B enoit B .(1 972)\ Statistical M ethodology for N onperiodic Cycles: From the Covariance toR /S A nalysis,"A nnals ofEconomic and SocialM easurement,1 /3,259-290 . [1 9]M andelbrot, B enoit B .(1 975)\ L imit T heorems on the Self-N ormalized R ange for W eakly and Strongly D ependent P rocesses," Z eitschrift vir W hrscheinlichkeitstheorie verwandte G ebiete,31 ,271 -285. [20 ]M andelbrot,B enoitB .andJ. R .W allis (1 969)\ SomeL ong-R unP roperties ofG eophysicalR ecords,"W aterR esources R esearch,5,321 -340 . [21 ]M ills,T erence(1 999)T heEconometricM odellingofFinancialT imeSeries, Second Edition,CambridgeU niversityP ress,Cambridge. [22]O jah,Kaluand D avid Karemera(1 999)\ R andom W alks andM arketE±ciencyT ests ofL atinA mericanEmergingEquityM arkets:A R evisit,"T he FinancialR eview,34,57-72. 12 [23]P agan, A drian (1 996)\ T he Econometrics ofFinancialM arkets," Journal ofEmpiricalFinance,3,1 5-1 0 2. [24]P arke, W illiam R .(1 999)\ W hat is Fractional Integration?," R eview of Economics andStatistics,81 ,4,632-638. [25]P oterba,J. M .andL . H .Summers (1 988)\ M eanR eversioninStockP rices: EvidenceandImplications,"JournalofFinancialEconomics,22,27-59. [26]U rrutia, J. L .(1 995)\ T ests of R andom W alk and M arket E±ciency for L atin A merican EquityM arkets,"JournalofFinancialR esearch,1 8,29930 9. 13 T able1 : D escriptive Statistics ofthe W eeklyR eturns ofthe JSE Indexes A superscripta,band cdenotes thestatisticalsigni¯canceatthe1 0 %, 5% and 1 % levels ofsigni¯cance,respectively.Q R (k)denotes theB ox-L jung Q -statisticthatis computed forthereturns usingklags and Q R 2 (k)denotes the B ox-L jungQ -statisticthatis computed forthesquared returns usingklags. A llShare G old SampleSize M ean Standard D eviation 60 6 0. 1 0 9b 1. 272 60 6 -0 . 0 30 5. 570 Industrial Financial 60 6 0. 1 27b 1. 497 60 6 0. 1 59b 1. 494 Skewness iid t-statistic G M M t-statistic -0 . 797 0. 427 (-7. 987)c (4. 283)c (-3. 322)c (0 . 0 0 2) -0 . 525 (-5. 272)c (-1 . 658)a -1 . 678 (-1 6. 825)c (-0 . 784) Kurtosis iid t-statistic G M M t-statistic 3. 828 (1 9. 1 23)c (2. 676)c 3. 80 3 (1 8. 999)c (3. 0 54)c 1 4. 486 (72. 50 3)c (2. 657)c L ags A utocorrelations 1. 1 87 (5. 898)c (2. 660 )c 1 2 3 4 5 0. 0 89b 0. 0 77a 0. 011 0. 0 71 a -0 . 0 261 0. 0 63 -0 . 1 1 4c -0 . 006 0. 0 74 -0 . 1 1 4c 0. 1 32c 0. 1 0 5c 0. 007 0. 0 68 a 0. 0 20 0. 1 57c 0. 0 71 a 0. 013 0. 004 0. 0 39 Q R (5) Q R (1 0 ) 1 2. 0 45c 1 3. 846c 21 . 71 8 c 29. 799c 20 . 498 c 29. 651 c 1 9. 246c 36. 428 c Q R 2 (5) Q R 2 (1 0 ) 30 . 98 c 42. 77c 30 . 0 4c 40 :82c 28:45c 63. 01c 110. 54c 1 70 . 00c 14 T able2: V ariance R atios forthe W eeklyR eturns ofthe JSE Indexes T hestandard errors undertheassumption ofhomoskedasticreturns arereported in parentheses and thestandard errors undertheassumption ofheteroskedastic returns arereportin brackets.O ne,twoand threeasterisks denotes thatthetest statisticis statisticallysigni¯cantusingtheChowand D enning(1 993)critical values atthe1 0 %,5% and1 % levelofsigni¯cance,respectively.A superscript a,band cdenotes thestatisticalsigni¯canceatthe1 0 %,5% and 1 % levels of signi¯cance,respectively. k-horizon Index 2 4 8 16 A llShare 1. 0 88 (2. 1 75)b [1 . 455] 1. 20 3 (2. 682)c a [1 . 931 ] 1. 265 (2. 20 4)b [1 . 753]a 1. 261 (1 . 460 ) [1 . 238] A llG old 1. 0 63 (1 . 561 ) [1 . 427] 0. 973 (-0 . 358) [-0 . 326] 0. 928 (-0 . 595) [-0 . 533] 0. 925 (-0 . 421 ) [-0 . 379] Industrial 1. 1 33 (3. 287)c¤¤¤ ¤ [2. 229] 1. 30 6 (4. 0 29)c c¤ ¤ ¤ [2. 90 5] 1. 447 (3. 721 )c [2. 923]c¤¤¤ 1. 453 (2. 534)b [2. 0 69]b¤ Financial 1. 1 60 (3. 945)c¤¤¤ [1 . 457] 1. 31 4 (4. 1 27)c a [1 . 749] 1. 370 (3. 0 78)c [1 . 50 6] 1. 224 (1 . 256) [0 . 677] 15 T able3 a: Estimates of d usingthe G eweke and P orter-H udakM ethod W ereporttheestimateofdand its standard errorin parenthesis where thenumberofweeklyobservations is 60 6.W eobtain thesameconclusion when usingdailydata,i. e.thereis noevidenceoflong-run predictability. ® Index 0. 45 0. 5 0. 55 A llShare 0. 1 97 (0 . 634) 0. 0 63 (0 . 244) 0. 002 (0 . 011) A llG old 0. 1 33 (0 . 427) -0 . 0 40 (-0 . 1 56) -0 . 0 65 (-0 . 291 ) Industrial 0. 0 97 (0 . 31 2) -0 . 003 (-0 . 0 1 3) -0 . 0 56 (-0 . 251 ) Financial 0. 258 (0 . 829) 0. 0 628 (0 . 244) -0 . 0 46 (-0 . 20 7) 16 T able3 b: Estimates of ± usingthe A kiakloglou and N ewbold M ethod W e¯tA R (5)models tothereturns and usetheresiduals (b "t)tocompute theregressors in (1 6).T heestimates of± and thet-statisticarereportedin the tableabove.A superscripta,b and cdenotestatisticalsigni¯canceatthe 1 0 %,5% and 1 % levels ofsigni¯cancerespectively. m Index 3 4 5 A llShare 0. 1 39 (0 . 0 48) 1. 0 88 (0 . 40 8) 0. 622 (0 . 478) A llG old 0. 383 (1 . 0 49) 0. 31 1 (1 . 0 38) 0. 398 (1 . 323) Industrial 1. 40 2 (1 . 223) 0. 257 (0 . 331 ) 0. 236 (0 . 449) Financial 2. 256 (2. 449)b 1. 427 (1 . 91 8)a 0. 879 (1 . 847)b 17 T able4: R escaled R ange Statistic T hecriticalvalues fortheR /S statisticareobtained from T ableII ofL o(1 991 ). Index R /S Statistic M odi¯ed R /S Statistic A llShare 1. 1 22 0. 995 A llG old 0. 942 0. 947 Industrial 1. 386 1. 1 85 Financial 1. 228 1. 0 57 18