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