Technical Trading Rules and Regime Shifts in Foreign Exchange

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Technical Trading Rules and
Regime Shifts in Foreign
Exchange
Blake LeBaron
SFI WORKING PAPER: 1991-08-044
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www.santafe.edu
SANTA FE INSTITUTE

TECHNICAL
TRADING
RULES
AND
REGIME
SHIFTS
IN
FOREIGN
EXCHANGE
Blake
LeBaron
Department
ofEconomics
UniversityofWisconsin-Madison
BLAKEL@vms.macc.wisc.edu
May1991
Revised:
October
1991
Abstract
This
paperperforms
tests
onseveraldifferentforeignexchangeseriesusingamethodologyinspired
by
technical
trading
rules.Movingaveragebasedrulesareusedasspecification
tests
on
the
processforforeign
exchangerates.Severalmodelsforregimeshifts
and
persistenttrendsaresimulated
and
comparedwith
resultsfrom
the
actual
series.
The
resultsshow
that
thesesimplemodelscan
not
capture
someaspects
of
the
seriesstudied.Finally,
the
economicsignificance
of
the
trading
ruleresultsaretested.Returnsdistributions
fromthetradingrulesarecomparedwithreturnsonriskfreeassetsandreturnsfromtheU.S.stockmarket.
Acknowledgments
The
author
is
grateful
to
Robert
Hodrick,Simon
Potter,
seminarparticipants
at
the
Mid-WestInternational
EconomicsMeetings,
The
University
of
Iowa,andCarnegie-MellonUniversity
for
commentson
an
earlier
draft.
This
researchwas
partially
supported
by
the
EconomicsResearch
Program
at
the
Santa
Fe
Institute
whichisfunded
by
grants
from
Citicorp/Citibank
and
the
RussellSageFoundation
and
bygrants
to
SFIfrom
the
John
D.
and
Catherine
T.
MacArthur
Foundation,
the
NationalScienceFoundation(PHY-8714918),
the
U.S.
Department
of
Energy(ER-FG05-88ER25054),
and
the
AlexC.WalkerEducational
and
Charitable
Foundation-Pittsburgh
NationalBank.
The
author
also
is
grateful
to
theNationalScienceFoundation
(SES-9109671)for
support.
I.
Introduction
Techniquesforusing
past
pricestoforecastfuturepriceshasalong
and
colorfulhistory.Since
theintroductionoffloatingrates
in
1973theforeignexchangemarketbecameanotherpotential
targetfor"technical"analystswho
attempt
to
predictpotential
trends
in
pricingusingavast
repertoire
of
toolswithcolorfulnamessuchaschannels,
tumble~,
steps
and
stumbles.These
markettechnicianshavegenerallybeendiscreditedin
the
academic
literature
since
their
methods
are
sometimesdifficultto
put
to
rigoroustests.This
paper
attempts
tosettlesomeofthese
discrepancies
through
the
useofbootstrappingtechniques.
Forstock
returns
manyearlystudiesgenerallyshowedtechnicalanalysistobeuseless,while
for
foreignexchange
rates
thereisnoearlystudyshowing
the
techniques
to
beofnouse.Dooleyand
Shafer(1983)foundinterestingresultsusingasimplefilterruleonseveraldailyforeignexchangerate
series.In
later
workSweeney(1986)documentstheprofitabilityofasimilarruleon
the
Deutsche
Mark.In
an
extensivestudy,Schulmeister(1987)repeatstheseresultsforseveraldifferenttypesof
rules.Also,Taylor(1990)finds
that
technicaltradingrulesdo
about
aswell
as
some
of
his
more
sophisticated
trend
detectingmethods.
Whilethesetestswereproceeding,
other
researchersweretrying
to
usemoretraditionaleco­
nomicmodels
to
forecastexchangerateswithmuchlesssuccess.
The
most
important
ofthesewas
Meese
and
Rogoff(1983).Theseresultsshowed
the
randomwalktobe
the
best
outofsampleex­
change
rate
forecastingmodel.Recently,resultsusingnonlineartechniqueshavebeenmixed.Hsieh
(1989)findsmost
of
the
evidencefornonlinearitiesindailyexchangerates
is
comingfromchanging
conditionalvariances.Diebold
and
Nason(1990)andMeese
and
Rose(1990)foundnoimprove­
mentsusingnonparametrictechniquesin
out
ofsampleforecasting.However,LeBaron(1990)
and
Kim(1989)showsmall
out
ofsampleforecastimprovements.DuringsomeperiodsLeBaron(1990)
foundforecastimprovements
of
over5percentinmeansquarederrorfor
the
GermanMark.Both
thesepapersreliedonsomeresultsconnectingvolatilitywithconditionalserialcorrelationsofthe
series.
Thispaperbreaksoffof
the
traditionaltimeseriesapproaches
and
usesatechnicaltradingrule
methodology.
With
the
bootstrap
techniquesofEfron(1979),someof
the
technicalrulescanbe
put
to
amorethoroughtest.This
is
doneforstockreturnsinBrock,Lakonishok,andLeBaron(1990).
1
This
paper
willusesimilarmethodsto
study
exchangerates.Theseallow
not
only
the
testingof
simple
random
walkmodels,
but
the
testingofanyreasonablenullmodel
that
canbesimulated
on
the
computer.Inthissense
the
tradingrulemovesfrombeingaprofitmakingtooltoanew
kindofspecificationtest.Thetradingruleswillalsobeusedasmomentconditions
in
asimulated
method
of
momentsframeworkforestimatingsimplelinearmodels.
Finally,
the
economicsignificanceoftheseresultswillbeexplored.Returnsfrom
the
trading
rulesappliedto
the
actualserieswillbetested.Distributionsofreturnsfrom
the
exchange
rate
serieswill
be
comparedwiththosefromriskfreeassetsandstockreturns.These
test
are
important
indetermining
the
actualeconomicmagnitude
of
thedeviationsfrom
random
walkbehavior
that
areobserved.
Section
II
willintroduce
the
simplerulesused.Section
III
describesthenullmodelsused.
SectionIVwillpresentresults
for
thevariousspecificationtests.SectionVwillimplementthe
tradingrules
and
compare
return
distributionsandsectionVIwillsummarize
and
conclude.
II.
Technical
Trading
Rules
Thissectionoutlinesthetechnicalrulesusedinthispaper.
The
rulesarecloselyrelatedto
thoseusedbyactualtraders.All
the
rulesusedhere
are
ofthemovingaverage
or
oscillatortype.
Here,signalsaregeneratedbasedon
the
relativelevelsofthepriceseries
and
amovingaverage
of
past
prices,
L-!
ma,
=
(l/L)
L
P'-i'
i=O
Foractualtradersthisrulegeneratesabuysignalwhenthecurrentpricelevelisabovethemoving
average
and
asellsignalwhen
it
isbelow
the
movingaverage.!This
paper
willusethesesignals
to
study
variousconditionalmoments
of
the
seriesduring
buy
and
sellperiods.Estimatesofthese
conditionalmoments
are
obtainedfromforeignexchangetimeseries,
and
theseestimatesarethen
comparedwiththosefromsimulatedstochasticprocesses.SectionIVof
the
paper
differsfrommost
tradingrulestudieswhichlook
at
actualtradingprofitsfromarule.Actual
trading
profitswillbe
exploredinsection
V.
1
There
are
many
variations
of
this
simple
rule
in
use.
One
is
to
replace
the
price
series
with
another
moving
average.
A
second
modification
is
to
only
generate
signals
when
the
price
differs
from
the
moving
average
by
a
certain
percentage.
Many
other
modifications
are
discussed
in
Schulmeister(1987),
SweeneY(1986),
and
Taylor(1990).
2
III.
Null
Models
for
Foreign
Exchange
Movements
Thissectiondescribessomeofthenullmodelswhichwillbeusedforcomparisonwiththe
actualexchange
rate
series.Thesemodelswillbe
run
through
the
sametradingrulesystemsas
the
actual
data
and
then
comparedwiththoseseries.Severalofthesemodelswill
be
bootstrapped
inthespirit
of
Efron(1979)usingresampledresidualsfrom
the
estimatednullmodel.Thisclosely
followssomeof
the
methodsusedinBrocket.al.(1990)for
the
DowJonesstockpriceseries.
The
firstcomparisonmodelusedistherandomwalk,
log(p,)
=
log(p,_
tl
+
f,
.
Logdifferencesof
the
actualseriesareusedasthedistributionfor
f,
and
resampledorscrambled
withreplacement
to
generateanewrandomwalkseries.
The
newreturnsserieswillhave
all
the
sameunconditionalpropertiesastheoriginalseries,
but
anyconditionaldependencewillbelost.
The
secondmodelusedis
the
GARCHmodel(Engle(1982)andBollerslev(1986)).Thismodel
attempts
to
capture
someof
the
conditionalheteroskedasticity
in
foreignexchange
rates?
The
modelestimatedhereisoftheform
z,
~
N(O,l).
ThismodelallowsforanAR(2)processinreturns.Thespecificationwasidentifiedusingthe
Schwartz(1978)criterion.OnlytheJapaneseYenseriesrequiredthetwolags,
but
for
better
comparisonsacrossexchangerates
the
samemodelisused.
3
Estimationofthismodel
is
doneusing
maximumlikelihood.
Simulationsof
this
modelfollowthosefor
the
randomwalk.Standardizedresidualsofthe
GARCHmodelareestimatedas,
2
For
more
extensive
descriptions
of
these
results
on
exchange
rates
see
Hsieh{1988,1989)
and
other
references
contained
in
Bollerslev
ct.
a1.(1990).
3
Other
specifications
with
changing
conditional
means
related
to
volatility
(GARCH-M)
were
also
tried,
but
these
turned
out
insignificant.
This
agrees
with
some
of
the
results
found
in
Domowitz
and
Hakio(1985).
3
Theseresidualsare
then
scrambled
and
the
scrambledresiduals
are
thenused
to
rebuildaGARCH
representationfor
the
data
series.Usingtheactualresidualsforthesimulationsallows
the
residual
distributiontodifferfromnormality.Bollerslev
and
Woldridge(1990)haveshown
that
theprevious
parameter
estimateswillbeconsistentundercertaindeviationsfromnormality.Thereforethe
estimatedresidualswillalsobeconsistent.4
The
third
modelhasbeenproposedforforeignexchangemarkets
in
a
paper
byEngleand
Hamilton(1990).
It
suggests
that
exchangeratesfollowlongpersistentswingsfollowinga2
state
markovchain.
It
isgivenby,
T,
=
(flo
+
fllS,)
+
(aD
+
a,S,)z,
P(S,
=
11S'_1
=
1)
=
P
P(S,
=
0IS'_l
=
1)
=
1-P
P(S,
=
0IS'_l
=
0)
=
q
P(S,
=
11S'_1
=
0)
=
1-
q
z,
~
N(O,1).
This
modelallows
both
the
mean
and
varianceforexchange
rate
returnstomovebetweentwo
differentstates.Sincethismodeliscapableofgeneratingpersistenttrends
it
presentsastrong
possibilityforgenerating
the
resultsseenusing
the
tradingrules.Estimationisdoneusingmaximum
likelihood.Forthismodel
the
simulationswillusenormallydistributed
random
numbersfroma
computerrandomnumbergenerator.
4
The
convergence
of
the
bootstrap
distribution
has
not
been
shown
for
GARCH
models.
Brock,
Lakonishok,
and
LeBaron(1990)
use
a
similar
technique
for
stock
returns.
Their
results
are
supported
through
large
computer
simulations.
4
IV.
Empirical
Results
A.
Data
Summary
The
data
usedinthispaperareallfrom
the
EHRAmacro
data
tape
from
the
FederalReserve
Bank.WeeklyexchangeratesfortheBritishPound
(BP),
GermanMark(DM),andJapaneseYen
(JY)
aresampledeveryWednesdayfrom
January
1974throughFebruary1991
at
12:00pmEST.
Returnsarecreatedusinglogfirstdifferencesoftheseweeklyexchangeratesquotedindol­
lars/fx.Table1presentssomesummarystatisticsforthese
return
series.Allthreeseriesshow
littleevidenceofskewnessandareslightlyleptokurtic.Thesepropertiesarecommon
for
many
highfrequencyassetreturnsseries.
The
first
10
autocorrelationsaregiven
in
therowslabeled,
Pn.
The
Bartlett
asymptotic
standard
errorfortheseseriesis0.033.
The
BP
showslittleevidenceof
any
autocorrelationexceptforlags4
and
8,whilethe
DM
showssomeweakevidenceofcorrelation,
and
the
JY
showsstrongevidenceforsomeautocorrelation.TheLjung-Box-Piercestatisticsare
shownonthe
last
row.Thesearecalculatedfor
10
lags
and
aredistributed
X2
(1O)
underthenullof
i.i.d.
The
p-valuesareincludedforeachinparenthesis.The
BP
and
JY
seriesrejectindependence
whiletheDMseriesdoesnot.
The
interest
rate
seriesusedarealsofromtheEHRAmacro
data
tape.Forthedollarthe
weeklyeurodollar
rate
isused.Forthe
pound,
theinternationalmoneymarketcallmoney
rate
is
used.
For
the
mark,
the
Frankfurtinterbankcallmoney
rate
isused,
and
for
the
yen,theTokyo
unconditionallenderrate.Weeklyratesareconstructedexpostfromthecompoundedratesfrom
WednesdaythroughTuesday.Theseratescanonlybeviewedasproxiesfor
the
desirablesituation
ofhavingasetofinterestratesfromthesameoffshoremarket
at
thesamematurity.
At
thistime
that
is
not
available.
B.
Random
Walk
Comparisons
Inthissectionsimulationsareperformedcomparingconditionalmomentsfrom
the
technical
tradingruleswitha
bootstrapped
randomwalkgeneratedfrom
the
actual
returns
timeseries
scrambledwithreplacement.Threemovingaverageruleswillbeused,
the
20
week,30week,
and
50
weekmovingaverages.Thesearefairlycommonlengthsusedbytraders.Wewillsee
that
the
resultsarenotverysensitivetothelengthsused.
The
movingaveragerulesforceusto
start
the
5
studyafteracertainnumberofweekshavegoneby.Forthis
paper
alltestsforall
the
rulesbegin
afterweekfifty.Thisgives
the
samenumberofweeklyobservationsforallthreerules.
Table2presents
the
resultscomparingtheactualseriesfor
the
British
Pound
with500simu­
lated
random
walks.Sixcomparisonstatisticsarecomputed
in
thistable.
First,
the
columnlabeled
Buyrefers
to
the
conditionalmeanduringbuyperiods.Thisis,
N-l
m,
=
(liN,)
L
Tt+l
I
:,
t=O
where
N,
arethenumberofbuy'ssignalsinthesample
and
It
isaindicatorvariableforabuy
signal
at
time
t.
The
secondcolumn,labeled
CT"
looks
at
the
standard
deviationofthissameset
of
returns
(buy's).Thisis,
N-l
((liN,)
L
(Tt+l-
m,)2In
1
/
2
.
t=O
Thisgivesasimpleideaofhowriskythebuyorsellperiodsmightbe,
and
tellsussomething
about
what
ishappeningtoconditionalvariances.Thethirdcolumn,labeledFractionBuy,is
just
the
fraction
of
buyweeks,
N,IN.
The
nexttwocolumns,Sell
and
CT,
repeat
the
previousdescriptions
for
the
sellperiods.Let
m,
bethemeanduringthesellperiods.
The
finalcolumn,Buy-Sell,refers
to
the
differencebetween
the
buy
and
sellmeans,
m,
-
m,.
Thistablepresentsseveralresultsforeachtest.Thefirstisthefractionofsimulatedrandom
walks
that
generateagivenstatisticgreater
than
that
fortheoriginalseries.Thiscanbethought
of
as
asimulatedp-value.For
the
20
weekmovingaveragerulesthisresultisgivenin
the
firstrow
of
thetable.Forthe
BP
series
we
see
that
8percentofthesimulationsgeneratedameanreturn
greater
than
that
from
the
actualseries.
The
nextrow,SimulationMean,showsthemeanof
m,
forthe500simulatedrandomwalks,andthethirdrow,
Xrate
Mean,shows
m,
for
the
exchange
rate
series.For
the
BPseriesthetablereportsamean1weekbuy
return
of0.091percentwhich
isgreater
than
the
simulatedmeanof-0.012percent.
The
simulationsshow
that
thisdifference
is
weaklysignificantwith8percentof
the
simulationsgeneratinga
m,
greater
than
0.091percent.
The
secondrowshows
the
resultsforthe
standard
deviationsof
the
buy
returns,
CT,.
The
columnshows
that
56pecentof
the
simulationshad
standard
deviationsgreater
than
that
inthe
originalseries.Thisclearlyshowsnosignificantdifferencebetween
the
simulationsand
the
original
series.
In
otherwords,while
the
buysgeneratealargermeantheydo
not
havealargervariance.
6
The
next
columnreports
that
the
fractionofbuys
to
sellsfor
the
actual
series,row3,
is
0.486.
This
doesnot
appear
to
beunusuallylargeorsmallrelative
to
the
simulated
random
walks.
For
the
sells,
m,
for
the
British
pound
seriesis-0.134percentwhichcompareswith-0.014
percentfor
the
simulation.Table1shows
that
98
percent
of
the
simulated
random
walksgenerated
m,
statisticslarger
than
-0.134indicating
that
the
sellperiodreturnsfor
the
originalseriesare
unusuallysmallwhencomparedwith
the
random
walk.
The
nextcolumn,
u"
shows
that
these
returns
arenotdifferentfrom
the
entiresampleintermsofvolatility.
The
finalcolumnreports
the
difference
m,
-
m,.
Forthisrule
the
differenceis
about
0.2
percent,
and
none
of
the
simulated
random
walksgeneratedsuchalargedifferencebetweenbuy
and
sellreturns.
The
next
6rowsof
the
table
repeat
the
sameresultsfor
the
other
tworules,
the
30
and
50
weekmovingaveragerules.
The
resultsfortheserulesaresimilar
to
the
firsttwowith
the
buy
meansunusuallylarge
and
the
sellmeansunusuallysmall.Therestillappears
to
be
noeffectin
volatility.5
The
finalsetoftestsperformajointtestbasedonallthreerules.Anaverageistakenfor
the
statisticsgeneratedfromeachof
the
threerules.For
the
mean
buysthiswouldbe,
m,
=
1/3(m,(1,20)
+
m,(1,30)
+
m,(1,50)).
Finding
the
distributionofthis
statistic
wouldrequireknowing
the
jointdistributionacrossall
the
rules.
The
resultsforeachruleareclearlyfarfromindependentsothiswouldbeadifficultjob.
With
the
simulatedrandomwalks
the
rulescannowbecomparedwithresultsfor
the
same
average
statisticsover
the
500simulated
random
walks.Thissectionof
the
tableshows
that
the
pattern
foreach
of
the
individualrules
is
repeated
in
the
averagerules.
Agoodquestion
to
ask
at
this
point
ishowgeneraltheseresultsarefordifferentmoving
averages.
This
paper
hasusedonly3differentmovingaveragerules.Thesearechosen
to
be
close
to
thoseusedby
actual
traders.
It
isquitepossible
that
there
may
be
some
data
snoopingproblems
herein
that
theseruleshavealreadybeenchosenbecauseoftheir
past
performancein
the
data.
This
problemispartiallyaccountedfor
in
figure1whichdisplays
the
buy-selldifferencesforseveral
5
This
is
generally
the
case
for
all
the
exchange
rate
tests
used
here.
It
differs
from
some/of
the
results
in
Brock
et..
a1.
(1990)
where
stock
returns
were
found
to
be
more
volatile
during
se'u
periods
than
during
buy
periods.
7
differentlengthsofmovingaverages.
It
isclearfromthisfigure
that
the
resultsarenotoverly
sensitive
to
thelengthofthemovingaveragechosen.
Thenexttwotables,3
and
4,
repeattheresultsfortheDM
and
JY
series.Turningtothe
averagerows
we
seeverysimilarresultstotable
2.
The
buy-selldifferencesarelargefor
both
with
p-values
of
o.
For
the
JY
series
the
standard
deviationsduring
the
buy
and
sellperiodsare
not
unusually
smallorlarge.For
the
DMseriessomeweakdifferences
appear
between
the
standard
deviations
during
the
buy
and
sellperiods.Fortheaverageacrosstherulesusingthe
buy
standard
deviations
the
simulatedp-valueis0.87,indicating
that
87
percentof
the
simulationsweremorevolatilethan
the
actualexchange
rate
series.For
the
sellsthisvalueis0.12,indicating
that
12
percentofthe
simulationsweremorevolatile
than
theoriginalseries.Thisshowssomeweakevidence
that
the
buy
periodswerelessvolatile
than
average
and
thesellsweremorevolatile
than
average.
The
results
are
pretty
weakfor
the
averagerule,
but
checkingtheindividualrulesstrongerrejectionsarefound
forthe
30
and
50
weekmovingaveragesindividually.Thisresultmovescounter
to
asimplemean
varianceconnectionfortheexchange
rate
fromadollarperspective.Thehigherconditionalreturns
from
the
buyperiodshouldbecompensatingformorerisk,
but
theseresultsshow
that
forthe
DM
the
risk(intermsofown
standard
deviation)islower.Whilethisispuzzling,measuringthe
riskiness
of
aforeignexchangeseriesismorecomplicated
than
estimating
the
standard
deviation,
sostrongconclusions
about
riskpremiarequiremoreadequatemodeling
of
the
exactrisk-return
trade
off.
Anothercheckforchangesintheconditionaldistributionsofreturnsisperformedintable
5.
In
thistableskewness
and
kurtosisareestimatedforthe
returns
during
the
buy
and
sellperiods.
It
ispossible
that
thesehighermomentsmightgivea
better
indicationof
the
riskinessofreturns
duringeach
of
the
periods.Thistablecombines
the
resultsfor
the
3seriesintoonetable.The
individualtestsaresummarizedwithasinglerow
entry
givingtheirsimulatedp-valueandthe
averages
are
presentedinthreerowsforeachexchangerate.Thistableshowslittledifferencein
thehighermomentsfromtheactualseriesbuy
and
sellperiods
and
theirsimulationcounterparts.
Table6considers
the
stabilityoftheseresultsovervarioussubsamples.
It
isquitepossible
that
theserulesmaybepickingupcertainnonstationaritiesin
the
data
series.
The
rulesthemselves
are
probablyverygood
at
checkingforchangesinregime.
If
theseregimechangesarerelatively
8
infrequentthensplittingthesampleintotwo
and
repeating
the
testsmakes
it
lesslikely
that
the
ruleswilldetect
any
differencesbetweenthebuy
and
sellperiods.Table6presentsresultsforsuch
anexperiment,whereeachseries
is
brokeninhalfandtheprevious
random
walksimulationsare
repeatedforeachsubsample.
For
the
BP,
the
resultsarebasicallyunchangedacrossthesubsamples.However,thetrading
ruleresultslookslightlylesssignificantin
the
secondsubsample.
The
simulatedp-valueforthe
averagebuy-selldifferencemovesfrom0
to
0.052.Also,
the
averagebuy-selldifferencefallsfrom
0.37percent
to
0.195percent.TheDMseriesshowssimilarresultsforthebuy
and
sellmeansin
the
twodifferentsubsamples.
The
p-valuefortheaveragebuy-selldifferencemovesfrom0.004in
the
firstsubsample
to
0
in
the
secondsubsample.
The
averagebuy-selldifferenceincreasesfrom
0.26percent
to
0.34percent.Forthe
standard
deviations
the
resultslookdifferent.Forvolatility,
the
smallvolatilityduringbuyperiodsiscomingentirelyfrom
the
firstsubperiod.For
the
average
standard
deviations
the
p-valuefor
cr,
is0.994forthefirstsubsample
and
0.330forthesecond
subsample.
The
resultson
cr,
alsoaremuchstrongerduringthefirstsubsamplewithap-valueof
0.01during
the
firstsubsample
and
0.566during
the
secondsubsample.
The
resultsfor
the
JY
serieschangeverylittlefrom
the
firstto
the
secondsubsample.
The
meanbuy-selldifferencefalls
from0.4percent
to
0.3percent.
The
p-valueforthisnumbergoesfrom0
to
0.012.
C.
GARCH
Comparisons
Table7shows
the
parameter
estimatesfor
GARCH(l,l
)-AR(2)modelforeachofthe3ex­
change
rate
series.
The
estimatesshowverysimilarestimatesfor
the
varianceparameters,
(3
and
"'1,
for
the
threeexchange
rate
series.
The
AR(2)parametersshowsomesignificantpersistencein
exchange
rate
movementsforallthreeseries,
but
both
the
Yen
and
the
Markshowasomewhat
larger
amount
of
persistencewith
both
the
AR(l),
andAR(2)parameterssignificant.
Standardizedresidualsfromthismodelarerunbackthrough
the
samemodeltogenerate
simulatedtimeseriesforthethreeexchange
rate
series.Resultsofthesesimulations
are
presented
intable8.Thistableshows
that
theGARCHmodelcombinedwiththeAR(2)causessomeincrease
in
the
meanbuys
and
somedecreasein
the
meansells.Mostofthisisprobablycomingfromthe
persistence
in
the
AR(2).However,themagnitudeofthesedifferencesis
not
asgreatas
that
for
the
actual
series.
9
For
the
BP
the
averagebuy-selldifferenceforthethreetestsis0.07percentwhichcompares
with0.29
percent
for
the
actualseries.
The
simulatedp-valuehereis0.01.For
the
BP
theGARCH
modelleaves
the
previousresultsunchanged.Also,thereare
no
effectsonvolatilityaspreviously
mentioned.
For
the
DM
and
JY
seriestheGARCHmodelhasaslightlystrongereffect.
The
simulations
generateaveragebuy-selldifferencesof0.10
and
0.13percentrespectively.
The
"p-values"forthese
differences
are
now0.054,
and
0.028respectively.
The
addedpersistenceoftheAR(2)hascaused
alargebuy-selldifferencefortheseseries.Whilethisdoeshaveasmallimpacton
the
resultsfrom
the
simulationsthedifferencesremainsmallrelativeto
the
buy-selldifferencefor
the
actual
series.
D.
Regime
Shift
Bootstrap
Some
of
the
resultsfortheGARCHmodelsuggest
that
whilethismodelismovingintheright
direction,
the
persistencegeneratedis
not
strongenoughtogenerate
the
tradingruleresults
that
areseenin
the
data.
The
rulesusedcontinue
to
generate
buy
orsellsignalsafter
the
pricehascut
through
the
movingaverage,not
just
in
the
neighborhood
of
the
movingaverage.
Longrangepersistencecouldbegeneratedusiugtheregimeshiftingmodelused
by
Engle
and
Hamilton(1990).
In
thismodelconditionalmeans
and
variancesfollowatwo
state
markovprocess.
The
parameter
estimatesforthismodelaregivenintable
9.
Foronlyoneof
the
threeseries,
the
JY,
are
both
the
conditionalmeanparameterssignificantlydifferentfromzere.For
the
BP
series
they
are
both
insignificantlydifferentfromzero.Thereisalsoasign
pattern
reversalon
the
JY
series.Forthisserieshighvarianceperiodsarehighmeanperiods.Fortheothertwoseriesthis
resultisreversed.
It
seemsdoubtful
that
themagnitudesof
the
regimeshiftparameterswillbelargeenoughto
generate
the
conditionalmeandifferences.Forexample,for
the
BP
seriestheconditionalmean
for
5,
=
0is0.05percent,
and
forthe
5,
=
1period
it
is-0.02percent.
It
isdifficult
to
seehow
thiswillgenerateabuy-sellspreadof0.29percent.Thisisconfirmed
in
tablewhichshowsthe
resultsforsimulationsofthismodelusinganormalrandomnumbergeneratortogenerateerrors.
Thereislittleevidenceofthismodelcapturingwhat
the
tradingrulesarepickingupforany
of
the
series.For
the
DM
and
BP
series
the
buy-selldifferencesareactuallynegative.Forall
the
series
the"p-values"for
the
buy-selldifferencesareallclose
to
zero.
10
Thisshould
not
rule
out
thismodelingeneral,
but
at
theserelativelyhighfrequencies(weekly)
it
does
not
seem
to
capture
what
isgoingon.Theremaybesomenumericalproblemsinestimation
as
the
probabilities,p
and
q,arecloseto1
at
thistimehorizon.InEngleandHamilton(1990)
the
conditionalmeanestimatesaresignificantandlarger
than
thosefoundhere.Thismaybedue
to
the
useofquarterlydata.
It
remainstobeseenwhetherotherestimationtechniquescanhelp
repair
theseresultsfor
the
regimeshiftmodel.
E.
Interest
Rate
Differentials
The
useof
the
previoussimpleprocessesforforeignexchangemovementsignoresmuchofthe
informationavailableinworldfinancialmarkets.Thissectionincorporatessomeofthisinformation
into
furthersimulations.
The
relation
that
willbeusedhere
is
uncoveredinterestparity.Thisrelationcanbewritten
as
where
i
and
i'
are
the
domestic
and
foreigninterestratesand
s,
isthelogof
the
exchangerate.In
ariskneutralworld
the
interest
rate
differentialovertheappropriatehorizonshouldbeequalto
the
expecteddrift
of
the
exchangerate.
Whileuncoveredparity,
and
theoriescloselyrelated
to
it,
havebeenrejected
by
severalstudies
it
is
important
to
see
if
thislongrangepersistentdriftcouldbecausing
what
the
trading
rulesare
pickingup.Forthis
test
amodeloftheform,
where
f,
isLLd.noisewillbeused.Onemajorproblem
is
getting
the
interestratesandtheir
timingcorrect.Thisisaproblemwhich
is
extremelydifficult
to
getexactlyright.Fortheweekly
exchangeratesusedhereweeklyeurorateswouldbethemostusefulseries
to
have.Thisstudyis
constrainedby
what
isavailableon
the
EHRAtapes.Forthedollarweeklyeuroratesareavailable
at
dailyfrequency
and
willbeusedas
the
riskfreedollar
rate
foreachweekbeginning
at
the
close
onWednesday.Unfortunately,
the
othercurrenciesdo
not
havesuchratesavailable.
The
weekly
11
rates
are
constructedfromdailyexpostovernightratesfromWednesday
to
the
followingTuesday.
Assuming
the
expectationshypothesisholds
at
the
veryshortendof
the
term
structure,
6
it
,7
=
E
Etit+i,l
i=O
or,
6
it
,7
=
2::=
it+i,l
+
et,
i=O
where
E,
(e,)
=
O.
The
expecteddrift
term
i,
-
i;
istherefore
i,
-,"
+
e,
andwhere,"istheexpost
rate
constructedfrom
the
overnightrates.Therefore
where
E,
(!J')
=
O.
The
timeperiodstudied
is
shorteneddueto
data
availability.For
the
BP
and
DMtheseries
now
start
in
January
1975,
and
forthe
JY
theseriesbeginsinOctober1977.
The
lengthsofthe
BP,DM,
and
JY
seriesare832,832,and690weeksrespectively.
Rather
than
immediatelyadjustingtheseseriesfor
the
interestdifferential,aslightlydifferent
approachistaken
at
first.Representativeseriesoftheform,
8,+1
=
8,
+
Jio'
+
f,
willbesimulated.
The
drift,
Jio'
isobtainedfromtheappropriateinterestdifferential.Anestimate
of
the
residualseries,
f,
is
obtainedbyremoving
the
driftfromtheactualexchangeratechanges.
Thisisthenscrambledwithreplacement,andanewseriesisgeneratedusing
the
originaldrift
series
and
the
scrambledresiduals.Thisgivesusrepresentativeexchange
rate
seriesreflectingthe
appropriate
informationfromtheinterestrates.
Thesesimulationsarethen
run
through
the
sametradingruletests
run
in
previoussections.
Resultsofthesetestsarepresentedintable11.
The
resultsarecomparabletothosefoundfor
the
random
walksimulationsintables2through4.Forallthreeseriesnoneof
the
rulesgenerate
buy-selldifferenceswhichareaslargeasthosegeneratedfromtheoriginalseries.
The
adjustment
fortheinterestdifferentialappearstohavehadlittleeffectonthetradingruleresults.
12
Table
12
repeats
someof
the
earlierGARCHsimulationsaccountingforinterestdifferentials.
Inthiscase
the
moretraditionalapproachofsubtracting
the
expecteddriftfrom
the
exchangerate
returnsisdone.AGARCHmodelisthenfit
to
these"zerodrift"termsandsimulatedbackusing
scrambledstandardizedresidualsasinsectionIVC.Comparingtable
12
withtable8showsvery
few
differences.Adjustingtheexchange
rate
seriesusingtheexpecteddrifthasverylittleimpact
on
the
GARCHsimulations.
The
large(small)returnsduring
bny
(sell)arestill
not
replicatedwell
by
the
simulatednullmodel.
F.
Simulated
Method
of
Moments
'Estimates
The
previoustestshave
not
incorporatedthetradingrulediagnostictestsintotheestimation
procedure.Thissectionpresentsamethodwherethetwocanbe
brought
togetherinonecombined
procedure.
Oneproblemwiththetradingrulemeasuresis
that
it
isdifficult
to
deriveanalyticresults
forthesemeasures.Onetechniqueforestimatingparametersusingconditionswhichcanonlybe
simulatedissimulatedmethodofmoments.Thistechniquewasdevelopedforcrosssection
data
byMcFadden(I989)
and
PakesandPollard(I989).
It
is
extended
to
timeseriescasesinDuffieand
Singleton(I989)
and
IngramandLee(I991).
Wewillfollow
the
procedureoffittingalinearprocessto
the
data
usingaset
of
moment
conditionswhichincludes
the
tradingrules.
The
tradingrules
must
'firstbemodifiedtofitintoa
momentconditionframework,Define
r,
asthereturnsseriesofinterest.Also,let
p,
betheprice
at
timetwhere
r,
=
log(p,)
-log(p,_
Jl.
Again,use
the
movingaverageoflengthL
at
timet,
L-1
mat(L)
=
(1/
L)
I:
Pt-;·
i;;;;
0
Onefirstguessfor
trading
rulerelatedmomentmightbe,
where
S(
x)
=
1if
x
::::
1
and
S(
x)
=
-1
if
x
<
1.
Thiswillnotdoforsimulatedmethodofmoments
since
the
firstderivativesofthismomentwill
not
necessarilybecontinuousin
the
parametersof
the
13
process
T,.
The
conditionmustbereplacedwitha"smooth
substitute".
The
hyperbolictangent
doesagoodjob
of
being
just
suchafunction.
6
Replacetheaboveconditionwith
E{tanh((I/
jL)(..E!..::2...-)
-
I)T,}.
mat_l
Thisconditioncannowbeadded
to
amore
standard
setofmomentconditions.7
The
estimationprocedurewill
attempt
tofitanAR(2)toeachof
the
exchange
rate
series.
Whenusinganymethodofmomentsestimatorchoosing
the
momentconditions
to
use
is
not
alwaysatrivialprocedure.Here
the
choiceofmomentswillfollow
the
goaloftryingtoseewhether
asimplelinearmodeldoesofgoodjobofreplicatingsomesimplelinearproperties
of
the
data
(autocovariances)aswellasthe
trading
ruleresults.Thisgoaldoesnotintend
to
get
the
tightest
estimatesof
the
parameterson
the
model.Forthisreasonthesetofmomentcondtitionswillbe
rather
smallrelativetootherstudies.
The
actual
data
willbealignedtosimulated
data
usingthe
mean,variance,
the
firstthreelaggedautocovariances,andonetradingrulemoment.Thisgivesa
total
ofsixmomentconditions.For
the
trading
rulemomentcondition
the
30
weekmovingaverage
isused.
The
resultsaregenerallysimilaracross
the
otherrules.
8
Therearetwofinaldetailsleftforestimation.
The
variancecovariance
matrix
isestimated
usingtheNewey-West(1987)weightingusing
10
lags.
The
laglengthhasbeenmovedfrom5
to
50
and
the
resultshave
not
changedgreatly.Thisis
important
forthiscasesince
the
moving
average
may
generateverylongrangedependencein
the
estimatedmoments.Lastly,
the
number
ofsimulationsissetto
50
times
the
samplesize.Formostoftheseseriesthisgivessimulation
samplesin
the
rangeof40,000
to
45,000.
Results
of
the
estimationaregivenintable13.Thistableshows
the
estimatedparameters
and
the
chi-squaredgoodnessoffitestimatefor
the
AR(2).For
the
BP
seriestheresultsshowa
6
7
This
condition
brings
in
the
problem
of
a
free
parameter.
This
parameter
is
set
to
1/10
the
standard
deviation
of
the
price-moving
average
ratio.
Experiments
with
this
parameter
have
show
the
results
to
be
insensitive
to
changes
in
the
paramater
ranging
from
1
to
1/100
standard
deviations.
8
This
technique
also
allows
the
use
of
several
trading
rule
conditions
simultaneously.
Dependence
across
rules
is
captured
in
the
variance
covariance
matrix.
14
weak,
but
insignificant
AR(I)
parameter
combinedwitharejectionofthemomentconditions
as
indicatedby
the
X
2
test.
The
AR(2)is
not
ableto
match
upwith
both
the
covariancesandthe
trading
ruleresults.
The
nextrowspresentresultsforsimilarestimationremoving
the
tradingrule
momentcondition.Similar
parameter
estimatesareobtained
but
now
the
goodnessoffitstatistic
is
not
is
onlysignificantat
the
18
percentlevel.
The
trading
ruleconditionhasclearlyaddedan
important
restrictionforthistimeseries.
The
rowlabeledDMrepeatsthisprocedurefor
the
DMseries.Inthiscase
the
modelestimates
twolarger
AR
coefficients
and
the
goodnessoffittestisonlysignificant
at
the
13
percentlevel.
The
AR(2)is
not
stronglyrejectedhere.
Part
ofthereasonforthiscanbeseen
in
table
1.
There
issomecorrelationinthisseries
at
the
firsttwolagswhichallows
the
estimated
AR
coefficientsto
belarger.When
the
ruleisremoved
the
chi-squarestatisticstillremainssmallwithasignificance
levelof
73
percent.
Forthe
JY
seriestheAR(2)specificationisrejected
at
the3percentlevel.Inthiscasethe
modelisestimatingthelargest
AR
parametersofthethreeseries.However,these
appear
to
not
be
enoughto
match
thetradingrulecondition.Thisisagaindemonstratedbyremovingthiscondition.
Afterthisisdone
the
chi-squaredstatisticdropsto2.1whichhasasignificancelevelof0.15.
The
nextrowsinthetablerepeattheseresultsfor
the
zerodriftseries.Theseseriesgenerate
resultssimilar
to
thosefortheoriginalseries.
The
simulated
method
of
momentsprocedurehasadded
to
the
earlierresults.
The
procedure
rejectedthesimplelinearspecificationfortheforeignexchangeseriesfor2of
the
3series.Thisre­
jectionfollowedfromaprocedure
that
combined
standard
autocovariancemomentswithconditions
basedon
the
trading
rules.
v.
Economic
significance
of
Trading
Rule
Profits
The
tests
run
in
the
previoussectionhaveshown
the
movingaverage
trading
rulestobeable
to
detectperiodsofhigh
and
lowreturns.These
returns
are
statisticallylargewhencompared
withseveraldifferentstochasticprocesses
for
the
exchange
rate
series.Theseresultsareinteresting
in
attempting
to
model
the
exactdynamics
in
theforeignexchangemarket,
but
theydonotgive
us
the
economicsignificanceoftheserules.Inotherwords,weretheforeignexchangemarkets
soinefficient
that
largeamountsofmoneywerelyingaround
to
be
madeinarelativelyriskfree
15
manor
bytradersimplementingthesestrategies?Thissectionwillmake
an
attempt
to
measurethe
trading
ruleresults.Transactionscosts
and
interestrateswillbeaccountedfor,
and
some
attempts
willbemade
to
measure
the
riskinessof
the
strategiesrelativetootherassets.
The
movingaveragetradingruleswill
be
implementedassuggested
by
theprevioustests.When
the
currentpriceisabovethelongmovingaverageabuyisindicated
and
when
it
isbelowasell
is
indicated.
The
implementationtestsperformedherewillconcentrateon
the
30
daymovingaverage
alone.Whenabuyisindicatedinacurrency
the
trader
takesalongpositionin
that
currency
and
depositsthisinforeignbonds.
In
the
rulesusedhere
the
trader
alsowillborrowdollarsand
investthesefundsin
the
foreigncurrency.
The
trader
willtakea50%leveraged
position?
This
generallyfollowstheprocedureusedinDooleyandShafer(1983).Sweeney(1986)takesaslightly
morecautiousrouteofneverborrowingandmovingonlyfromdomesticbondstoforeignbonds
conditionalon
the
signal.Thisstrategyleaves
the
trader
exposedtoforeignexchangeriskonly
part
ofthetime.
There
isobviouslyacontinuousrangeofadjusting
the
leverage
parameter
which
movestheoutcomeof
the
strategy
both
interms
of
risk
and
return.
In
this
study
the50%leverage
strategy
willusedforcomparabilitywithotherstudiesandfor
the
purposeofriskcomparisonswith
the
stock.market.
Tradingisdoneonceaweek.When
the
rulesignalsachangeinpositiona
trade
is
made.
Transactionscosts
are
assumedtobe
0.1
%of
the
sizeof
the
trade.Thisappears
to
be
areasonable
estimate
and
isused
in
Dooley
and
Shafer(1983).Somestudiesareslightlyabovethisnumber
(Sweeney(1984)uses
1/8%),
whileothersclaim
that
thisisamaximumforforeignexchangetrading.
The
dailyeurodollar
rate
series
and
callmoneyovernightratesareusedwithcompoundingoccurring
at
dailyfrequencies.
An
interest
rate
differential
of
3%
peryearisused
to
estimate
the
borrowing
ratesfrom
the
lendingratesfromthetape.Thisisprobablyan
upper
bound
on
the
borrowingand
lendingspread
and
is
estimated
from
the
currentprime
rate
-CDspread.Resultswill
be
compared
withthosefrombuying
and
holdingstocks
in
theU.S.market.
The
CRSPvalueweightedindex
includingdividendswill
be-
used
to
representthisasset.AlltestsbegininOctober1977
and
end
in
December1989.
9
This
means
that
an
investor
with
$1
who
receives
a
buy
signal
will
borrow
$1
domestically
and
invest
$2
in
the
foreign
currency.
The
reverse
is
followed
for
a
sell.
16
Table
14
presentssomesummarystatisticscomparingtheresultsfor
the
variousassets.
The
rowlabeled
BP
gives
the
tradingstrategyfor
the
pound.
The
tableshows
that
the
strategyexecuted
36
trades
and
yieldedanaverage
return
of16.7percentperyearcontinuouslycompounded.
It
had
aweekly
standard
deviationovertheperiod
of
2.25percent.
The
columnlabeled
f3
estimatesthe
CAPM
beta
for
the
dynamicstrategyusing
the
CRSPportfolioas
the
marketproxy.Whilea
static
CAPM
basedonlyondomesticsecuritiesisprobably
not
agoodrepresentationofrisk
it
is
stillinteresting
to
observehowcorrelated
the
strategyiswith
the
stock
market,
and
howmuch
potentialthereisfordiversification.Forallcurrencystrategies
the
f3
isnegative
and
veryclose
to
zero.
The
last
three
columnspresentresultsforabuy
and
hold
strategy
in
theforeigncurrency
and
bonds.For
the
poundthis
is
9.9percentwithaweekly
standard
deviationof1.57percent.
Thisshouldbecomparedwiththereturntoonlyholdingdollarbonds(reportedin
the
lastrow)
of9.5percentwithaweekly
standard
deviationof0.05percent.
The
nextthreerowspresentresultsfortheDM,
JY,
and
CRSP
seriesrespectively.Allthe
serieshavesimilar
standard
deviation
and
beta
riskcharacteristics.
The
DMunderperformsCRSP
by
about
2.6
percent,
and
the
JY
exceeds
the
CRSPseries
by
about
5percent.Ineachcase
the
strategiesdramaticallydominatethebuy
and
holdportfolios.
Twocurrenciesgive
returns
in
excessoftheCRSPreturn.
The
important
economicquestion
iswhetherthesestrategiesofferan
important
newsecurityintermsofrisk
and
return.Thisa
difficultquestion
to
answerwithoutanappropriatemodelforrisk
or
the
exactstochasticprocess
foreitherforeignexchangeorstocks.Afairlystraightforwardtechniquewillbeusedto
try
to
getsomeinitialanswerstothisquestion.Returnswillbemeasuredoverfixedhorizonschoosen
at
random
out
of
the
entiresample.In
other
words
fix
the
horizon
at
1year
and
estimatereturns
at
randomlychoosen1yearperiodsduring
the
sample.Thiswillgenerateajointdistributionofstock
and
exchange
rate
returns
whichcanbecompared.!O
Resultsfor500simulations
at
the
1yearhorizonarepresentedintable15.For
the
BPseries
the
simulationsgaveanaverageannual
return
of19.5percentwitha
standard
deviationof13.7
10
One
drawback
of
this
technique
is
that
the
first
and
last
part
of
the
series
will
be
under
represented
in
sim­
ulations.
One
solution
might
be
to
think
of
the
series
as
rolling
around
back
onto
itself
on
a
circle.
However,
this
imposes
a
severe
pasting
together
of
disjoint
parts
of
the
sedes.
Another
solution
might
be
to
use
the
rn-dependent
bootstrap
of
Kunsch(1989).
Both
of
these
possibilities
are
left
for
the
future.
For
the
present
the
reader
should
realize
that
the
simulation
does
not
adequately
sample
parts
of
the
series.
17
percent.Thiscompareswitha
return
of16.2percentwitha
standard
deviationof
18
percentfor
the
CRSP
series.
The
tablealsopresentssomeotherriskmeasures.
The
first,prob(<
RF
-
5%),
reports
the
estimatedprobabilityofgettinga
return
ofless
than
5%
belowtheriskfreerate.
Investors
appear
tospendalotoftimediscussingtheprobabilities
of
largedownwardmovesor
drawdowns.This
number
attempts
tocapturesomeofthisrisk.FortheBPseriesthishappens
in
15
percent
of
the
the
simulationsascomparedwith
29
percentfor
the
CRSPseries.
The
next
columnreports
the
probability
of
the
exchange
rate
return
fallingbelowCRSP.Thisis
46
percent
fortheBP.
The
nextcolumn,
T<RF,
estimates
the
fractionoftime
that
the
compounded
return
on
the
strategywasbelow
the
compounded
return
onariskfreebond.FortheBPseriesthisis
34
percent.
The
finalcolumnreportstheaverage
beta
and
the
standard
deviationof
the
estimated
beta
across
the
simulations.
Beta
isestimatedweeklyforeachsimulation.Thisagainshows
that
thereisverylittlecorrelationbetweenthestrategy
and
the
CRSP
series.Resultsfor
the
buy
and
hold
strategy
for
the
BP
are
showninthenextrow.Thisgivesamean
return
of11.0witha
standard
deviationof15.7.Buy
and
holdfallsbelow
the
dynamicstrategyinmean
return,
but
it
showslittleimprovementinriskiness.
The
distributionofthese
returns
alongwiththe
CRSP
distributionisshown
in
figure
2.
These
are
the
1yearholdingperiodsimulatedreturns.Thisfigureclearlyshowsstrongevidence
that
the
BP
seriesmayfirstorderstochasticallydominateitsequivalentbuy
and
holdposition.
The
comparisonwith
CRSP
ismoredifficult,
but
the
graphsuggests
that
thepoundstrategymay
secondorderstochasticdominateCRSP.Boththesecomparisonsawaitmoredetailedstatistical
testingY
Resultsfor
the
DMseriesaregiveninthenexttworowsof
table
15.Thisseriesgivesamean
return
less
than
CRSPwithsimilarriskcharacteristics.
It's
returns
areagainmuchlarger
than
11
First
order
stochastic
dominance
is
obtained
when
F(,,)
-G(,,)
~
0
"
for
the
distribution
functions
F
and
G,
where
the
inequality
is
strict
over
a
set
of
positive
measure.
Any
consumer
prefering
more
to
less
will
prefer
the
distribution
G.
Second
order
stochastic
dominance
is
obtained
when
JX
(F(s)
-
G(s))ds
~
0
-=
In
this
case
only
risk
averse
consumers
will
prefer
G,
Rothschild
and
Stiglitz(1970).
18
the
equivalentbuy
and
holdstrategy.Figure3plots
the
distributionfor
the
DMstrategies.There
isagainaclearindication
that
the
strategyfirstorderstochasticallydominates
the
buyandhold
strategy.
No
simplecomparisonscanbemadebetweentheDMstrategy
and
GRSP.
Resultsfor
the
JY
aregivenin
the
nexttworows.
The
JY
outperformsGRSP
by
6percentand
its
buy
and
holdby10percent.
It
hasalarger
standard
deviation,
but
its
other
riskmeasuresare
equivalenttoGRSP.Figure4shows
the
distributions.Onceagain
it
appears
that
the
strategyfirst
orderstochasticallydominatesbuyandhold.Thestrategyappearsclose
to
firstorderdominating
GRSPexceptforasmallsection.However,
it
showsstrongevidenceforsecond
order
stochastic
dominance.
Forallthreecurrencies
the
betas
areverylow.Thissuggeststhepossibilityfordiversification.
The
nextrowlabeled
CRSP+BP
presentsresultsforaporfolioformedby
starting
out
invested
half
in
stocks
and
halfin
the
BP
dynamicstrategy.
The
portfolioincreases
returns
and
reduces
standard
deviationovertheoriginalGRSPportfolio.
It
iseasy
to
selectanoptimalportfoliousing
currenciesdetermined
by
looking
at
theresultsexpost.
The
nextrowtestsa
strategy
that
might
havebeenfollowedhadtheinvestor
not
knowntherelativelypoorperformance
of
the
DM.Inthis
strategy
wealthissplitequallybetweena
buy
and
holdGRSPportfolio
and
the
dynamicportfolios.
The
half
in
the
dynamicforeignexchangeportfoliosissplit
1/3
to
eachcurrency.Thisstrategy
performs,verysimilar
to
the
BP+GRSP
strategy.Thisresultshows
that
there
is
probablylittle
diversificationgainacrossforeignexchangestrategiesthemselves.
Resultsforallthesestrategies
and
GRSPareplottedinfigure5.
The
twodynamicstrategies
areclose
to
each
other
and
appear
closetosecondorderdominating
the
GRSP
returns
alone.This
isconsistentwith
the
propertiesof
the
dynamicforeignexchangestrategieswhichsuggested
that
theywerezero
beta
securitiesexhibitingsimilarrisk-returncharacteristicsto
the
stockportfolio.
Theseresultsarefurthertestedintable16.Thistablecompares
the
previousdistributions
usingamyopic1yearinvestorwith
crra
utility.
The
coefficientofrelativeriskaversionissetto
4.
The
table
finds
a
that
sets
Eu(aW
R,)
=
Eu(WR2
),
(
)
1
1-~
UX
=
(1_I')x
,
19
whereR
,
is
the
return
givenby
the
labelsontheleftsideoftherows,andR
2
isthe
return
given
in
the
columns.Foreachcurrency
it
isclear
that
thethisconsumerwouldwilling
to
giveupclose
to
8percentof
the
investedwealth
to
shifttothedynamicstrategyfromBH.Comparisonswith
CRSPsuggest
the
consumerwouldbewillingtogiveup4-5percentofwealthforeachstrategy
exceptfor
the
DMwhereCRSPispreferred.Thisimprovementholdsfor
the
threeexchangerate
CRSPportfolio.
The
last
columncompares
the
diversifiedportfoliowitheachof
the
strategies.
Interestingly,
the
portfolioshowslittleimprovementover
the
BP
and
JY
strategiesseparately.
Finding
an
optimal
portfolioexpostis
not
aconfirmationof
an
inefficientmarket.
It
should
alwaysbeeasy
to
findportfolioswhichdominate
the
market
portfolio
in
anexpost
data
search.The
evidenceshowssomeperformanceimprovementsforsomecurrency
and
currency-stockportfolios
whencomparedto
the
stockportfolio.Thisevidenceshouldbeveiwedwithsomecautionas
it
awaitsfurtherstatisticaltesting.Allthetradingrulesdooffersimilarperformancecharacteristics
to
themarketportfoliowithno
beta
risk.Tothestockmarketinvestorwonderingwhetherto
speculatein
the
foreignexchangemarkettheevidence
at
thispointappearssomewhatuncertain.
However,foranyeconomicagentwhosejobrequiressome
amount
of
liquidity
in
variousforeign
exchangemarkets
the
recommendationisclear.Theseagentsarecomparing
the
riskfreeratesof
return
inallmarkets
and
willhave
to
maintainsomeexposuretoforeignexchangerisk.Thereare
verydramaticimprovementsinmovingfrombuy
and
holdstrategies
to
the
trading
rulesforthese
agents."
There
areseveralproblems
that
couldmovetheseconclusions
in
eitherdirection.First,the
data
usedmay
not
representinterestrates
that
traderscouldactuallyuse.Also,theremaybe
sometimingproblemsintermsofsettlements.Forexample,
the
rulesasimplemented,assume
that
traderscangettheclosingpriceonthedayofthesignal.Thismay
not
alwaysbethecase.
Also,settlementproceduresare
not
consideredhere.
13
Finally,measurement
of
riskwithrespect
toaU.S.stockportfolioprobablymissesmuchof
the
exposure
to
internationalportfoliorisk
that
12
This
may
explain
the
extensive
use
of
technical
trading
advice
by
many
market
participants.
13
An
experiment
was
performed
to
test
the
robustness
of
the
results
to
timing.
The
testing
programs
were
modified
so
that
investors
could
not
get
the
interest
rates
at
time
t,
but
could
get
the
rates
given
one
day
later.
Results
of
this
experiment
are
not
presented
since
they
are
almost
identical
to
those
from
the
original
series.
20
the
exchange
rate
portfoliosareexposedto.Estimating
betas
onaworldportfolioorusinga
multifactormodelmightbemore
appropriate
here.
There
aresomeproblemsintheanalysiswhichworkinfavorof
the
trading
rules.First,
the
rulesusedareverysimplecomparedto
what
mosttradersuse.Also,most
traders
wouldoperate
at
the
dallyfrequencyorhigher.
14
Second,
the
comparisonseries,the
CRSP
index,maybedifficult
to
obtaininpractice.No
attempt
wasmade
to
adjustfortransactionscostsonthisserieseventhough
using
the
CRSPindeximplies
that
dividendsarebeingcontinuouslyreinvested.
The
abilityofthe
averageinvestor
to
trackthisindexshouldbemorecarefullyconsidered.
There
hasbeensomerecentevidence
that
theusefulnessoftechnical
trading
strategieshas
diminishedovertime(Sweeney
and
Surarjaras(1989)).Tocheckthepossibility
of
a
trend
intrad.ing
ruleprofitsovertimeaplotismadeof
the
trading
rulereturnsmeasuredovertwoyearhorizonsfor
thethreecurrenciesrolling
the
horizonforward1quarterforeachpointplotted.Thisisplottedin
figure
6.
Thereissomeevidenceforadropoffinprofitsinrecentyears.However,whenanalyzing
the
entireseries
it
isunclearwhetherthisperiodis
at
allunusual.Therehavebeenearlierperiods
when
the
rulesdid
not
performverywell.
The
timeperiodaround1982-1983appears
to
havealso
beenrelativelypoor.
It
isinteresting
that
thesemightbeperiodsinwhich
the
2yearhorizonis
reachingintoperiods
just
after
the
Plaza
Agreementwhen
the
dollarchangeddirection.
VI.
Conclusions
This
paper
haspresentedevidencesupportingthepremise
that
exchangeratesdonotfollow
a
random
walk.Moreover,thesedeviationsaredetectedbysimplemovingaveragetradingrules.
Theserulesfind
that,
ingeneral,
returns
duringbuyperiodsarehigher
than
returns
duringsell
periods.Volatilityappears
to
beindistinguishableduringthesetwoperiods.Also,skewnessand
kurtosisshownodiscernible
patterns
overbuyandsellperiods.
Theseresultsaresupportive
of
earlierworkinDooley
and
Shafer(1983),Schulmeister(1987),
Sweeney(1986),Taylor(1980),Taylor(1986),
and
morerecentlyTaylor(1990).These
other
authors
performextensivetestson
the
profitability
of
thesetests
and
find
that
in
general
the
rulesmake
moneyevenwhenadjustedfortransactionscosts,interest
rate
differentials,
and
verysimplemea­
suresofrisk.
14
Most.
of
the
rules
used
here
were
repeated
at
daily
frequency
with
little
change
in
the
results.
21
In
this
paper
the
rules
are
firstusedasspecificationtestsforseveraldifferentprocesses.
The
GARCH,regimeshifting,
and
interest
rate
adjustedmodelsareunabletogenerateresultsconsistent
with
the
actualseries.
In
eachcase
it
isstillpossible
that
amodifiedversionof
the
modelcould
becapableofgeneratingresultsconsistentwiththeactual
data,
but
thisawaits
further
experi­
mentation.Twoanswersfortheseresultsarethefollowing.
First,
it
ispossible
that
the
seriesare
nonstationary
and
are
punctuated
bystrongchangesinregime
that
cannotbecapturedbythese
simplemodels.Second,noneof
the
modelsconsideredhereallowforanyconnectionbetween
trend
and
volatilitychanges.ThispossibilityisconsideredinTaylor(1986),
and
results
in
Bilson(1990),
Kim(1989),
and
LeBaron(1990)suggest
that
there
may
besomeconnection.
The
finalsectionof
the
paper
runssomeexperimentsto
test
the
economicsignificanceofthese
results.
The
trading
rulesareimplementedonthe
data
astheywouldbeusedinpractice.Estimates
fortransactioncosts
and
interest
rate
spreadsareused
to
measure
the
realized
returns
from
the
strategies.
For
the
threecurrenciestestedthetradingrulestrategiesgenerated
return
distributions
similar
to
thosefrom
the
CRSPstockindexwithverylowcorrelationwith
the
market.This
suggestsportfoliosformedbycombiningthestrategieswith
the
CRSPindexmaydominatethe
stockindexonitsown.
Whiletheseresultsareinterestingtheyshouldstill
be
viewedwithsomecaution.
There
are
stillseveralinterest
rate
andtimingissues
that
are
not
exactlyworkedout.Also,
the
useofother
riskfactors
than
CAPM
beta
maybe
important.
Asinanytradingrulestesttherearefurther
questions
about
the
parametersused
and
whetherthepricesusedwereactuallytradeable.Given
theseissues
and
the
lackofastatistical
test
on
the
distributioncomparisonstheresultscannot
betakenasclearevidence
that
everyeconomicagentismissingabigopportunity.However,for
onegroup
of
agents
the
resultsare
pretty
strong.Forpeoplewhoareinvolvedinforeignexchange
markets,eitherin
trading
goodsorsecurities,andwhomaintainpositionsinforeigncurrencies
there
appear
to
be
major
gainsoverbuyandholdstrategies.Thisiseasilyseenin
table
15
by
comparing
the
buy
and
holdstrategieswiththosefor
the
rules.This
may
explainthelargenumber
oftechnical
trading
servicesavailableintheforeignexchangemarket.'5
15
See
Frankel
and
Froot(1990)
for
some
evidence
on
the
number
of
chartists.
22
The
resultsinthis
paper
mayeventuallyleadtosome
better
explanationsforseveraleffects
inforeignexchangemarkets.Amongtheseare
the
movementsinforwardandfuturesmarketsfor
foreignexchange.'6Also,resultsfromsurvey
data
foundinDominguez(1986)
and
Frankeland
Froot(1990)may
be
relevanttosomeoftheresultsfound
hereP
Lastly,foreignexchangemarkets
differfromstockmarketsin
that
centralbanksplayan
important
role.
The
behavioroftheselarge
economicagentsmaydiffergreatlyfrom
that
ofordinarytraders.Theseagents
may
evenbewilling
to
losemoney
to
satisfyotherobjectives.
This
paper
hasshown
that
technicaltradingrulesmayprovideausefulspecification
test
for
examiningforeignexchangemarkets.This
paper
usestheserules
to
demonstratesomeof
the
shortcomingsofcommonparametricmodelsforforeignexchangemovements.Someevidenceis
givenontheeconomicsignificanceoftheseresults,andshows
that
thestrategiesgeneratereturns
similar
to
thosefromadomesticstockportfolio.
Further
testswillbenecessarytocompletely
answer
the
questionsraised
about
theeconomicsignificanceoftheseresults.
16
See
Hodrick(1987)
fora
survey
of
these
results.
17
These
papers,
using
survey
data,
find
that
short
range
forecasts
are
more
trend
following
while
longer
range
forecast
are
more
mean
reverting.
23
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26
Table1
SummaryStatistics
Description
BP
DM
JY
SampleSize893893893
Mean*lOO-0.01620.0686
0.0875
Std.*1001.43981.43501.4012
Skewness0.21070.35320.3785
Kurtosis
5.59314.3735
5.1425
PI
0.04880.06360.1105
pz
-0.02480.06090.0962
P3
0.03670.00600.0592
P.
0.09590.0414
0.0446
P5
0.0164-0.02000.0338
Pa
-0.0135-0.0570-0.0002
P7
0.0070-0.0028
-0.0359
PS
0.08620.06250.0060
P9
-0.03050.0146-0.0036
PIO
-0.00470.0414-0.0833
Bartlett
0.03350.0335
0.0335
LBP20.1615.53
26.41
p-values
(X
2
(10))
(0.027)(0.115)(0.003)
Summary
statisticsfor
BP
(BritishPound),
DM
(GermanMark),
JY
(JapaneseYen)weeklyexchangerates
from1974-February1991.
Table2
BP
RandomWalkBootstrap
Rule
Result
BuyFractionBuy
Sell
U,
Buy-Sell
(1,20)
Fraction>
Xrate
0.080000.560000.460000.980000.420000.00000
SimulationMean-0.000120.014340.47810-0.000140.014290.00002
XrateMean
0.000910.014260.48624-0.001340.014420.00225
(1,30)Fraction>Xrate0.010000.63000
0.340001.000000.370000.00000
SimulationMean-0.000180.014380.47364-0.000100.01427-0.00007
XrateMean
0.001350.014210.50406-0.001840.014520.00319
(1,50)
Fraction>
Xrate
0.010000.640000.370001.000000.180000.00000
SimulationMean
-0.000190.014360.46809-0.000110.01425-0.00009
XrateMean0.001450.014100.49466-0.00182
0.01487
0.00327
Average
Fraction>
Xrate
0.010000.660000.390000.990000.320000.00000
SimulationMean-0.000160.014360.47333
-0.000120.01427-0.00005
XrateMean0.00124
0.014190.49496-0.001660.014600.00290
Buyreferstothemean1weekreturnduringbuyperiods,
Ub,
the
standard
deviationofthesereturns,and
FractionBuyis
the
fractionofbuyweeks
out
oftotalweeks.Selland
u,
are
the
sameforthesellreturns.
Buy-Sell
is
thedifferencebetweenthebuymeanandsellmean.
The
rowlabeled
Fraction>
Xrateshows
the
fractionofthe500simulationswhichgenerateavalueforthestatisticlarger
than
that
fromtheactual
series.Simulationmeanisthemeanvalue
for
thestatistic
for
thesimulatedrandomwalks,andXrateMean
isthevaluefromthe
orig~al
series.
Table3
DM
RandomWalkBootstrap
RuleResult
BuyFractionBuy
Sell
Buy-Sell
(1,20)
Fraction>
Xrate
0.030000.680000.320000.990000.310000.00000
SimulationMean
0.000700.014310.562730.00065
0.014220.00005
XrateMean
0.001770.013980.58601-0.00112
0.014540.00288
(1,30)
Fraction>
Xrate
0.040000.900000.270001.000000.080000.00000
SimulationMean
0.000670.014340.578160.000700.01421-0.00003
XrateMean
0.001690.013520.61877-0.001120.015260.00281
(1,50)
Fraction>
Xrate
0.050000.960000.550000.960000.010000.00000
SimulationMean
0.000690.014340.600500.000660.01417
0.00003
XrateMean
0.00164
0.013300.59786-0.000950.015550.00259
AverageFraction·
>
Xrate
0.030000.870000.430001.000000.120000.00000
SimulationMean
0.000680.014330.580240.00067
0.014200.00002
XrateMean
0.001700.013600.60085-0.001060.015120.00276
Table4
JY
RandomWalkBootstrap
BuySellSell
FractionBuy
Buy
ResultRule
,
-
(1,20)
Fraction>
Xrate
0.000000.520000.73000
I
1.00000
0.590000.00000
SimulationMean
0.000870.013950.591320.00100
0.01392-0.00013
XrateMean
0.002500.013880.54817-0.001140.013680.00363
(1,30)
Fraction>
Xrate
0.000000.49000
0.790001.000000.530000.00000
SimulationMean
0.000890.013970.614370.001000.01388-0.00012
XrateMean
0.002600.013910.55156-0.001160.013720.00376
(1,50)
Fraction>
Xrate
0.010000.400000.740000.980000.450000.00000
SimulationMean
0.000860.01393
0.648140.001050.01392-0.00018
XrateMean
0.002130.014020.59431
-0.000710.013970.00284
Average
Fraction>
Xrate
0.000000.470000.720001.000000.520000.00000
SimulationMean
0.000870.013950.61761
0.001020.01390-0.00014
XrateMean
0.00241
0.013940.56439-0.001000.013790.00341
Table5
SkewnessKurtosis
Rule
ResultBuySkew
Buy
Kurt.
SellSkew
Sell
Kurt.
I
BP
(1,20)
Fraction>
Xrate
0.572000.266000.336000.48000
(1,30)
Fraction
>
Xrate
0.504000.306000.42200
0.48800
(1,50)
Fraction>
Xrate
0.61800
0.330000.266000.48000
Average
Fraction>
Xrate
0.564000.296000.326000.48000
SimulationMean
0.198995.41680
0.217495.45927
XrateMean0.134465.94169
0.335415.51124
DM
(1,20)
Fraction>
Xrate
0.328000.202000.446000.61800
(1,30)
Fraction>
Xrate
0.300000.168000.35400
0.66600
(1,50)Fraction>
Xrate
0.30600
0.076000.346000.79800
Average
Fraction>
Xrate
0.312000.136000.350000.72400
SimulationMean
0.341524.282220.353444.27991
XrateMean0.44344
5.553640.394103.43740
JY
(1,20)
Fraction>
Xrate
0.340000.528000.31400
0.62000
(1,30)
Fraction>
Xrate
0.364000.524000.334000.62800
(1,50)
Fraction>
Xrate
0.354000.54200
0.392000.70800
Average
Fraction>
Xrate
0.330000.53000
0.346000.66800
SimulationMean0.383545.089460.407575.05235
XrateMean0.490694.96434
0.541884.63171
Table6
Subsamples:
Random
Walk
Rule
Result
Buy
<Tb
Fraction
Buy
Sell
<T,
Buy-Sell
BP
First
Half
(1,20)
Fraction>
Xrate
0.022000.976000.160000.998000.090000.00200
(1,30)
Fraction>
Xrate
0.012000.946000.106000.998000.058000.00000
(1,50)
Fraction>
Xrate
0.010000.888000.124001.000000.052000.00000
Average
Fraction
>
Xrate
0.006000.952000.11800
1.00000
0.06400
0.00000
SimulationMean-0.000800.011680.39614-0.000590.01172-0.00022
Xrate
Mean0.001030.010250.51048-0.002670.012880.00370
BP
Second
Half
(1,20)
Fraction>
Xrate
0.206000.200000.612000.694000.694000.18400
(1,30)
Fraction
>
Xrate
0.070000.244000.462000.834000.648000.05000
(1,50)
Fraction>
Xrate
0.074000.366000.498000.804000.510000.04400
AverageFraction>
Xrate
0.094000.246000.518000.788000.630000.05200
SimulationMean
0.000110.01647
0.52470
0.000500.01652-0.00038
Xrate
Mean0.001480.017190.51900-0.000470.016090.00195
DM
First
Half
(1,20)
Fraction>
Xrate
0.108000.960000.180000.992000.058000.00800
(1,30)
Fraction>
Xrate
0.196000.996000.122000.966000.008000.02000
(1,50)
Fraction>
Xrate
0.134000.996000.206000.994000.002000.00600
AverageFraction
>
Xrate
0.134000.994000.174000.992000.010000.00400
SimulationMean
0.000080.012590.521430.000260.01265-0.00018
Xrate
Mean0.001010.010670.61935-0.001640.014760.00264
DM
Second
Half
(1,20)
Fraction>
Xrate
0.104000.254000.450000.950000.698000.03000
(1,30)
Fraction>
Xrate
0.046000.392000.454000.996000.600000.00200
(1,50)
Fraction>
Xrate
0.040000.400000.734000.948000.416000.00600
Average
Fraction
>
Xrate
0.048000.330000.562000.986000.566000.00000
SimulationMean0.001030.015690.626560.001410.01577-0.00037
Xrate
Mean0.002550.015990.60792-0.000840.015490.00339
Table6continued
Rule
Result
Buy
Ub
Fraction
Buy
Sell
17,
Buy-Sell
JY
First
Half
(1,20)
Fraction>
Xrate
0.002000.754000.528000.998000.51600
0.00200
(1,30)
Fraction>
Xrate
0.00000
0.782000.536000.998000.414000.00000
(1,50)
Fraction>
Xrate
0.012000.792000.41000
0.982000.210000.00000
Average
Fraction
>
Xrate
0.000000.792000.488000.998000.330000.00000
SimulationMean
0.000070.012460.528960.00039
0.01243-0.00032
Xrate
Mean0.002170.011780.53548-0.001850.012740.00402
JY
Second
Half
(1,20)
Fraction>
Xrate
0.10000
0.432000.690000.958000.908000.01600
(1,30)
Fraction>
Xrate
0.114000.464000.798000.940000.91400
0.03200
(1,50)
Fraction>
Xrate
0.082000.260000.802000.964000.942000.01400
AverageFraction
>
Xrate
0.088000.386000.772000.972000.950000.01200
SimulationMean
0.001390.015210.674450.001850.01532-0.00046
Xrate
Mean0.00262
0.015500.60469-0.000330.013550.00295
Table7
GARCH(I,I)
ParameterEstimates
x,
=
a
+
blx'_1
+
b,x,_,
+
<,<,
=
hi/'z,
h,
=
"'0
+
"'I<LI
+
(3h'_1
z,
~
N(O,
1)
Xrate
"'0
(3
"'I
b
l
b,
a
BP
2.29400.72870.16800.08320.0324-3.4473
(0.3504)(0.0363)(0.0303)(0.0391)(0.0393)(4.5733)
DM
1.41310.75390.18890.06040.09356.8368
(0.3480)(0.0319)(0.0289)(0.0378)(0.0349)(4.2092)
JY
1.34600.76100.18750.11790.08326.3114
(0.2403)(0.0321)(0.0308)(0.0380)(0.0389)(4.1427)
Estimation
is
bymaximnmlikelihood.Nnmbersinparenthesisareasymptotic
standard
errors.
Table8
GARCH
Bootstrap
Rule
ResultBuy
"b
FractionBuySell
",
Buy-Sell
BP
(1,20)
Fraction>
Xrate
0.176000.552000.394000.818000.530000.09400
(1,30)
Fraction>
Xrate
0.048000.574000.276000.958000.504000.00800
(1,50)
Fraction>
Xrate
0.020000.588000.304000.970000.424000.00800
AverageFraction>
Xrate
0.056000.570000.316000.944000.486000.01000
SimulationMean0.000080.014740.46559-0.000580.014920.00066
XrateMean
0.001240.014190.49496-0.001660.014600.00290
DM
(1,20)
Fraction>
Xrate
0.28400
0.78200
0.40400
0.92800
0.64600
0.07600
(1,30)
Fraction>
Xrate
0.258000.858000.322000.942000.494000.07000
(1,50)
Fraction>
Xrate
0.232000.878000.556000.934000.422000.05000
Average
Fraction>
Xrate
0.250000.850000.426000.944000.522000.05400
SimulationMean0.001220.015820.589330.000210.015580.00101
Xrate
Mean0.001700.013600.60085-0.001060.015120.00276
JY
(1,20)
Fraction>
Xrate
0.134000.734000.682000.920000.642000.04600
(1,30)Fraction>
Xrate
0.086000.718000.738000.944000.606000.02600
(1,50)
Fraction>
Xrate
0.132000.670000.634000.896000.568000.03800
Average
Fraction>
Xrate
0.114000.71000
0.69200
0.932000.602000.02800
SimulationMean
0.001460.01611
0.596720.00016
0.01519
0.00130
Xrate
Mean0.00241
0.013940.56439
-0.001000.013790.00341
Resultsfromsimulations
of
500GARCHmodels.Thesemodelsaregeneratedfrom
estimated
parameters
and
standardizedresidualsfrom
maximum
likelihood.
Table9
RegimeShift
Parameter
Estimates
x,
=
(1'0
+
I'IS,)
+
("'0
+
""S,)Z,
peS,
=
11S'_1
=
1)
=
p
pes,
=
O[S'_1
=
1)
=
1-
p
pes,
=
0IS,_1
=
0)
=
q
pes,
=
l[S'_1
=
0)
=
1-
q
z,
-
N(O,
1)
Xrate
"'0
*
1000
"'1
*
1000
1'0
*
1000
1'1
*
1000
P
q
BP
2.781112.21390.4923-0.71190.99330.9260
(0.2447)(0.3578)(0.3851)(0.6374)(0.0033)(0.0342)
DM
6.74229.04071.1889-0.63880.99400.9738
(0.4188)(0.5350)(0.5313)(0.7815)(0.0034)(0.0136)
JY
4.897311.8531-0.76462.45870.93870.8773
(0.2892)
(0.5093)(0.3632)
(0.7655)
(0.0157)(0.0266)
Estimationis
by
maximumlikelihood.Numbers
in
parenthesis
are
asymptoticstandarderrors.
Table
10
RegimeShift
Bootstrap
Rule
Result
Buy
0"1
Fraction
Buy
Sell
0",
Buy-Sell
BP
(1,20)
Fraction>
Xrate
0.048000.482000.53800
0.972000.566000.02000
(1,30)
Fraction>
Xrate
0.004000.512000.394000.994000.49400
0.00200
(1,50)
Fraction>
Xrate
0.004000.620000.424000.99800
0.270000.00000
Average
Fraction>
Xrate
0.008000.530000.430000.994000.44200
0.00000
SimulationMean
-0.000250.014200.48442-0.00007
0.01449-0.00018
Xrate
Mean0.001240.014190.49496
-0.001660.014600.00290
DM
(1,20)
Fraction>
Xrate
0.036000.532000.538000.996000.600000.00000
(1,30)
Fraction
>
Xrate
0.042000.700000.43400
0.996000.326000.00600
(1,50)
Fraction
>
Xrate
0.056000.780000.64400
0.982000.230000.00800
Average
Fraction
>
Xrate
0.028000.68400
0.552000.994000.370000.00000
Simulation
Mean0.000640.014050.608950.00080
0.01481-0.00016
Xrate
Mean0.001700.013600.60085
-0.001060.015120.00276
JY
(1,20)
Fraction>
Xrate
0.008000.73400
0.642001.000000.428000.00000
(1,30)
Fraction>
Xrate
0.004000.650000.732000.998000.45200
0.00000
(1,50)
Fraction>
Xrate
0.024000.572000.666000.97800
0.412000.00400
AverageFraction>
Xrate
0.004000.63000
0.684000.996000.422000.00200
SimulationMean
0.000890.01422
0.597010.000870.013640.00002
XrateMean
0.002410.013940.56439
-0.001000.013790.00341
Resultsfromsimulations
of
500regime-shiftmodels.Thesemodels
are
generatedfrom
estimated
parameters
andcomputergeneratednormalrandomnumbers.
Table
11
Interest
Rate
Drift
RuleResult
Buy
O"b
FractionBuy
Sell
0",
Buy-Sell
BP
(i
,20)
Fraction>
Xrate0.012000.312000.406000.982000.650000.00200
(1,30)
Fraction>
Xrate
0.010000.616000.284000.992000.296000.00000
(1,50)
Fraction>
Xrate
0.006000.71400
0.328000.994000.152000.00000
Average
Fraction>
Xrate0.00600
0.552000.320000.994000.330000.00000
SimulationMean-0.000140.01468
0.47660-0.000110.01461-0.00002
XrateMean0.001550.014570.50731-0.001650.014910.00320
DM
(1,20)
Fraction>
Xrate
0.032000.462000.544000.990000.606000.00400
(1,30)
Fraction>
Xrate'
0.110000.756000.540000.960000.212000.02400
(1,50)
Fraction>
Xrate
0.046000.822000.75000
0.972000.166000.00800
Average
Fraction>
Xrate0.040000.704000.63000
0.982000.294000.00400
SimulationMean0.000280.01541
0.538980.000480.01544-0.00020
Xrate
Mean0.001570.015050.51050-0.001350.015820.00292
JY
(1,20)
Fraction>
Xrate
0.022000.354000.76200
0.996000.638000.00200
(1,30)
Fraction>
Xrate
0.002000.350000.866000.996000.664000.00000
(1,50)
Fraction>
Xrate0.124000.57600
0.812000.978000.594000.01200
AverageFraction>Xrate0.024000.424000.814000.992000.642000.00000
SimulationMean0.000770.015210.596420.000980.01533-0.00021
Xrate
Mean0.002260.015310.52717-0.001210.015020.00348
Resultsfromsimulations
of
500replications
of
seriesgeneratedwithconditionaldriftequal
to
giveninterest
rate
differentials.
r,
=
J1.,
+
f,
where
J1.,
corresponds
to
the
interest
rate
differential
at
time
t.
Table
12
GARCHZeroDrift
RuleResult
Buy
(fb
FractionBuy
Sell
(f,
Buy-Sell
BP
(1,20)
Fraction>
Xrate
0.064000.368000.362000.902000.660000.03600
(1,30)
Fraction>
Xrate
0.030000.514000.278000.934000.420000.00600
(1,50)
Fraction>
Xrate
0.01400
0.548000.342000.978000.332000.00400
Average
Fraction>
Xrate
0.02400
0.480000.318000.944000.444000.00600
SimulationMean
0.000510.014800.51214-0.000350.01519
0.00085
XrateMean0.00197
0.014620.54196-0.001490.015020.00346
DM
(1,20)
Fraction>
Xrate
0.156000.698000.270000.916000.60800
0.05000
(1,30)
Fraction>
Xrate
0.138000.792000.18200
0.894000.486000.05400
(1,50)
Fraction>
Xrate
0.034000.78800
0.388000.950000.466000.00800
Average
Fraction>
Xrate
0.092000.762000.298000.92400
0.526000.02600
SimulationMean0.000520.01576
0.47758-0.000690.015130.00121
Xrate
Mean0.001590.013850.51942-0.001760.014620.00335
JY
(1,20)
Fraction>
Xrate
0.024000.360000.568000.96200
0.528000.00600
(1,30)
Fraction>
Xrate
0.016000.436000.556000.970000.47800
0.00200
(1,50)
Fraction>
Xrate
0.194000.352000.598000.932000.636000.04000
AverageFraction>
Xrate
0.050000.372000.57600
0.97000
0.54600
0.00000
SimulationMean0.000600.015220.49250-0.000520.01513
0.00112
XrateMean0.002010.015430.47334
-0.001930.014990.00394
Resultsfromsimulations
of
500GARCHmodels.Thesemodelsaregeneratedfrom
estimated
parameters
and
standardized
residualsfrommaximumlikelihood.Modelsareestimatedandsimulatedusingforeign
exchange
returns
serieswithinterest
rate
differentialsremoved.
Table
13
SMMEstimation
r,
=
/1
+
Pl(r,_1-
/1)
+
P2(r,_2-
/1)
+
,n,
<,
-
N(O,
1)
SeriesCondition
(f
PI
P2
x2
BP
Rule-0.0461.4070.043-0.0178.261
(0.054)(0.074)(0.035)(0.041)(0.016)
BP
No
Rule-0.0231.4500.031-0.0291.793
(0.057)(0.073)(0.034)(0.039)(0.181)
DMRule0.0991.4100.0510.0423.989
(0.052)(0.061)(0.031)(0.043)(0.136)
DM
No
Rule0.0711.4270.052
0.0450.120
(0.054)
(0.062)(0.031)(0.043)(0.729)
JY
Rule0.1521.3640.1040.1036.819
(0.055)
(0.062)(0.039)(0.042)
(0.033)
JY
No
Rule0.1251.4050.1000.0882.066
(0.059)(0.062)(0.039)(0.040)(0.150)
BPZD
Rule0.0751.463
0.063
-0.017
8.540
(0.059)(0.078)(0.035)(0.041)(0.014)
BPZD
No
Rule
0.043
1.4920.043-0.0221.773
(0.063)(0.078)(0.035)(0.040)(0.183)
DMZDRule0.0211.4040.0630.0473.035
(0.055)
(0.063)
(0.031)(0.045)(0.219)
DMZD
No
Rule0.0111.4280.0630.0480.009
(0.058)(0.063)(0.031)(0.045)(0.924)
ZYZDRule
0.0611.5010.098
0.113
5.734
(0.070)(0.062)(0.040)(0.043)(0.057)
JYZD
No
Rule
0.006
1.5090.1000.0831.230
(0.073)(0.064)
(0.042)(0.044)(0.541)
Parametersestimated
by
simulatedmethod
of
moments.Numbersinparenthesisareasymptoticstandard
errorsfor
the
parametersandthep-valueforthechi-squaredgoodnessoffittest.Momentsusedarethe
mean,variance,3autocovariances
and
the
30
daymovingaveragetradingrule.
The
chi-squaredstatistic
has6-4
=
2degrees
of
freedomwhen
the
tradingrule
is
used
and
5-4
=
1degrees
offreedom
when
it
is
not
used.
The
variance-covariance
matrix
is
estimatedusingtheNewey-West(1987)techniquewith
10
lags.
Table
14
RuleImplementationSummary
Series
TradesReturn/yearReturn/weekStd/week
(J
BH
BH/weekBH(std)
BP
36
16.70.352.25-0.079.90.201.57
DM
43
12.6
0.262.19-0.086.20.131.54
JY
26
20.1
0.41
2.17-0.038.40.171.52
CRSPVW
15.20.322.18
$
RF
9.5
0.180.05
Thistablesummarizes
the
resultsofthetradingrulesoverthefullsample.
(J
istheestimatedCAPM
beta
for
thetradingstrategyestimatedusingweeklydata.t(return-CRSP)isat-statisticforequality
of
thereturns
forthestrategy
and
CRSP.
BH
standsfor
the
thebuyandholdstrategyin
the
foreigncurrencyholding
foreignbonds.
Table
15
1YearHorizon
Series
Return/year
Std
Prob(<RF-5%)Probe
<CRSP)
T<RF
(3
BP
19.513.7
0.150.460.34-0.04
(0.31)(0.26)
BP
BH
11.015.70.420.59
0.500.06
(0.39)(0.16)
DM14.318.7
0.320.500.46-0.06
(0.34)(0.18)
DMBH
5.614.70.550.65
0.62
0.07
(0.37)(0.19)
JY
22.224.00.190.370.39
-0.04
(0.35)(0.21)
JY
BH
9.416.60.45
0.590.560.03
(0.37)(0.17)
CRSP16.2
18.0
0.29
0.41
(0.35)
CRSP+BP
17.911.20.14
0.460.360.47
(0.32)(0.13)
3FX+CRSP
17.412.8
0.180.380.370.47
(0.32)(0.10)
Simulationresultsforoneyeartradingborizon.Results
of
1000
simulationofrandomlyselected2year
intervalsduringthesample.Probe<RF-5%)
is
theprobabilityofunderperforming
the
domesticriskfree
ratebymore
that
5%.probe<CREip)istheprobability
that
thestrategyunderperformstheCRSPindex.
T
<RF
is
thefraction
of
time
that
thecumulativereturn
on
thestrategyspendsbelowthecummulativereturn
ontheriskfreeasset.
{3
is
againthe
beta
estimatedagainsttheCRSPindex.
CRSP+BP
isaportfoliowhich
isstartedwithportfolioweightsof
1/2
and
1/2
on
CRSPandthetradingstrategyrespectively.
3FX+CRSP
isaportfoliowhichisstartedwithweightsof
1/2
on
abuyandholdstockposition,and
1/2
onaequally
weightedpositionin
the
threeforeignexchangestrategies.
Table
16
UtilityDistributionComparisons:1YearHorizon
Series
BP
DM
JY
BP+CRSP
3FX+CRSP
BH
0.92
0.93
0.91
CRSP
0.95
1.01
0.96
0.96
0.96
3FX+CRSP
0.99
1.05
1.00
0.99
Utility
comparisons
of
myopic
1
year
erra
investors.Fraction
of
wealthathatwouldmakeaninvestor
indifferentbetweentherowruleusedon
aW
andthecolumnruleon
W.
Degree
of
relativeriskaversionis
fixed
at
4.
0.004
0.0035
Q)
g
0.003
Q)
"
~
0.0025
'H
-.-I
Q
0.002
.-j
.-j
aO.0015
I
:>,
g
0.001
0.0005
o
British
Pound
Buy-Sell
Differences
For
Various
Moving
Averages
-'"
/'
"
"-
V
............
V
./"
~
V
IV
5
1015
20
25
3035
40
Moving
Average
Length
(weeks)
Figure
1
45
50
Fig.
2:
Simulated
Return
Distribution
-1
Year
Period
./i
::/"
II!
1//
1/
/.
/11
//
l
VI
I
/
//
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
o
-0.2
o
0.2
0.4
0.6
Return
(per
year)
0.8
BP
BPBH
CRSP
/,/..;....-
~
"/
I
/j
1/
I
II
//
//
/
J
~/
Fig.
3:
Simulated
Return
Distribution
-1
Year
Period
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
o.
1
o
-0.2
0
0.2
0.4
0.6
0.8
Return
(per
year)
DM
DMBH
CRSP
Fig.
4:
Simulated
Return
Distribution
-1
Year
Period
/1
,/
/
II
I
1/
I"
I
/I
II/!
t
VI
II'
...u
1
0.9
o.
8
0.7
0.6
0.5
0.4
0.3
0.2
o.
1
o
-0.2
o
0.2
0.4
0.6
Return
(per
year)
0.8
JY
JYBH
CRSP
Fig.
5:
Simulated
Return
Distribution
-1
Year
Period
/
I-'"
/I
1/
II
J
I
/I
fIJ
I
II
/~
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
o.
1
o
-0.2
o
0.2
0.4
0.6
Return
(per
year)
0.8
BP+CRSP
3FX+CRSP
CRSP
0.6
0.5
c:
0.4
H
::l
.j..l
QJ
0.3
i:<:
.-l
m
0.2
::l
c:
c:
~
0.1
0
-0.1
Rolling
Strategy
Returns
2
Year
Periods
~
'V\
I(
~
vv
\\1
"
\I
A
\
I
J
~
V:
~~\
V-
i""
"'\j
....
r'V
\
~/
V
00
mOM
NM
~~
w
~
00
~~
0000
00
00
0000
00
0000
Year
Figure
6
BP
DM
JY