Determinants of the Implied Shadow Exchange Rates from a Target Zone
Rangvid, Jesper; Sørensen, Carsten
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1998
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Rangvid, J., & Sørensen, C. (1998). Determinants of the Implied Shadow Exchange Rates from a Target Zone.
Institut for Finansiering, Copenhagen Business School. Working Papers / Department of Finance. Copenhagen Business School No. 1998-15
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WP 1998-15
Determinants of the implied Shadow Exchange Rates from a Target Zone af
Jesper Rangvid & Carsten Sørensen
INSTITUT FOR FINANSIERING, Handelshøjskolen i København Solbjerg Plads 3, 2000 Frederiksberg C
tlf.: 38 15 36 15 fax: 38 15 36 00
DEPARTMENT OF FINANCE, Copenhagen Business School Solbjerg Plads 3, DK - 2000 Frederiksberg C, Denmark
Phone (+45)38153615, Fax (+45)38153600 www.cbs.dk/departments/finance
ISBN 87-90705-14-9
from a target zone
Jesper Rangvidand Carsten Srensen
Departmentof Finance
Copenhagen Business School
Rosenrns Alle 31
DK-1970 Frederiksberg
Denmark
December1998
We thank Gabriele Becker at BIS and Heino Bohn and Niels Lynggard Hansen at Danmarks
Nationalbankforprovidinguswiththedatausedinthepaper. CommentsandsuggestionsfromTom
Berglund, Stephen Christophe, HaraldUhlig, twoanonymous referees and participants at the EFA
meeting inFontainebleau, theESEM-EEAmeetings inBerlin,andtheFMAmeetingin Chicagoare
appreciated. CarstenSrensenacknowledgesnancialsupportfrom theDanishNatural Scienceand
SocialScienceResearchCouncils.
The paperprovidesa continuoustime modelof thedynamicbehaviorof exchange ratesand
interestrates when exchange rates aremanaged withina target zone withthe possibilityof
realignments. Inthecaseofarealignmenttheexchangeratejumpstoashadowexchangerate.
The timingof realignmentsismodellledbyaCoxprocess withan intensitythat dependson
thelocationoftheexchangerateinthetargetzonebandaswellasthedistancetotheshadow
exchange rate. We set up an approximate maximum likelihood estimation approach and
provide parameter estimates for six ERM target zones. Moreover, in the empiricalanalysis
we lter outthe shadow exchange rates and investigate which fundamental macroeconomic
factorsareabletoexplaintheshortrunandlongrunbehaviorofthelteredshadowexchange
rates.
Thepaperinvestigatesthebehaviorofexchangeratesand interestrateswithinatarget zone
by analyzing empirically the Exchange Rate Mechanism (ERM) within the European Mo-
netary System during the period form March 1979 to June 1997. The target zones within
theEuropean MonetarySystem providean interestingand uniquecaseforexamining issues
concerning how monetary policy aects the behavior of exchange rates, interest rates, and
other important macroeconomic variables. In particular, the exchange rate regime experi-
enced some stableand unstable periods. Thus, at several occasions there were \speculative
attacks" on a number of currencies which eventually lead to the de facto suspensionof the
target zoneregimeinAugust 1993 wherethebandswereextendedto thirtypercentformost
of thecurrencies involved.
It remainsan open questionwhether the unstable periodsof theERM within theEuro-
pean Monetary System were caused by macroeconomic variables and exchange rates being
fundamentally misalignedorwhetherthespeculativeattackswerebasicallydrivenbyagents
suddenly changing beliefs with respect to the sustainability of a given target zone (see e.g.
Obstfeld, 1996,and Velasco, 1996).
We use a two step procedure in order to analyze how the behavior of the exchange
rates and interest rates withinthe ERM can be explained by fundamental macroeconomic
factors. In the rst step, we present and estimate a bilateral continuous time model where
the exchange rate jumpsto a shadow exchange rate whenever a realignment happens. The
shadowexchangeratecanbethoughtofasthenatural,orfundamental,leveloftheexchange
rate if there was no target zone regime. An important feature of the modelling is that we
can lterouttheshadowexchange rate fromthe continuous timeempiricalanalysis. In the
secondstep,weinvestigate whetherbasicmacroeconomic variablescanexplaintheshortrun
and long runmovementsof thelteredshadowexchangerates.
Thecontinuoustimetarget zonemodelshares somebasicideaswiththemodel inChris-
tensenetal.(1998)inthesensethatitallowsforrealignmentsdescribedbyaPoissonprocess
withastochasticintensity;aso-calledCoxprocess. Basically,theexchangerate isrestricted
to movewithinabandandcanonlyleavethebandbyarealignmentjump. Theintensityfor
arealignmentdependsonthepositionoftheexchangeratewithinthebandand thedistance
to the shadow exchange rate. We will assume \uncovered interest rate parity" in order to
lter out the shadow exchange rate. The target zone model in this paper mainly diers
from themodelinChristensenet al. (1998)bythefunctionalformofthedriftand volatility
of themanagedexchangerate aswellastherealignment intensity.
We setup an estimationapproach forthecontinuoustimemodel andprovideparameter
estimates forthedierentEuropeancurrenciesthathave beenengagedintheERMsincethe
startinMarch1979. Theshadowexchangerate islteredoutusingtheparameterestimates
obtained byan approximatemaximum likelihood procedure. In theapproximatemaximum
likelihoodprocedurewesubstitutethetruetransitiondensityoftheexchangerate,described
byastochasticdierentialequation,withitsdiscretetimecounterpart. Moreover,weassume
that the model facilitates a sequential cut, in the sense of Engle et al. (1983), in order to
focus on the estimation of the parameters necessary to lter out the shadow exchange rate
at thevariousobservationdates.
The empirical target zone analysis in this paper is especially analogue to the empirical
analyses in Bekaert & Gray (1998) and Li (1998).
3
In particular, Bekaert & Gray (1998)
and Li(1998) also incorporate stochasticrealignment riskbutexplicitlyspecifythemacroe-
conomic variables which are assumed to determine therealignment probabilities; moreover,
Li (1998) provides a smallsurvey of related literaturethat she buildson and extends. This
diersfrom the setup in thispaperwhere theimplied shadow exchange rate is intended to
summarizetherelevantinformationaboutmacroeconomicvariablesintheformalcontinuous
time target zonemodel.
After ltering out the shadow exchange rates at the various parameter estimates, we
analyze whether macroeconomic variables can explain the observed path of the dierence
between theshadow exchange rate and the managedexchangerate (thedegree of misalign-
ment)during theperiod surroundingthe ERM turbulencein 1992-1993. Conceptually,this
investigation is in the spirit of the approach in Svensson (1993) and a number of papers
extendingonhisapproach,e.g. Lindberget al.(1993), Chen&Giovannini(1994, 1997), and
Rose & Svensson(1994); however, these authorsanalyze expected realignmentswhereas we
1
Intheappendix,wedemonstratetheconsistencyof\uncoveredinterestrateparity"withaspecicequi-
libriumpricingstructureintheeconomyanddescribebrieyhowthemodelisusedforpricingexchangerate
contingentclaims.
2
Thelteringofthe shadowexchange ratediersfrom thesimplelinearlterinFloodet al.(1991)and
Flood&Rose(1995)especiallyduetotheformalmodelofanon-credibletargetzoneusedinthispaper.
3
WhereasthespecicestimationprocedureinBekaert&Gray(1998)alsoreliesonamaximumlikelihood
approach,Li(1998)usesaBayesianapproach.
look upon the degree of misalignment. Using six macroeconomic variables, we run simple
regression analysesinorderto investigate whichvariablesare ableto explainthelteredde-
grees of misalignment. In linewiththeresults inCampa&Chang (1998), we ndexchange
reserves to be an important determinant of the degree of misalignment across the involved
ERM exchange rates.
Inadditionto theregressionanalysis oftheshort-runbehaviorof thedegree of misalign-
mentduringthe1992-1993 ERM turmoil,wefurthermore tryto establishsome basicresults
concerningthelong-run determinantsof thelevels ofthelteredshadowexchangerates. To
thisend,werelyonthestandardmonetaryapproachto exchangerates. Especially,weinves-
tigate whether theshadow exchange ratescan on thelong runbe described by relative (i.e.
thehomecountryversusGerman)moneysuppliesandrelativelevelsofproductionbymeans
of a cointegration analysis. The use of cointegration analysisis natural inthiscontext asit
takesthepossibilityof non-stationarityinthelevels oftheseriesintoaccount. Basically,we
ndamonetaryequationto be validonthelongrunforall oftheinvolvedshadowexchange
ratesexcept theFrenchfranc versusGerman mark.
The paper is organized as follows. In Section 2 we describe the continuous time target
zonemodel. WesetupanapproximatemaximumlikelihoodestimationapproachinSection3.
In Section4wereporttheparameterestimates of thetarget zonemodel andinvestigate the
macroeconomicdeterminantsofthelteredshadowexchangerates. Section5concludes. The
appendixdescribestherelevantpricingkernelintheeconomyandhowtopriceexchangerate
contingent claims.
2 The continuous time target zone model
The dynamics are in the following continuous time model described by a two-dimensional
wienerprocess (W
1t
;W
2t
),witha correlationcoeÆcient ,and ajumpprocessN
t
whichcan
bethoughtofasaPoissonprocesswheretheintensityisdescribedbyastochasticprocess
t
;
formallyN
t
isaso-calledCoxprocess. Thebasicideainthetargetzonemodellingistorestrict
the exchange rate to move within a band unless a realignment occurs. Within the band,
the exchange rate is described by a standard stochastic dierential equation. Furthermore,
realignments occur just as jumpsin a Cox process. The intensity for jumps, in particular,
4
Themodelallowsus tolter outtheprobabilities ofdevaluations aswell, though,wefocus hereonthe
seriesfortheshadowexchangerates.
the exchange rate is relativelyto some \fundamental" which we willrefer to asthe shadow
exchange rate. The shadow exchange rate can be interpreted as the natural level for the
exchangerate ifthetarget zone regimeis suspended.
2.1 The exchange rate dynamics
As described above, the exchange rate is restricted to move within a band in time periods
withoutrealignments. Wheneverarealignmenthappens,theexchangeratejumpstoashadow
exchange rate. The dynamics of the (log-) shadow exchange rate, f
t
, are described by a
stochastic dierentialequation ontheform,
df
t
=dt+dW
1t
(1)
where and areconstant parameters. Theparameter describes theexpectedchange in
theshadowexchangerate whereas is thevolatilityof theshadow exchange rate.
We willusethe convention thatthe exchange rate is thepriceof theforeign currencyin
units of the domestic currency. Moreover, for all specic exchange rates in the paper, we
use theconvention that the German markis theforeign currencyso thatthe exchange rate
always refersto thepriceof theGerman markindierent localcurrencies.
Thedynamics ofthelogarithm to theexchangerate,x
t
,aredescribed by,
dx
t
=[ a(
t x
t )+b
t (f
t x
t
)]dt+Æ s
4(u
t x
t )(x
t l
t )
(u
t l
t )
2
dW
2t +
t dN
t
(2)
where
b
t b(x
t
;
t
;l
t
;u
t )=
8
>
>
>
<
>
>
>
:
ut xt
ut t
if
t
<x
t
<f
t
xt lt
t lt
if f
t
<x
t
<
t
otherwise
and a, , and Æ are constant parameters. The processes
t , l
t
, and u
t
are the(log-) central
parity, the (log-) lower bound of the band, and the (log-) upper bound of the band and,
in general,
t
= 1
2 l
t +
1
2 u
t
. The jump component in (2) incorporates the possibility of re-
alignmentsinto themodel. In particular,
t
describesthejump sizeof theexchange rate in
case of a realignment. We will describe the precise modelling of realignments in the next
section. In this section, we will focus on the basic ideas behind the modellingof exchange
rate movements and theinterpretationand signicance of the parameters a, , and Æ when
theexchangerate moves withinthetarget zone.
wise, the parameter describes the degree of tendency towards the shadow exchange rate
throughthefunctionb(x
t
;
t
;l
t
;u
t
);thishappensinsuch awaythatthetendencyevaporates
at theboundarieswheneverthetendencytowardstheshadowexchangeratewouldotherwise
pullitoutsidetheexchangerateband. Ourspecicationofthedriftratefortheexchangerate
diersfromthespecicationinChristensenetal. (1998)wherethere isno tendencytowards
the shadow exchange rate whenever the exchange rate is close to the boundaries. In our
model,the tendencytowards theshadowexchangerate isnotzero if,say,theexchangerate
is close to thelowerboundaryand theshadowexchange rate isabove the current exchange
rate. This seemsa more appealing characteristic since, inthiscase, thetendency towards a
shadowexchangerate shouldtendto pulltheexchange rate awayfrom thelowerboundary.
Themodelisformulatedinsuchawaythatthevolatilityoftheexchangerateevaporates
attheboundaries. Thetendencytowardsthecentralparitywillthusalwayspulltheexchange
rateintothebandwheneveritisclosetoaboundary{andsometimeshelpedbythetendency
towards the shadowexchange rate,asdiscussedabove. The onlyway theexchangerate can
leavethebandisbyajump. TheparameterÆ isavolatilityparameterfortheexchangerate
withinthebandand themodelisspeciedsuch thatÆ can be interpretedasthevolatilityof
theexchangerate when located inthecenter ofthetarget zone.
[INSERTFIGURE 1AND FIGURE2 ABOUTHERE]
The drift and volatility of the managed exchange rate as a function of the exchange rate
within a specic target zone bandis illustrated inFigure 1 and Figure 2, respectively. The
drift rate is a stochastic process, sinceit is a function of the shadow exchange rate f
t , and
Figure 1thusonlyillustratestheform ofthedriftrate forvedierentvaluesoftheshadow
exchange rate. As illustrated by Figure 2, the volatility of the managed exchange rate is
highest in thecenter of thetarget zone butevaporates rapidlyclose to theboundaries. On
the other hand,the driftrate of themanagedexchangerate is always negative (positive)at
the upper(lower) boundarysothatthemanaged exchange rateis pulledtowards thecenter
of thetarget zonewhen locatedat theboundaries;fora given level ofthe shadow exchange
ratef
t
thedriftrateislinearforsome segmentsbutnon-linearforothersegmentsinorderto
obtainthisfeatureattheboundaries. Basically,whentheexchangerateisnotunderpressure
the drift is linear. This is e.g. the case ifthe shadow exchange rate is at the center of the
target zoneasinthecase f
t
=0 inFigure1.
The shadowexchangerate f
t
isdened asthenatural leveloftheexchangerateifa realign-
ment of the band should occur. Realignments usually occur whenever at least one of the
currencies in theERM is underpressure. The involved currencies willusuallynot be under
pressure versus German mark, if the exchange rate is close to the lower boundary of the
band. Below, we willstartoutby describingthetimingof realignmentsand thenthemodel
forwhat happensat a realignment date.
The arrival of realignmentsis modelled as jumpsina Cox process. This generalizes the
jump feature of the target zonemodellingby Ball& Roma (1993) and Dumaset al. (1995)
wheretherealignmentintensityisassumedconstant. Wewillassumethatthejumpintensity
forarealignmentinanyspeciccurrencyagainsttheGerman markdependsonthedistance
to the centralparityof thetarget zoneaswellasthedistance to theshadowexchangerate.
Formally,the jumpintensityforarealignment is,
t
(x
t
;
t
;l
t
;u
t
;f
t )=
0 +
1
max[0;(f
t x
t )
x
t
t
u
t l
t
] (3)
where
0 and
1
areconstantnon-negativeparameters. Thisspecicationcapturesthebasic
characteristics of the probability of a realignment as described above and has some other
desirableimplicationsas we willdiscussbriey inthefollowing.
If the exchange rate is located above (below) both the central parity and the shadow
exchangerate,thespecicexchangeratewillnotbeunderpressure. Inthiscase,thetendency
towardstheshadowexchangeratewillalsoadjusttheexchangeratetowardsthecentralparity
and,hence, thedivergencefromtheshadowexchangerateisnotathreattothemaintenance
of the target zone. The intensityis, inthiscase, a non-negative constant,
0
,and potential
realignments will occur asoccasional general adjustments of the central parities within the
ERM orasa consequenceof anothercurrencybeing misalignedunderthepresent parities.
However, if the exchange rate is located in between the central parity and the shadow
exchangerate the specic currencymight beunder pressuresincethe tendencytowards the
shadowexchangeratewillpulltheexchangeratefurtherawayfromthecentralparity. Inthis
case, the intensity describing the probabilityof a realignment is more signicant and given
by
0 +
1 (f
t x
t )(x
t
t )(u
t l
t )
1
. The realignment intensity depends on the current
distanceto thecentralparityandthedistanceto theshadowexchangerate. Thedistance to
the shadow exchange rate, in particular, reects how far the current exchange rate is from
the natural level of the exchange rate if the target zone was suspended and thus describes
Ifa realignment occurs,there willbe a discretechange inthelower andupperboundary
for the exchange rate band and usually a discrete change in the central parity as well.
5
Moreover, therewillbeno actionstaken bycentralbanksto prevent theexchange ratefrom
jumping to its natural level asreected bythe shadow exchange rate and, hence, there will
usuallyalso bea discretejumpin theexchange rateat a realignment date.
Thelowerandupperboundaryof thebandarestochasticprocessesthatcan bemodelled
as Coxprocesses withstochastic intensitiesand stochastic jump sizes. Likewise,the central
paritywillbeaprocessofthistype. Foralltheprocesses,thearrivalofjumpsisdescribedby
N
t
. However, we willnotmake anyspecic assumptionsonthe jumpsizes fortheprocesses
as this is not crucialfor the estimation approach and the ltering of the shadow exchange
ratesas describedinSection3.
6
Finally, we will assume that whenever a realignment occurs, the exchange rate will not
necessarily jumpexactlytotheshadowexchangerate. Rather, the(log-) exchangerate will
jump to the(log-) shadow exchange rate plussome noisedescribed byserially independent
andnormallydistributedstochasticvariableswithmeanzeroandstandarddeviation!. The
jump sizesinequation (2)thushave theform:
t
=(f
t x
t )+
t
where
t
NID (0;! 2
).
2.3 The interest rate spread
Wewillfollowastandardapproachintheeconomicliteratureandassume\uncoveredinterest
rate parity" inthe sensethat theshort interest rate spreadis equalto theexpected change
inthelog-exchangerate. \Uncovered interestrate parity"isoftenrejectedinempiricaltests
based on oating exchange rates. However, Svensson (1992) argues that the assumption
of \uncovered interest rate parity" is more appropriate for managed exchange rates in a
target zone, and this is to some extent supported by empirical results in e.g. Bossaerts &
Hillion (1991) and Rose &Svensson(1995). In addition, imposing\uncovered interest rate
parity" places some restrictions on the pricing kernel (for example marginal utilities of a
representative agent) in the economy that is relevant for general asset pricing and pricing
of exchange rate contingent claims. In the appendix we demonstrate the existence of a
5
Weconsiderallchangesineithert,lt,orutas indicatorsofarealignment. Thus,weconsiderachange
inthewidthofthebandasarealignmentevenwhenthecentralparityiskeptconstant.
6
Wewill assumethat themodelfacilitatesasequentialcutinthesenseofEngleetal.(1983);thedetails
aregiveninSection3.
that theshortinterestrate spreadis equalto the expected changein thelog-exchangerate;
moreover, the appendixdescribeshowexchange ratecontingent claimscan be pricedwithin
thecontinuoustime target zonemodel.
By the \uncovered interest rate parity", the interest rate spread in the economy must
equal thedrift rate in the(log-) exchange rate plusa component representing the expected
(log-) exchange rate movement due to a realignment; see equation (2). Let r D
t
denote the
domesticinterestrate and letr F
t
denote theforeigninterestrate. From (2), we thenhave:
r
t r
D
t r
F
t
=a(
t x
t )+b
t (f
t x
t )+
t (f
t x
t
): (4)
The rst two terms inthe descriptionof theinterest rate spread, r
t
,arerelated to the drift
in theexchange rate ifthe possibilityof a realignment is ignored. The thirdtermis related
to the expected change in the exchange rate due the possibilityof a realignment; the term
can be interpreted as a rate of probability for a realignment,
t
, times the expected jump
size, f
t x
t .
Theinterestratespreadisanincreasingfunctionoftheshadowexchangerate. Thebasic
intuitionisthatthedomesticcurrencywilldepreciateforhighlevelsof theshadowexchange
rateand,inequilibrium,rationalinvestorsmustbecompensatedbyarelativelyhighdomestic
interest rate. Obviously, it is a key feature of the modelling that the depreciation of the
domesticcurrency,forrelativelyhighlevels oftheshadowexchange rate,mayhappeneither
continuouslyorbya realignmentjump.
Likewise, whenever the exchange rate is below (above) the central parity, the tendency
towards the central parity will tend to depreciate (appreciate) the domestic currency, and
thusinducea high(low)interestrate spread.
Note that, if there was no tendency towards a shadow exchange rate (and, hence, the
secondtermandthirdtermontherighthandsideofequation(4)wereabsent),theexchange
rateandtheinterestratespreadwouldinfactbeperfectlynegativelycorrelated. Thenegative
correlationisentirelydueto thetendencyawayfromtheboundariesandtowardsthecentral
parity; this is similar to the implications of the Krugman (1991) model but is not in line
with the results from empirical tests as noted in e.g. Bertola & Caballero (1992). In this
perspective,thetendencytowardstheshadowexchangerateinourmodelallowsforapositive
correlationbetween theexchange rate and theinterestrate duringperiodswithpressureon
theexchangerate.
An important feature of the model is that there is an invertible relationship between the
interest rate spread and the shadow exchange rate given knowledge of all the other state-
variablesintheeconomy (i.e.x
t ,l
t ,u
t ,
t ).
We will use this observation to lter out the shadow exchange rate at dierent time
points and in the estimation approach in the following section as well. The relationship
between the shadow exchange rate and the interest rate spread inthe economy is given by
(4). Asdiscussedabove, theinterestrate spreadismonotonicallyincreasingasafunctionof
the shadow exchange rate and, hence, we can invert the relationship and write the shadow
exchange rate as a function of the interest rate spread. The interest rate spread is in fact
a linear function of the shadow exchange rate for some values of the state-variables and a
second order polynomialfor other values; thiscan be seen by inserting the denitions of b
t
and
t
in equation (4). By inverting therelationship, thesolutionfor theshadow exchange
rate hasthefollowingform:
f
t
= 8
>
<
>
: x
t +
r
t +C
t
+
0
if(r
t +C
t )(x
t
t )0
x
t +
B
t +
p
B 2
t +4A
t (r
t +C
t )
2A
t
otherwise
(5)
where
A
t
=
1
x
t
t
u
t l
t
B
t
= 8
<
:
u
t x
t
u
t
t
+
0 ifr
t +C
t
>0;x
t
>
t
x
t l
t
t l
t
+
0
otherwise
C
t
=a(x
t
t ):
In theestimationapproach below, we willthinkof and referto f
t
asan observable variable,
similar to the other state-variables involved, though the observability is always understood
to be transmitted through (5). Likewise, equation (5) describes how the shadow exchange
rate forthesubsequent macroeconomic analysisis lteredout.
3 Target zone model estimation approach
Inthissectionwedescribean approximativemaximumlikelihoodestimationmethodforthe
continuoustimetarget zonemodel insection2.
Thedataareobservedatdiscreteequidistanttimepoints,t
i
,i=0;:::;I and=t
i t
i 1 .
Whenever convenient, we use the simplifying notation: x
i x
t
i ,
i
t
i
and so on. At
lower and upper boundary of the exchange rate band as well as the central parity. Given
a set of parameters,there is an invertiblerelationship betweenthe interest rate spread and
the shadow exchange rate, as solved for in (5). We will therefore occasionally refer to the
observed shadow exchange rate rather than the observed interest rate spread, though it is
implicitlyunderstoodthatthisobservabilityis transmittedthroughequation(5). Thestate-
vector S
i
=(x
i
;f
i
;
i
;l
i
;u
i
)thusdescribestheobservedstate oftheeconomyat thei'thtime
point.
Thetarget zone modelin Section 2is Markovian. Hence,if theconditional densities for
the state-vector are analytically known, the log-likelihood function can be formulated and
usedasabasisforndingmaximumlikelihoodparameterestimates. Theconditionaldensities
implied by the continuous time target zone model are, however, not known and maximum
likelihoodestimationistherefore notdirectlyaccessible. The approachtaken belowisbased
on approximativemaximum likelihoodestimationinthesensethat theconditionaldensities
are proxied by their counterparts from a discretized version of the stochastic dierential
system in(1) and (2). The discretized versionis based on Euler discretizationschemes and
hasthe form,
f
i+1
= f
i
++z~
1;i
(6)
x
i+1
= 8
>
<
>
: x
i +[a(
i x
i )+b
i (f
i x
i
)]+Æ r
4(u
i x
i )(x
i l
i )
(u
i l
i )
2
~ z
2;i ifJ
i+1
=0
f
i+1 +
i+1
ifJ
i+1
=1
(7)
where (~z
1;i
;z~
2;i
) are seriallyuncorrelated and bivariatenormallydistributedvariables,
(~z
1;i
;z~
2;i
)NID(0 ;); =
1
1
!
and J
i+1
is an indicator functionfor a realignment;J
i+1 1
fli+16=lig[fui+16=uig
. Hence, J
i+1
takesonthevalueone ifthere isarealignment andzero otherwise. Ifthereisa realignment,
the exchange rate jumps to the shadow exchange rate plussome noise and this happens in
theendof theperiod;J
i+1
=1thusindicates arealignment immediatelybeforetimet
i+1 .
Wewillusethefollowingapproximationoftheprobabilityofnorealignmentoccurringin
theinterval fromt
i to t
i+1 :
E
t
e R
t
i+1
t
i
udu
e
i
(8)
The left handside of (8) is an exact characterization of the probabilityof no realignment;
see forexampleLando(1994). Ananalyticalsolutiontotheexpectationsisnotknown. The
of (8)with itsleft point approximation.
We can solve analyticallyfor theconditional densities implied by thediscretizedsystem
describedby(6),(7),and(8)andthiswillbeusedinthenalformulationofthelog-likelihood
problembelow.
Similartothecontinuoustimemodel,thediscretizedversionisMarkovian. Thelikelihood
function is a function of the model parameters (as wellas the empirical observations) and
is described by the joint density of the observations. The parameters of the model can be
summarizedas=
1 [
2
where,inparticular,
1
isthesetof parameters involved inthe
formalmodellingof theexchangerate dynamicsin(1), (2),and (3)while
2
areparameters
describing jump sizes of
t , l
t
, and u
t
. The joint density function for the state-variables
conditional on theinitialstate S
0
can be writtenon thefollowingform,
p(S
1
;:::;S
I jS
0
;) = I
Y
i=1 p(S
i jS
i 1
;) (9)
= I
Y
i=1 p(x
i
;f
i
;J
i jS
i 1
;)p(
i
;l
i
;u
i jx
i
;f
i
;J
i
;S
i 1
;) (10)
= I
Y
i=1 p(x
i
;f
i
;J
i jS
i 1
;
1 )p(
i
;l
i
;u
i jx
i
;f
i
;J
i
;S
i 1
;) (11)
where p(j;) is used as notation for the conditional densities of the discretized model.
In the rst equality in (9), we have used the Markovian properties of the model while the
second equality is based on an application of Bayes Rule. The last equality follows from
the observation that p(x
i
;f
i
;J
i jS
i 1
;) = p(x
i
;f
i
;J
i jS
i 1
;
1
); which essentially just states
the factthat theconditional densitiesforthe exchange rate,theshadowexchange rate,and
whether a jump occursor not arefunctions of the parameters in (6) and (7) as well as the
jump probabilitiesin(3) and (8)only.
Wewillassumethatthedensityfunctionfacilitatesasequentialcut;seeEngleetal.(1983).
Stated formally, we assume that p(
i
;l
i
;u
i jx
i
;f
i
;J
i
;S
i 1
;) = p(
i
;l
i
;u
i jx
i
;f
i
;J
i
;S
i 1
;
2 ).
Thebasicintuitionbehindthisassumptionisthatthelocationofthebandimmediatelyafter
a realignment depends onlyon theexchange rate and theshadowexchangerate but noton
theparametersdescribingtheirdynamicsin(6)and(7).
7
Fortheproblemofmaximizingthe
7
Sincetherealignment occursimmediatelybefore theobservationofthe newcentral parityand thenew
bandwidth,thisassumptionisconsistentwithe.g.themodellinginChristensenetal.(1998)whereboththe
exchangerateandcentralparityjumptotheshadowexchangerateandthebandwidthisheldconstant.
1
impliesthatwecan ignoretheconditional densitiesp(
i
;l
i
;u
i jx
i
;f
i
;J
i
;S
i 1
;) in(11) since
they do not depend on
1 .
8
Specically, the likelihood function for the estimation of the
parameters in
1
isthusobtainedbyadoptingthedescriptionofthedensityfunctionin(11)
and ignoring these conditional density terms. The remaining part of (11) depends entirely
on the conditional densities implied by the discretized system in (6) and (7) as well as the
jump probabilitiesin(3) and (8).
Asnotedabove,the relevantlikelihoodfunctionis afunction oftheparameters in
1 as
wellastheempiricalobservations. Theexpressionin(11)describesthelikelihoodof theim-
pliedstate-vectors S
i
=(x
i
;f
i
;
i
;l
i
;u
i
) whiletheempiricalobservations are(x
i
;r
i
;
i
;l
i
;u
i ).
In order to obtain the likelihood in terms of the empirical observations we will apply the
change-of-variable formula which involves the Jacobian determinant of the transformation
betweenthe empiricalobservationsand thestate-vector S
i
. In ourcase, theJacobiandeter-
minantreducestothepartialderivativeoftheimpliedshadowexchangeratef
i
(asdescribed
in(5)) with respect to theinterest rate spreadr
i
. Usingthe change-of-variable formula, the
relevant log-likelihoodfunctionforestimationcan be writtenonthe followingform,
l(
1 ) =
I
X
i=1 log
@f
i
@r
i
p(x
i
;f
i
;J
i jS
i 1
;
1 )
(12)
= I
X
i=1 log
@f
i
@r
i
+
I
X
i=1 logp(x
i
;f
i jJ
i
;S
i 1
;
1 )+
I
X
i=1
logp(J
i jS
i 1
;
1
) (13)
where
p(J
i jS
i 1
;
1 )=
8
<
: e
i
1
ifJ
i
=0
1 e
i 1
ifJ
i
=1
and
p(x
i
;f
i jJ
i
;S
i 1
;
1 )=
1
2jV
i j
1=2 e
1
2 m
i V
i m
0
i
with
m 0
i
= 8
>
>
>
>
>
>
<
>
>
>
>
>
>
: 0
@
f
i f
i 1
x
i x
i 1 [a(
i 1 x
i 1 )+b
i 1 (f
i 1 x
i 1 )]
1
A
ifJ
i
=0
0
@ f
i f
i 1
x
i f
i
1
A
ifJ
i
=1
8
Moreover,(xt;ft;Ji)isweaklyexogenous forinferenceabout2;seeEngleetal.(1983),Denition2.5.
V
i
= 8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
: B
B
@
2
Æ
4(u
i x
i )(x
i l
i )
(u
i l
i )
2
Æ r
4(u
i x
i )(x
i l
i )
(u
i l
i )
2
Æ
2 4(u
i x
i )(x
i l
i )
(u
i l
i )
2
C
C
A ifJ
i
=0
0
@
2
0
0 w
2 1
A
ifJ
i
=1
The parameter estimates of the target zone model reported in the following section are
obtained bymaximizingthelog-likelihoodfunction(13) withrespectto
1 .
4 Empirical results
Inthissectionwe willdescribethedataandestimationresults. Furthermore,wewilldiscuss
theparameterestimatesand presentlteredshadowexchangerates. Finally,we willanalyze
issues concerning macroeconomic determinants of misalignment on the short run and long
run.
4.1 Data
Thedataconsistsofweeklyobservationsonexchangeratesand1-monthEuromarketinterest
rates. Datawascollectedforthesevencountries whichhavebeeninvolvedintheERM since
March 13, 1979. The exchange rate data consists of exchange rates for the six currencies:
Belgian franc(BEF), Danishkroner (DKK),Frenchfranc (FRF), Irishpound(IEP),Italian
lira(ITL),andDutchguilder(NLG)versustheGermanmark(DEM).Thedatafortheshort
interestratesconsistsoftheinterest ratespreadsbetweenthe1-month Euromarketinterest
ratesinthesame sixEuropean countriesand the1-monthEuromarketratefortheGerman
mark. The Euromarketinterestratesarebid ratescollectedat around 10a.m. SwissTime;
theinterestrate datawas suppliedbytheBank forInternationalSettlements(BIS).
The data seriesspan from March 13, 1979 until June 25, 1997 (November 4, 1981 until
June 25, 1997 for the Irish pound). The data is availableon a daily basis, but since there
might be small discrepancies in theexact time of the day forthe sampling of the exchange
rate observations and the Euro market interest rate observations, we have chosen to use
weekly data. The data consists of observations on Wednesdays (Thursday if Wednesday is
not available) and we have 955 weekly observation dates for all currencies except the Irish
poundforwhichwe have 817 weekly observation dates.
In implementing the approximate maximum likelihood method as described above, we ex-
perienced some identication problems when running the problem unrestricted for the six
individual currencies.
9
As a consequence, we x two parameters to be equal for all the six
currencies intheresultsreportedbelow.
Thetwoparameterswendreasonabletoxa prioriaretheparameters! and
0 which
areparametersthatdescribethetimingaswellaswhathappensatarealignmentdate. Since
onlyafewrealignmentsareobservedforeachexchangerate(twofortheDutchguilderversus
Germanmarkexchangerateto elevenfortheItalianliraversusGermanmarkexchangerate)
it is obviously diÆcult to obtain reliable estimates on these parameters; the parameter
1
alsodescribesthetimingofrealignmentsduringperiodsofspeculativepressureonthetarget
zone and is estimated as part of the general approximate maximum likelihood estimation
problemoutlined inSection 3. Our specic reasoningand theconcrete choice of parameter
valuesfor! and
0
aredescribed inthefollowing.
At a realignment date, the(log-) exchange rate willjump to the(log-) shadow exchange
rate plussome \noise"; theparameter! describesthestandarddeviationof thisnoiseterm.
In the estimation results reported below, we x this standard deviation at a 0.25 percent
level, i.e. ! = 0:0025, which reects our subjective view of the part of a realignment jump
whichcan beattributed to \noise".
10
The parameter
0
describes the intensity of a realignment when the specic currency
is not under pressure versus the German mark; see the discussion of equation (3). When
the currency is not under pressure, the realignment intensity is constant as in a standard
Poissonprocess. Moreover, a reasonable(maximumlikelihood)estimator of the intensityin
aPoissonprocessisthenumberofjumpsdividedbytheprocesstime. Hence,thisparameter
iscalibratedbytheratioofnumberofjumpstotheamountoftimespentnotunderpressure
forthetotal ofthe sixcurrencies. The positionoftheexchangerate withinthebandisused
as a proxy for whether the currencyis under pressureor not. Ifthe exchange rate is below
the centralparity, we view the exchange rate asnot being under pressure. Thisis based on
the reasoningthat theGerman mark isusually the\strong"currencywithinthe ERMand,
9
Thisproblem shows up wheninvertingthe Hessianmatrixofthe log-likelihood problemat theoptimal
parameterestimatesinordertoobtainstandarderrors.
10
Thise.g.impliesthatifthereisarealignmentjumpandtheshadowexchangerateisF
t e
f
t
=1,there
isa95%probabilitythattheexchangeratewillbeintheinterval[0.9951,1.0049] aftertherealignmentjump.
the upperboundary ofthe band. The numberof realignments when theexchange rate is in
the lower part of the bandin thedata sample is three (one for DKK, one forITL, and one
forIEP). The total timespent inthelowerpart of thebandis44.71 years and theestimate
of theintensityparameteris obtainedas
0
=3/44.71 = 0.0671.
Theestimatesobtainedbymaximizingthelog-likelihoodfunctionin(13),withrestrictions
on ! and
0
,arepresentedinTable1.
11
[INSERTTABLE1 ABOUTHERE]
We will provide our interpretation of the parameter estimates in the following discussion.
Moreover,theimpliedshadowexchangeratesinthesubsequentempiricalanalysisareltered
outusingthetabulated parameterestimates.
Thecredibilityofthetargetzoneisreectedthroughatleasttheparameters,a,and
1 .
The parameter determines the driftrate of the shadow exchange rate and thusindicates
the sustainability of a given target zone. Specically, if the parameter is dierent from
zero, it isnotlikelythata specic target zonebandcan besustainedoverlongtimeperiods
since the shadow exchange rate will tendto driftoutside theband and will thusinevitably
putthe target zoneunder pressureat some timepoint. Theparameter estimates inTable 1
indicate that the drift rate of the Dutch guilder shadow exchange rate is not signicantly
dierent from zero (= 0.0010) whiletheparameterestimates of fortheother currencies
versus German mark are all positive. The highest estimate of is obtained forthe Italian
liraversusGerman mark( =0.0619) whichindicatesa lowsustainabilityofthisparticular
targetzone;thisisnotasurprisingresulthavinginmindthefrequenttargetzoneadjustments
of theItalianliraversusGerman markintheestimationperiod.
Highcredibilityofatargetzoneisalsoassociatedwithahighparametervalueofa,which
describes the tendency towards the central parity, and a low parameter value of
1
, which
describestherealignmentintensityduringperiodswheretheexchangerateisunderpressure.
The highest parameter estimate of a and thelowest parameter estimate of
1
are obtained
for the Dutch guilder(a = 0.1560 and
1
= 56.29) which again indicate high credibilityof
the target zone for the Dutch guilder versus German mark.
12
The lowest estimate of the
11
Thebandfor theItalian lirawassuspended inthe periodfrom September17, 1992untilNovember25,
1996. Intheestimationprocedure,wehandledthisproblembyimposingaverywidebandintheperiod;1000
percentoneachsideoftheexchangerate.
12
ThendingthattheDutchguilderversusGermanmarkexchangerateshowsthehighestdegreeofmean
existence ofa target zonethushastheweakestimplicationsforthemean-revertingbehavior
of the Italian lira against German mark exchange rate within the ERM band. Likewise,
the highest estimate of
1
is obtained forthe Italian lira (
1
=1162.83) which in addition
suggests that this target zone is considered very likelyto be adjusted whenever the Italian
lirais closeto theboundariesof theband.
As indicated in the above discussion, the parameter estimates in Table 1 indicate two
polar case: a credible Dutch guildertarget zone and a non-credible Italian lira target zone.
In thisperspective,we willinterpret theother parameterestimates.
Theparameterdescribesthedegreeoftendencyoftheexchangeratetowardstheshadow
exchange rate that takes place continuously and not as a discontinuous realignment jump.
ThelowestparameterestimateofisobtainedfortheDutchguildertargetzone(=0.9299)
while the highest estimate is obtained for the Italian lira target zone ( = 5.3490). A low
estimateofcan,inourview,beinterpretedasindicatingawell-functioning(credible)target
zone in the sense that the tendency towards thecenter of the target zone (as described by
theparametera)tends todominatethetendencytotheshadowexchangerate forlowvalues
of .
Theparameter describesthedegree of correlationbetweenthemanagedexchangerate
and the shadow exchange rate. A low estimate (2. lowest) of is obtained for the Dutch
guildertargetzone(=0.2804) whilethehighestestimateofisobtainedfortheItalianlira
target zone ( =0.9628). In particular, low estimates of indicatethat uctuations in the
shadowexchange rate have relativelylowimpact on the shortrunbehaviorof the managed
exchangerate;thisisthecasefortheDutchguilderversusGermanmark. Ontheotherhand,
non-credible target zones, such as the Italian lira target zone, tend to be associated with a
high instantaneousfeed-backbetweenthemanagedexchangerate and theshadow exchange
rate asreected inhighestimates of .
The parameterÆ describesthe volatilityof the exchange rate within theband whilethe
parameter describesthe volatility of the shadow exchange rate. The lowest estimate of Æ
isobtained forthe Dutch guildertarget zone(Æ =0.0113) whilethesecond highestestimate
is obtainedfor theItalianlira (Æ= 0.0563). Theparameter estimates of exhibit thesame
pattern; lowest for the Dutch guilder ( = 0.0240) and highest for the Italian lira ( =
0.0576). If the shadow exchange rate is interpreted as the exchange rate without a target
reversionisconsistentwiththendingsinAnthony&MacDonald(1998).
induces a volatile managed exchange rate within the target zone. Moreover, the estimates
in Table 1 indicate that a low volatility of the managed exchange rate is related to a high
credibilityof thetarget zone.
As noted above, the Dutch guildertarget zone and the Italian lira target zone provide
two polar cases. The estimates of the parameters for the Belgium franc target zone are
consistentlysimilar to the estimates of the Dutch guildertarget zone butless evident and,
hence, theresultsindicate a relativelyhigh credibilityof theBelgian franc target zone. For
the othertarget zones theresultsaremixed and lessclear, though,theparameter estimates
areingeneralinbetweenthetwopolarcases oftheDutchguildertargetzoneandtheItalian
liratarget zone. InthediscussionbelowwewillfocusontheFrench franctarget zonewhich
inourview seems to have some specialcharacteristics.
TheparameterestimatesoftheFrenchfranctarget zonearestrikingsincetheresultsare
somehowmixedinrelationtotheabovediscussionofcredibleversusnon-credibletargetzones.
In particular, the French franc versus German mark exchange rate exhibits high volatility
within the bandas reected in thehighest estimate of Æ (Æ = 0.0621). This suggests a low
credibility of the French franc target zone. On the other hand, the lowest estimate of the
correlationbetweenthemanagedexchangerateandtheshadowexchangerateisalsoobtained
for the French franc ( =0.2431) which,referring to theabove discussion, indicates a high
credibilityof the French franc target zone. Anyhow, the remaining parameter estimates of
the French franctarget zonemodel are inbetween thetwo polar cases of the Dutch guilder
and Italian lira target zones which all and all suggest that the French franc target zone is
lesscrediblethantheDutchguildertarget zonebutmorecrediblethantheItalianliratarget
zone.
Finally,notethat the French franc target zone hasa particular featurethat diersfrom
theothertarget zonessincetheFrenchfrancshadowexchangerateisseeminglysignicantly
less volatile than the managed exchange rate in the center of the target zone band ( =
0.0410 < 0.0621 = Æ). This is somehow surprising since a basic motivation for imposinga
target zone on the exchange rate is to reduce the overall exchange rate uncertainty. The
highestimateof Æ fortheFrenchfranc targetzonemaybecaused byan inexiblefunctional
form imposed on the volatilityterm of the managedexchange rate inthe particular case of
the French franctarget zone (see Figure 2). Forexample, Bekaert and Gray(1998) present
evidence indicating that thevolatilityof the managedexchange rate may be higherclose to
target zone. We willnotpursuethisissuefurtherinthiscontext.
4.3 Plots of the shadow exchange rates
AlthoughtheparameterestimatesinTable1haveinterestontheirown,anadditionalimpor-
tant purposewith thispaperisto providesome lteringestimates regardingthe positionof
theshadowexchangeratesduringtheestimationperiod. Theseareobtainedbyinsertingthe
obtained parameterestimates and byinvertingtherelationbetweentheinterestrate spread
and theshadowexchangerate asinequation(5). Thelteredshadowexchangeratesaswell
as exchange rate data, interest rate data, and ERM band data are plotted in Figure 3 to
Figure 8.
[INSERTFIGURE3 TO8ABOUT HERE]
At least two features of the plotsin Figure 3 to 8 deserve special attention. First, with the
exceptionoftheDutchguildershadowexchangerate,allshadowexchangeratesaregenerally
positioned\above" theexchange rate. Second,the plotof theItalian shadowexchange rate
isinterestingbecause therelativelyhighinterestrate spreadisnotreectedina highdegree
of misalignment,asmeasuredbyf
t x
t
. Hence,thenon-credibilityof thisparticular target
zoneinsome senseimplieslowstabilizingpropertiesofthetarget zoneandat thesame time
the frequent realignments exclude the possibility of large dierences between the shadow
exchangerate and themanagedexchangerate.
In addition to these general remarks on the basic properties of the ltered shadow ex-
change rates, we would like to point outthat the procedure used inthispaper can possibly
addanotherdimensiontothequestionofwhetherexchangerateshavebeenmisalignedbefore
periodsofcurrencyturmoilsand,ifso,agentshavebeenabletorecognizesuchmisalignments
andtrade domesticrelativetoforeignassetsat anincreasinginterestratespread. Thelitera-
ture on thisissue iswelldeveloped. Forinstance,Svensson(1993), Rose&Svensson(1994),
and Chen&Giovannini(1997)subtract theexpectedchangeof theexchangeratewithinthe
target zone band from the interest rate dierential to study credibility of the target zone
(the so-called drift-adjustment method); Campa & Chang (1998) compute realignment in-
tensitiesfromERM cross-rate optionsinorder to studytheeconomicdeterminantsofthese;
and Bekaert & Gray (1998) study credibilityof theFrench franc target zone by specically
incorporatingthepossibilityofjumpsinto theirempiricalmodel.
t
above and then look at whether exchange rates have been increasingly misaligned in the
runninguptoperiodsofcurrencyturmoils. Actually,inFigure9,thedegreeofmisalignment
(dened as f
t x
t
) forthe ERM currencies are shown. The plotsindicate that there have
beena numberof occasions whereexchange rateswere beingincreasinglymisaligned before
realignments { for instance,the paths of f
t x
t
preceding a numberof realignments in the
early 1980sare increasingforat least someof currencies.
[INSERT FIGURE9 ABOUTHERE]
Forthecurrencycrisesin1992-1993, on the otherhand, onlyfortheItalian liraand the
Irishpound,aclearincreaseinthespreadf
t x
t
beforetheSeptember1992 respectivelythe
February 1993 realignment isvisible inFigure 9. Perhapsto a lesserextent thisis also the
case for the Belgian franc and Danish kroner but is seemingly not the case for the French
franc. This indicates that the nancial markets considered only the Irish pound and the
Italianlira asbeingfundamentally misalignedduringthe ERM-turmoilsin1992-1993.
Furthermore,overthewholeperiod,thehypothesisofzeromisalignmentisclearlyrejected
(testsarenotshown 13
)andthisholdsevenifthehypothesisofnomisalignmentistestedover
theperiodaftertheBasle-NyborgagreementinSeptember1987,aftertheincreaseinnumber
of participantsintheERM in1990, orafter 1992,i.e. theERM currencies have throughout
theERM historygenerally beenmisalignedinrelationto theGermanmark{ andthisholds
for all exchange rates. For the Dutch guilder,though, the degree of misalignment seems to
workintheoppositedirectioninthesensethatforNLG,f
t x
t
isnegativethroughoutthelast
part of the sample. However, supplementing the resultson a general misalignment of ERM
exchangerates,itisnotedthatthedegreeofmisalignmenthasdecreasedinrecentyearswhich
could point towards increasing convergence betweenERM currencies before introducingthe
euro onJanuary1,1999.
4.4 Determinants of degree of misalignment
In additionto thequestionof currencycrises\predictability"asreected throughincreased
misalignment, a number of recent articles deals with the extent to which specic macroe-
conomic variables are correlated with measures of speculative pressures and realignment
expectations. Inthissection,wefollowtheapproachofEichengreenetal. (1994, 1995,1996)
13
Specically,wetestedwhetherthemeanofthedegreeofmisalignment,ft xt,isequaltozero.
end, we estimatedthe models,
(f
t x
t
)=cons:+
m m
t +
r r
L
t +
y y
t +
t +
R R
t +
q q
t +
t
(14)
overtheperiodJanuary1991toDecember1994 14
,wherem
t
denotesthelogarithmofrelative
(domestic versus foreign) nominal money (M1), r L
t
denotes relative interest rates on long
government bonds, y
t
denotes (the logarithm of) relative industrialproduction,
t
denotes
(the logarithm of) relative rates of ination,R
t
denotes(the logarithm of) relative levels of
oÆcial reserves excludinggold, q
t
measures the (logarithmof) bilateral real exchange rates
towards Germany, and
t
is a regression residual. The macroeconomic data was obtained
from the Main Economic Indicators, OECD. As macroeconomic variables are available up
to a monthly frequency only, we obtained seriesfor x
t and f
t
at the monthly frequency by
averaging overthe weekly observations.
[INSERTTABLE2 ABOUTHERE]
Table 2 shows parameter estimates of (14).
15
A potentially important result from the
table is thesignicant estimates of
R
forBelgium, France, Ireland, and Italy. Onaverage,
a one percentage decrease inrelativereserves areassociatedwitha 0.55 percentage increase
in thedegree of misalignment. Thishas potentially importantbearingsfor theliteratureon
speculativeattacks,asthemorerecentsecond-generationmodels(seee.g.Obstfeld,1996)are
oftenseenspeciedintermsofothervariablesthanoÆcialreserves, whichplayacrucialrole
inrst-generationmodelsofspeculativeattacks(seee.g. Krugman,1979orFlood &Garber,
1984). First-generation modelsversussecond-generation modelsare surveyed and discussed
ine.g. Obstfeld(1994, 1996), Krugman(1996), andFlood& Marion(1997).
Asecondvariablewhichseemstobeofsomeimportanceistherelativelonginterestrates,
whichissignicantforBelgium,France,Ireland, andItaly. A onepercentageincreaseinthe
spreadbetween a domesticand a foreignlong interest rate is associated withan increase in
the degree of misalignment of, on average, 0.50 percentage. This is consistent with Ozkan
14
Campa &Chang (1998) estimatetheir modeloverthe September1991 toMarch1995 period, whereas
Eichengreenetal. (1994,1995,1996)employexclusionwindowsoftwoyearsbeforeandafteracurrencycrises
intheiranalyses. Inourestimationof(14) weessentiallycombinethesetwoapproaches.
15
Assomerst-orderautocorrelationwasleftintheresidualsafterrunningthebasicOLSregressionof(14);
wecorrectedforthisfeatureinthenalregressions.
interestratesand probabilitiesofspeculative attacks.
Tocompare,Rose&Svensson(1994)andChang&Campa(1998)generallyndexplana-
tory powerof oÆcial foreign reserves, as do Chen& Giovannini(1997) and Bekaert & Gray
(1998) for France, whereas Rose & Svensson (1994) furthermore nd dierentials between
rates of ination to be signicant over the whole 1979 to 1992 period. In general, though,
the overall conclusion from thisanalysis (andprevious analyses) isthat traditionalmacroe-
conomic variableshave ahard timeinexplainingtheshort-runturbulenceinexchange rates
orrealignmentexpectationsduringtimesofcurrencyturmoilsasonlyrelativelyfewvariables
are foundtobe signicant.
4.5 Long-run determinants of the shadow exchange rates
Havingpointedtowards some characteristicsof thedegree ofmisalignment duringturbulent
periods of time, it seems natural to investigate whether the lacking explanatory power of
traditionalmacroeconomicvariablesisageneralcharacteristicfortheshadowexchangerates
which holds also over longer time spans. We investigate this by analyzing the standard
monetary approach to exchange rates. The basic monetary equation for example follows
from assuming a quantity equation in each country, m i
t p
i
t
= y i
t
, i = D;F, imposing
purchasingpowerparity(which,fortheshadowexchangerates,takestheformf
t
=p D
t p
F
t )
and reorganizing,
f
t
=m
t y
t
(15)
to whicha regression constant, v,and a stationary errortermcan beadded.
Wheninterpretingestimatesof( 15)itisnotedthatingeneralweexpect(15)toholdonly
foroatingexchangeratesand notnecessarily forexchange ratesina target zone. Consider
e.g. the basic Krugman(1991) model of a target zone. A prominent feature of this model
is that the target zone imposes a S-shaped form on the exchange rate withinthe band, i.e.
monetarydisturbancesaretransmittedonlypartiallytotheexchangerate(the\honeymoon
eect"). Inthis sense, a change in m
t
does notcause an instantaneouslyone-to-one change
in the exchange rate. In our model, however, f
t
is estimated exactly to give the behavior
of the exchange rate if no target zone is present. Therefore, a reasonable hypothesisis that
the shadow exchange rate behaves like a oating rate and that the relationship in (15) is
appropriate.
theshadowexchangeratestend to move inanon-stationary fashion(see also Figure3 to 8)
and for thisreason we employthe multivariate cointegration framework of Johansen (1988,
1991, 1995) in order to obtain estimates of the long run attractors for the ltered shadow
exchangerates.
16
Wethusestimated vector errorcorrection modelssuch as,
Y
t
=
0
e
Y
t 1 +
11
X
i=1 i
e
Y
t i
+ d846+"
t
(16)
with Y =(f
t
;m
t
;y
t )
0
; e
Y
t
=( f
t
;m
t
;y
t
;d901) 0
; and e
Y
t
=
e
Y 0
t
;1
0
; where d911 and d846 are
dummies and 0
is the matrix of cointegration vectors, the rank of which, if cointegration
can beestablished, islessthan thedimensionof Y
t
,i.e.lessthan three.
17
In testing for cointegration, we followed Johansen (1992) who advocates the use of a
sequentialtestingstrategy; theso-called\Pantula-principle",soasto test forrestrictionson
the constant. We ended uprestricting theconstant to enter thecointegration spaceonly,as
in(16).
[INSERTTABLE3 ABOUTHERE]
ThetestsforthenumberofcointegrationvectorsaregiveninTable3forthemodelswhere
the constant isrestrictedto thecointegration space.
18
Theresults pointtowardrejection of
boththehypothesesofat mostzeroand atmostonecointegration vector,i.e. wedecidedon
two cointegration vectorsin all models. Our next step wasto lookfora monetary equation
as a long-run attractor for the shadow exchange rates. We therefore tested whether f
t
16
Inaddition,thecontinuoustimetargetzonemodelsuperimposesnon-stationarityofft;seeequations(1)
and(6).
17
InJuly1990, formerEast andWestGermany joinedGEMU (GermanEconomicand MonetaryUnion)
having the German mark replacing the (East) German currency as legal tender. Obviously, this causeda
signicantpositivejumpinGermanmoneysupplyinuencingallmt=m D
t m
F
t
seriesnegatively. Inorder
totakethisjumpintoaccount,wefollowJuselius (1996)andincludeadummy(d911;takingonthevalue0
for periodsbetween1979:3 and 1990:12and thevalue 1inperiods between1991:1 and1997:6) as aweakly
exogenousvariable. Furthermore,theGermanproductionseriestakesamassivedipin1984:6whichinuences
ally
t
seriesandtotakethisintoaccountweincludedadummy(d846)takingonthevalue1inperiod1984:6
andzerootherwise.
18
We included twelve lags in ournal regression which seems as a reasonable numbergiventhe sample
frequencyof monthlyobservations. Reassuringthough,thebasic resultswerenotalteredwhenrunningthe
cointegration modelswithotherchoicesforthelaglength.
t t
cointegration vectors aregiven inTable 4.
[INSERTTABLE4 ABOUTHERE]
Itappears that forve outof thesixcurrencies under investigation,the hypothesisthat
the monetary equation acts as a long-run attractor for the ltered shadow exchange rates
cannot be rejected at standard signicancelevels. Only for theFrench franc thehypothesis
is rejected. Hence, the results support the basic monetary equation in the sense that we
cannot reject an unit elasticity of the shadow exchange rates with respect to the relative
money suppliesand the relative productionseries, respectively. Furthermore, theestimated
coeÆcientofthedummyd911associatedwiththeGermanunicationispositive. Thepositive
jump in m F
t
, and thus the negative jump in m
t
, dictates a negative jump in the shadow
exchangeratesifdescribedbythemonetaryequation in(15);however,these negative jumps
are to some extent adjustedforbythe dummyd911.
In general, the results in Table 4 indicate that the shadow exchange rates, on the long
run, do behave inaccordance witha standard theory of exchange rate determination. This
is consistent withthendingsof e.g. McDonald&Taylor(1993, 1994), Mark(1995), Chinn
&Meese (1995), andMark &Choi(1997) inthe context ofoating exchange rates.
5 Conclusions
We have investigated the dynamics of exchange rates and interest rate spreads in a target
zoneusingatwostepprocedure. Intherststepwepresentedacontinuoustimetarget zone
modelwheretheexchangeratewasrestrictedtomovewithinabandbutwasallowedtoleave
thebandbyajump. We estimatedthemodelforsixERM exchangeratesand extractedthe
relevant shadowexchangerates. Inthesecondstep,we regressedthedegree ofmisalignment
(thedierencebetweenthelteredshadowexchangerateandthemanagedexchangerate)on
sixmacroeconomicvariables. Basically,theresultsindicatedsomesupportofrst-generation
modelsinthesense that exchange reserveswerecorrelated with thedegree of misalignment
duringthe1992-1993 ERMturmoilfortheinvolvedexchangerates. Likewise,acointegration
analysis provided some evidence for the monetary equation as a long run attractor for ve
(out ofsix) shadow exchange rates.
In line with most of the literature on target zones, the analysis in this paper has been
restrictedto bilateralexchangerates. Thisis admittedlyasimplicationsincethe ERMisa
exchangerates beingtargeted simultaneously. This has, forinstance, theimplicationthat a
currencywhichisactuallywithinthebilateralbandtowardsanothercurrencycanbeaected
bytheinterventionagainst a thirdcurrency; seee.g. theanalyses inJrgensen &Mikkelsen
(1996) orFlandreau(1998). Thecontinuoustimeframeworkinthispapercan potentiallybe
extended to incorporate such features; ofcourse, at thecostsof highercomplexity.
Moreover, when estimating the parameters of the target zone model, one could poten-
tially extend the analysis by incorporating data on cross-rate options, as in e.g. Campa &
Chang (1996, 1998). Thiswouldbeinterestinginmodelsthat extendstheframework of the
present paper, but would require some additional numerical work as no close-form option
pricingformulasexist.
Finally,we believe thatthe specic modellingin thispapercan be extended indierent
directions. For example, one may allowfor a more general specication of the dynamics of
the shadow exchange rate. Likewise, the dynamics of the managed exchange rate may be
generalized toallowforstochastic volatilitywithintheband;i.e.theparameterÆ in(2)may
be replacedbya stochasticprocess.
In this appendix we will demonstrate the existence of a pricing kernel process that makes
themodelinSection2consistent with\uncoveredinterestrateparity" inthe sensethat the
interest rate spreadequals the expected change of the log-exchange rate. In fact, below we
willdemonstrate thatthe followingpricingkernelprocess providessuch consistency,
dK
t
= [r D
t +
t (
J
t
1)]K
t dt
1
2 Æ
t K
t dW
2t +(J
t 1)K
t dN
t
(17)
wherethe relative jumpsize(J
t
1) is stochasticand (
J
t
1) denotestheexpectedrelative
jumpsize. Moreover, Æ
t
is usedasnotationforthewholediusioncoeÆcient inthediusion
term for the log-exchange rate in (2) (see also (20) below). The relative jump size can be
chosen onthe form,
J
t 1=
(f
t x
t )e
1
2
t 1
8
! 2
e 1
2 (f
t x
t )
e 1
2 (f
t x
t )
1 and
J
t 1=
(f
t x
t )e
1
2 (f
t x
t )
e 1
2 (f
t x
t )
e 1
2 (f
t x
t )
1 (18)
where
t
=(f
t x
t )+
t and
t
NID(0;! 2
).
By denition of a pricingkernel process, we can priceany nancial asset by evaluating
therelevant expectations. Consideraclaimhavingstochasticpayoinunitsof thedomestic
currency, S
T
, at some future date T > t. Then the present value of this claim (in units of
thedomestic currency)mustsatisfy,
S
t
=E
t
K
T
K
t S
T
(19)
In a representative agent equilibrium set-up, the pricing kernel is identical to the marginal
utilityof therepresentative agent andprices arefoundbymarginalutilityweighted payos;
see e.g. Lucas (1978) for the discrete time case and Cox et al. (1985a) for the continuous
timecase. In general,themereabsenceof arbitrageensures theexistenceof apricingkernel
processsuchthat allassetprices satisfy(19); see e.g. DuÆe (1996).
Theaboveassumptionsontheformofthepricingkernelprocessareconceptuallydierent
fromthose inChristensenet al.(1997). Comparedtothepricingkernelprocess in(17), they
basicallyconsidera kernelprocesswherethediusiontermandthejumpsizearezero. This
impliesthattheinterestrate spreadequalsthe rateof changeintheexchangerate, X
t .
19
19
However, dueto Siegel'sparadox, it is well-knownthat the interest ratespread will notequal therate
ofchangefor(1=X
t
),i.e.theexchangeratequotedinversely. Inourcase, specifyingthemodelinlogarithms
ensuresthatthe\uncovered interestrateparity"isvalidregardlessofhowtheexchangerateisquoted.
