**Determinants of the Implied Shadow Exchange Rates from a** **Target Zone**

### Rangvid, Jesper; Sørensen, Carsten

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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.