Revisiting the Dollar-Euro Permanent Equilibrium Exchange Rate: Evidence from Multivariate Unobserved Components Models

Transcription

1 Revisiing he Dollar-Euro Permanen Equilibrium Exchange Rae: Evidence from Mulivariae Unobserved Componens Models Xiaoshan Chen * and Ronald MacDonald Deparmen of Economics Universiy of Glasgow Adam Smih Building. May 00 Absrac We propose an alernaive approach o obaining a permanen equilibrium exchange rae (PEER) based on an unobserved componens (UC) model. This approach offers a number of advanages over he convenional coinegraion-based PEER. Firsly we do no rely on he prerequisie ha coinegraion has o be found beween he real exchange rae and macroeconomic fundamenals o obain non-spurious long-run relaionships and he PEER. Secondly he impac ha he permanen and ransiory componens of he macroeconomic fundamenals have on he real exchange rae can be modelled separaely in he UC model. This is imporan for variables where he long and shor-run effecs may drive he real exchange rae in opposie direcions such as he relaive governmen expendiure raio. We also demonsrae ha our proposed exchange rae models have good ou-ofsample forecasing properies. Our approach would be a useful echnique for cenral banks o esimae he equilibrium exchange rae and o forecas he long-run movemens of he exchange rae. Key words: Permanen Equilibrium Exchange Rae; Unobserved Componens Model; Exchange rae forecasing. JEL Classificaions: F3; F47 * Correspondence o: Xiaoshan Chen. x.chen@lbss.gla.ac.uk. Tel.: +44(0) R.Macdonald@lbss.gla.ac.uk. Xiaoshan Chen acknowledges financial suppor from he ESRC (Award reference PTA ). We would like o hank Terence Mills Joe Byrne and Mario Cerrao for helpful commens on he paper.

2 . Inroducion There are a large variey of mehods available for calculaing a counry s equilibrium exchange rae from he inernal-exernal balance approach o he behavioural and permanen equilibrium approaches hrough o he new open economy approach (see MacDonald (000) and Driver and Wesaway (004) for a survey of he lieraure). All of hese approaches have heir own advanages and disadvanages which is why perhaps end users (such as cenral banks and praciioners) use a range of esimaes in coming o a view as o wheher an exchange rae is misaligned or no. In his paper we focus on an exension o he so-called permanen equilibrium approach which has appeared under differen guises in he lieraure. In sum his approach relies on decomposing an acual real exchange rae ino is permanen and ransiory componens and hen using he permanen componen as a measure of he equilibrium exchange rae. A variey of ime series mehods have been used o exrac he permanen componen ranging from Beveridge-Nelson (98) decomposiions (Huizinga 986; Cumby and Huizinga 990) o srucural vecor auoregression (Clarida and Gali 994) and coinegraion-based mehods (Clark and MacDonald 004). This paper proposes an alernaive approach o obain he PEER which is based on an unobserved componens model specificaion and offers a number of advanages over he convenional coinegraion-based PEER proposed by Clark and MacDonald (004). The firs advanage is ha we do no rely on he prerequisie ha coinegraion has o be found beween he real exchange rae and macroeconomic fundamenals o obain non-spurious long-run relaionships and esimaes of he PEER. Our approach faciliaes he esimaion of he long-run relaionship beween he inegraed variables using maximum likelihood and he use of he likelihood raio es o idenify he significance of hese long-run coefficiens even if coinegraion is rejeced. Secondly he impac ha he permanen and ransiory componens of he macroeconomic fundamenals have on he real exchange rae can be modelled separaely in he UC model. This is imporan for variables whose he long and shor-run effecs may drive he real exchange rae in opposie direcions such as he relaive governmen expendiure raio. Addiionally alhough our UC model does no require coinegraion amongs he real exchange rae and he fundamenals as a prerequisie in obaining he PEER he UC model can easily accommodae a coinegraion analysis using

3 he mehods of Nyblom and Harvey (000). We also provide an ou-of-sample forecasing exercise o es he validiy of mulivariae UC models agains a random walk process of he real exchange rae. The ouline of he remainder of his paper is as follows. In he nex secion we ouline our unobserved componen models of he PEER. In secion 3 we discuss he daa se and he ime series properies of he daa. Secion 4 conains our esimaes of he unobserved componen models while Secion 5 conains coinegraion based ess of he models. Our ou-of-sample forecasing resuls are conained in Secion 6 and Secion 7 concludes.. Unobserved componen models of he PEER This secion presens he UC model used o obain he dollar-euro permanen equilibrium exchange rae (PEER). We consruc four commonly used macroeconomic variables (he macroeconomic fundamenals) which are he erms of rade o he produciviy differenial pd he relaive governmen expendiure raio gov and he real ineres rae differenial rid o rack he underlying values of he dollar-euro exchange rae. In conras o he firs hree fundamenals which are expeced o have a long-run impac on he real exchange rae he fourh variable he real ineres rae differenial is hough o affec he real exchange rae in he medium o shor-run. According o Clark and MacDonald (999) he acual exchange rae may be deermined by he following equaion q = o + pd + gov + rid + () 3 4 ε In he porfolio balance models discussed by Faruqee (995) Fell (996) and MacDonald (997) ne foreign asses (NFA) are a cenral deerminan of he real exchange rae. Therefore we ried o incorporae his variable in our model specificaion. We ried boh he euro NFA which is measured as he cumulaed curren accoun balance as a percenage of GDP and he overall relaive NFA beween he US and he euro area. However neiher variable could be used successfully. This problem as explained in Maeso-Fernandez e al. (00) is ha he euro NFA is aggregaed on he basis of naional daa so ha i does no correc for inra-euro area curren posiions. Moreover for he bilaeral exchange rae analysed in his sudy one should use he NFA posiion beween he wo counries involved. However his daa is no available and canno be accuraely proxied by he overall relaive NFA beween he euro area and he US. Therefore he following analysis uses he above four fundamenals. 3

4 where q denoes he real exchange rae he coefficiens for j = 04 indicae he effecs ha individual fundamenals have on he real exchange rae and ε may conain a se of shor-run variables and a random error. The behavioural equilibrium exchange rae (BEER) approach of Clark and MacDonald (004) employs Johansen s coinegraion approach (Johansen and Juselius 990) o deermine wheher here is a coinegraion or long-run relaionship amongs he real exchange rae and macroeconomic fundamenals. If here is one coinegraion relaionship he curren equilibrium exchange rae q can be calculaed using he esimaed long-run coefficiens as follows: j q = ˆ o + ˆ pd + ˆ gov + ˆ rid () 3 4 and he error correcion erm ε measures he curren misalignmen. One way of calculaing he PEER q is o subsiue he susainable (equilibrium) values of he fundamenals (i.e. relaionship in equaion (). o pd gov and rid ) ino he pre-esimaed long-run q = ˆ o + ˆ pd + ˆ gov + ˆ rid. (3) 3 4 Therefore he oal misalignmen is given by ( o o ) ( pd pd ) ( gov gov ) ( rid rid ) ε = ε + ˆ + ˆ + ˆ + ˆ (4) 3 4 = ε + ˆ o + ˆ pd + ˆ gov + ˆ rid C C C C 3 4 which is he sum of he curren misalignmen and he ransiory componens of he fundamenals (i.e. C o C pd C gov and C rid ). The calibraion of he fundamenals a heir susainable levels is usually achieved by using some sor of saisical filer such as he Hodrick-Presco filer (997 HP henceforh). However hese filers are known o produce spurious cycles for non-saionary daa (Cogley and Nason 995; Murray 003; Doorn 006). In addiion we quesion wheher he long-run coefficiens should remain he same See he discussion in Éger (003). 4

5 when he acual values of he macroeconomic fundamenals are replaced by heir susainable values. Tha is o say we do no believe he susainable level (i.e. he permanen componen) of each fundamenal should be given he same weigh ˆi as is ransiory componen as suggesed by he convenional coinegraion-based PEER approach in equaions (3) and (4). Alernaively Clark and MacDonald (004) use he Granger and Gonzalo (995) decomposiion o esimae he PEER which is derived from a vecor error correcion model (VECM). This approach idenifies he common rends shared by he variables in he model and heir impac on he real exchange rae bu offers no direc measure of how he permanen componen of each fundamenal drives he movemens of he real exchange rae. Insead he UC model proposed in his paper focuses on he impac he permanen and ransiory componens of each fundamenal has on he real exchange rae. This provides a clearer economic inerpreaion han analysing he common permanen and ransiory componens in he macroeconomic fundamenals and he real exchange rae. We also argue ha he main consrain of he coinegraion-based PEER approach is ha i requires he presence of a coinegraing relaionship amongs he real exchange rae and he fundamenals. If his prerequisie is lacking we canno proceed wih he esimaion of he PEER. However in he following UC seing we can relax his consrain. The mulivariae UC model used in his paper o esimae he PEER is specified as follows C o o o C pd pd pd C gov = gov gov. C rid rid rid q 3 4 q ε q (5) I can overcome a number of he drawbacks in convenional esimaes of he PEER as oulined above. Firsly in he UC model each macroeconomic fundamenal is decomposed ino is permanen and ransiory componens. We do no arbirarily se he signal-noise raio as in he HP filer. Insead he weighs used for signal exracion are esimaed from he daa by maximising he likelihood funcion. The permanen and ransiory componens of each fundamenal are specified in equaions (6) and (7) respecively Y Y β η Y Y = + + ( ) η ~ NID 0 Y ηy (6) 5

6 Y C Y κ ~ NID 0 Y Y Y κy. (7) C = ( L) + κ ( ) To save space Y represens he permanen componens of o pd C gov and gov and C rid o pd gov and rid ) and Y C denoes he ransiory componens (i.e. rid (i.e. C o C pd ). The permanen componen Y is ofen referred o as he susainable (equilibrium) value of he fundamenal and is generally specified as having he same order of inegraion as he fundamenal. For example if a fundamenal appears o be saionary he permanen componen should also no conain a uni roo herefore = ρ + β + η η ~ NID Y ( 0 ηy ) (8) Y Y Y Y where he damping facor ρ saisfies 0 < ρ <. However if Y is inegraed of order wo I () a sochasic drif β can be included in he random walk process Y = + β + η η ~ NID Y ( 0 ηy ) (9) Y Y Y Y = + β ( ξy ) β β ξ Y Y Y Y ~ NID 0 Therefore Y becomes an I () process. In he special case ha ηy = 0 Y becomes a smoohed I () process. On he oher hand he ransiory componen Y C measures he exen o which he acual fundamenal differs from is susainable level which is modelled as a saionary auoregressive process. In his paper we adop he saionary AR() specificaion used by Clark (987) o model he ransiory componens for each macroeconomic fundamenal. Since he macroeconomic fundamenals are relaed o he real exchange rae as shown in equaion () he las row of equaion (5) is a generalisaion of equaion () where we allow he permanen and ransiory componens of each fundamenal o have differen coefficiens. This generalisaion is imporan as he permanen and ransiory componens of some fundamenals may have opposie effecs on he real exchange rae. For example according o Frenkel and Mussa (988) high governmen expendiure oday may lead o ax disorion and moneisaion of governmen deb in he fuure. This in urn can produce a real depreciaion of 6

7 a currency in he long-run. However in he shor o medium-run he real exchange rae can be posiively affeced if a rise in governmen spending increases ne domesic demand especially for non-raded goods. Therefore i is imporan for researchers o be able o separae he permanen and ransiory effecs of his variable on he real exchange rae. However his canno be achieved using he coinegraion-based BEER/PEER approach. Therefore as saed by Osba e al. (003) who used his approach o analyse he yen-euro exchange rae he sign on he long-run coefficien of he relaive governmen expendiure raio can be ambiguous as he esimaed coefficien is an average of boh he permanen and ransiory impac. However using he UC model oulined above can overcome his problem and we expec i o show 4 (he coefficien on he permanen componen) o be negaive and 40 (he coefficien on he ransiory componen) o be posiive. In he las row of equaion (5) he erm o + pd + 3 gov + 4 rid is ha par of he PEER ha can be explained by he four fundamenals a heir susainable values. This corresponds o equaion (3). componen q However we also include an unobserved random walk q = q + η η ~ NID(0 ). q q q η This is inended o model any remaining non-saionariy ha is no capured by he fundamenals used. Therefore he PEER is measured by o + pd + gov + rid + q in he UC model. The random walk componen q 3 4 can also be hough of as being applied in he manner of Harvey e al. (986) and Saranis and Sewar (00) who used an unobserved componen o model he variables ha are omied from he coinegraion relaionship. 3 If he sandard deviaion of he innovaion o he random walk process q η is zero q will reduce o a consan and he real exchange rae is said o be coinegraed wih he fundamenals (Nyblom and Harvey 000). However a significan advanage of he UC model over he coinegraion-based PEER approach is ha we do no need he real exchange rae o be coinegraed wih he fundamenals as a prerequisie o obain non-spurious long-run relaionships and in urn o calculae he PEER. 3 Harvey e al.(986) add an unobserved componen o he employmen-oupu relaion o accoun for he underlying produciviy rend. Saranis and Sewar (00) use an unobserved componen o capure any omied variables from he consumpion-income relaionship such as wealh. 7

8 Insead omied variables from he coinegraion relaionship can be reaed as an unobserved componen ha can be esimaed from he observed daa using Kalman filer. This allows he esimaion of long-run relaionships beween inegraed variables (i.e. he real exchange rae and permanen componens of macroeconomic fundamenals) using maximum likelihood and he significance of hese long-run coefficiens can be esed using he likelihood raio es. Finally as wih equaion (4) he oal misalignmen is a linear combinaion of ransiory componens C C C C 0o + 0 pd + 30gov + 40rid plus an irregular erm ε q ~ NID(0 q ε ). The UC model used in his paper can be recas ino sae-space form for esimaion. 4 hyperparameers in he UC model can be esimaed by maximum likelihood using he predicion error decomposiion produced by he Kalman filer. Since non-saionary variables such as o pd The gov and q appear in he sae vecor he Kalman filer requires a diffuse iniialisaion and we use he iniialisaion mehod developed by Koopman and Durbin (003). Esimaing he mulivariae UC model we can obain he unobserved permanen and ransiory componens and he coefficiens on he real exchange rae equaion simulaneously. 5 3 Daa The quarerly daa used in his paper covers he period 975Q o 008Q4. This sample encompasses he period of floaing exchange raes beween he euro area counries and he US afer a brief adjusmen phase following he collapse of he Breon Woods sysem. The real dollar-euro exchange rae (LQ) for welve euro area members prior o 999 (based on consumer prices) is compued as a weighed geomeric average of he bilaeral exchange raes of he eleven currencies (Belgium and Luxembourg already had a common currency) agains he dollar. The weighs are given by he share of exernal rade of each euro area counry in oal euro area rade (aking ino accoun hird marke effecs) for he period The consumer price indices and he bilaeral nominal exchange raes are re-based o 005=00. The real exchange rae used in he analysis is in is naural log-form. The consumer price 4 The sae-space represenaion of he model is available upon reques. 5 All he compuaions were preformed using he library of sae-space funcions in SsfPack 3.0 developed by Koopman e al. (008) and Ox 5 by Doornik (006). 6 Weighs used for each currency are 34.49% for Deusche mark 7.75% for French franc 3.99% for Ialian lira 9.6% for Duch guilder 7.98% for Belgian and Luxembourg franc 4.90% for Spanish pesea 3.76% for Irish pound 3.7% for Finnish markka.89% for Ausrian schilling.07% for Poruguese escudo 0.74% for Greek drachma. 8

9 indices and bilaeral nominal exchange raes were aken from IMF Inernaional Financial Saisics (IFS) lines 64 and rf respecively. A counry s erms of rade (LTOT) is compued as he raio of is expor prices o impor prices (for some counries expor uni value and impor uni value are used). The same weighs used o consruc he synheic euro-dollar exchange rae prior o 999 are used o compue he raio for he euro area. Finally he erms of rade differenial is compued as he raio of he euro erms of rade relaive o he US. All variables are rebased so ha 005=00 and he log of he erms of rade differenial is used. Expor and impor prices for Ausria Finland Germany and he US were obained from IFS lines 76 and 76.x respecively. Since he daa are no available for he remaining counries expor and impor uni values aken from IFS lines 74 and 75 are used. 7 The produciviy differenial (LPD) is measured as he raio of real GDP o he number of employed persons in he euro area compared wih he same raio for he US. This is a direc measure of he Balassa-Samuelson effec. This variable is also rebased o 005=00 and logged. Real GDP series were obained from IFS line 99bv; Annual employmen was from he OECD Labour produciviy growh daase. Quarerly daa have been convered from he annual daa. Relaive governmen expendiure (LGOV) is compued as he raio of governmen expendiure o GDP in he euro area relaive o he same raio for he US. Consisen wih he above variables his series is also rebased and logged. For he euro area he governmen expendiure o GDP raio is obained from he Area Wide Model consruced by Fagan e al. (00). The corresponding variable for he US is consruced using GDP and governmen expendiure a curren prices aken from IFS lines 99b and 9f. Finally he real ineres rae differenial (RID) is he difference beween real ineres raes for he euro area and he US. Daa on bond yields for he US and a geomeric weighed average of long-run ineres raes of counries consiuing he euro area are used. The expeced rae of inflaion is proxied by he annual rae of consumer price inflaion in he previous year. The nominal long-erm ineres raes for he euro area counries and he US were aken from IFS lines 6. 7 For France he daa series of expor and impor uni values are available from 990Q onwards herefore he erms of rade daa aken from he OECD daabase is used before 990Q. 9

10 3. Uni roo ess The saionariy of he daa used in his paper are examined using he convenional Augmened Dickey-Fuller (ADF) es which ess he null hypohesis ha an AR ( p ) process conains a uni roo agains he alernaive ha i is saionary. The es proposed by Kwiakowski e al. (99 hereafer KPSS) based on srucural ime series models is also used o cross-check he resuls of he ADF ess. In conras o he ADF es he KPSS ess he null hypohesis ha he innovaion of a random walk process has zero variance agains he alernaive ha he innovaion variance is posiive. Boh es saisics presened in he upper panel of Table sugges ha he level of he real ineres rae differenial is a saionary variable while he levels of he erms of rade and produciviy differenial are non-saionary around eiher a level or a rend. However he resuls are inconclusive for he levels of he real dollar-euro exchange rae and he relaive governmen expendiure raio. The same uni roo ess are also conduced for he firs differenced variables. Alhough he ADF ess srongly indicae ha all firs differenced variables are saionary he KPSS ess rejec saionariy in he firs differenced series of he produciviy differenial and he relaive governmen expendiure raio a he 5% and 0% levels respecively. Given he resuls obained from he uni roo ess apar from he real ineres rae differenial we rea all he oher variables as non-saionary. The evidence for a saionary ineres rae differenial has been widely suggesed in he lieraure including Hoffmann and MacDonald (009). 3. Model modificaion Given he properies of he daa suggesed by he uni roo ess we ensure ha he permanen componen o has he same order of inegraion as he logged acual daa. Since boh he ADF and KPSS sugges ha he erms of rade is an I () variable we model is permanen componen as a random walk wih a consan drif as specified in equaion (6). However as he ineres rae differenial appears o be saionary as indicaed by boh he ADF and KPSS saisics he permanen componen of his variable should also no conain a uni roo and herefore equaion (8) is used. In addiion evidence of I () processes for he produciviy differenial and relaive governmen expendiure is suppored by he KPSS es saisics bu rejeced by he ADF ess. This is explained by Nyblom and Harvey (00) who argue ha he ADF es oo ofen 0

11 rejecs an I () null because he process followed by he second differences of he observaions is close o being non-inverible. Furhermore hey suppor he use of an I () permanen componen for series such as real GDP as i can give a good fi when modelling wihin an unobserved componens framework. In his paper we used he I () specificaion in equaion (9) o model he permanen componens of he produciviy differenial and he relaive governmen expendiure raio. This specificaion is suppored by he KPSS saisics and using a sochasic drif allows us o model he gradual slowdown of euro area produciviy and governmen expendiure relaive o he US as observed in Figure where he esimaed permanen componens are ploed agains he observed daa. {Table abou here} 4 Esimaion resuls The parameer esimaes of he five-variae UC model (hereafer Model ) oulined in Secion are repored in Table. Inspecion of he auxiliary residuals allows us o deec wo ouliers occurring during 980Q and 008Q3 for he ineres rae differenial and he erms of rade respecively. Two dummies are used for hese ouliers and boh dummy variable coefficiens are saisically significan. In addiion boh he Ljung-Box saisics for auocorrelaion in one-sep-ahead predicion errors and he Jarque-Bera saisics for normaliy are insignifican. The upper panel of Table presens he coefficiens for he real exchange rae equaion. The posiive and significan parameer esimae of suggess ha an increase in he erms of rade differenial of he euro area relaive o he US will resul in he euro appreciaing agains he dollar. This can be by a subsiuion effec generaed by higher prices of expored goods relaive o impored goods. Since higher expor prices will iniially lead o higher wages in he radable secor relaive o he non-radable secor his will raise he overall price level in he domesic economy in he long-run which makes he domesic currency appreciae. A negaive parameer esimae is found for 0. This may reflec he income effec generaed by growing expor revenues ha may induce higher demand for non-raded goods. To resore inernal equilibrium in his siuaion he real exchange rae needs o depreciae. However he negaive income effec appears o be less predominan han he posiive subsiuion effec as 0 is boh smaller han and saisically insignifican.

12 According o he Balassa-Samuelson hypohesis (Balassa 964 and Samuelson 964) higher produciviy in he domesic relaive o he foreign economy is usually expeced o resul in an appreciaion of he domesic currency. Therefore we expec o be posiive. Whils has he expeced sign (posiive) he coefficien is saisically insignifican. This seems o conradic Alquis and Chinn (00) and Schnaz e al. (004) who boh find ha he direc measure of he Balassa-Samuelson effec has a significan and posiive long-run relaionship wih he dollar-euro exchange rae beween 985 and 00. However his difference may be due o he sample used in his paper being exended o 008. I can be observed from Figure ha he real dollar-euro exchange rae and relaive produciviy move in differen direcions over he exended sample period The euro appreciaed agains he dollar from 00 onwards whils he decline in he relaive produciviy of he euro area wih respec o he US coninued. The coefficien of he permanen componen of he relaive governmen expendiure raio 3 is negaive and significan. This is consisen wih he argumen of Frenkel and Mussa (988) ha a higher governmen expendiure raio will lead o a real depreciaion in he longrun. However he shor-run posiive effec of his variable on he real exchange rae remains unclear as 30 appears o be posiive bu insignifican. As oulined in Secion one of he advanages of using he UC model is ha i can separae he negaive long-run effec of he relaive governmen expendiure raio from is poenially posiive shor-run impac on he real exchange rae. As demonsraed in Secion 5 his canno be achieved using he VECMbased BEER/PEER approach as i only esimaes he average of hese wo effecs. As he ineres rae differenial appears o be saionary he permanen componen of his series should no conain a uni roo and i is hus specified o follow a saionary AR() process as in equaion (8). However we find ha he esimae of he AR() componen appears o be very small and less persisen han he AR() ransiory componen of his variable. In addiion he damping facor ρ rid and he consan β rid are insignifican. Therefore in he resuls presened in Table we resric 4 o be zero a priori. 8 On he oher hand a posiive esimae of 40 is found which is in line wih he heoreical predicion ha higher demand for he currency wih a relaively higher ineres rae will creae an appreciaion pressure on ha currency. However 40 appears o be insignifican suggesing 8 The log-likelihood value for he unresriced model is The null hypohesis ha 4 = 0 canno be rejeced by he log-likelihood raio es a he 0% level.

13 here is no srong evidence ha an increase in he ineres rae differenial in he euro area relaive o he US leads o a real appreciaion of he euro agains he dollar for he sample period sudied. {Table abou here} Finally here are signs of a coinegraing relaionship beween he macroeconomic fundamenals and he real exchange rae as he sandard deviaion of he innovaion o he unobserved random walk process q η appears o be small. Alhough he coinegraion beween he real exchange rae and he fundamenals is no required by he UC model o obain he PEER he UC model can easily accommodae he coinegraion analysis. We employ he es proposed in Nyblom and Harvey (000) o deermine wheher q η = 0. This es can be regarded as an assessmen of he validiy of he pre-specified coinegraing vecor which is discussed in deail in Secion 5. Secion 5 also performs more general ess of coinegraion proposed in Nyblom and Harvey (000) and Johansen and Juselius (990) o examine he possible coinegraion relaionship among hese variables. The permanen componen of each series is ploed agains is acual value in Figure. The esimaed PEER ploed agains he acual exchange rae is a linear combinaion of four non-saionary componens o + pd + 3 gov + q. The subsequen misalignmen is hen given by 0o C + C C C. 0 pd + 30gov + 40 rid + ε The resuls show ha he euro is q undervalued during he mid-980s as a resul of he dollar s srengh prior o he plaza agreemen. However srong signs of overvaluaion of he euro became apparen in which corresponds o a period of weakness of he dollar agains major European currencies. Laer on he euro was broadly a is equilibrium level in 999 bu considerably undervalued in he years immediaely afer is launch as a resul of financial marke uncerainy. Neverheless he euro moves closely around is equilibrium value from 003 onwards. Figure also shows ha apar from he real ineres rae differenial which is primarily driven by is AR() ransiory componen he flucuaions in he oher variables are mainly aribuable o heir permanen componens. {Figure abou here} 3

14 4. AR() specificaion for he ransiory componen of he real exchange rae Since none of he coefficiens of he ransiory componens are significan in Model a he 5% level suggesed by he likelihood raio ess repored in Table we model he ransiory componen of he real exchange rae independenly o follow a saionary AR() process. As such he las row of equaion (5) becomes q o pd gov rid q q (0) C = ε q q = q + q + κ and C C C where q q q κ ~ NID(0 ). All oher rows in equaion (5) q q κ remain he same. The parameer esimaes of he modified model (hereafer Model ) are repored in Table 3. Some changes are observed when comparing he parameer esimaes of Model wih hose of Model. Firsly he sandard deviaions of innovaions o he AR() ransiory componens are increased in Model. This is accompanied by a decline in he volailiy of he permanen componens of he macroeconomic fundamenals. The larges change is observed in he decomposiion of he produciviy differenial. Is AR() componen becomes large and persisen while he permanen componen becomes a smoohed I () process. Secondly a moderae increase in he size of and 3 is idenified in Model compared o Model. However he significance of hese coefficiens does no aler. {Table 3 abou here} Figure plos he unobserved componens esimaed from Model. The esimae of he PEER resembles he one obained from Model. Consequenly he AR() specificaion of he oal misalignmen is also consisen wih ha obained from Model. {Figure abou here} Alhough Models and are no nesed hey are defined over he same five variables. Therefore he Bayes facor mehodology discussed by Kass and Rafery (996) can be used 4

15 o indicae which model is favored by he daa. 9 This approach compares he log-likelihood values of he wo models aking ino accoun he number of parameers used in each. Kass and Rafer sugges using he Schwarz crierion for comparing models ( ) S = l( D M ) l ( D M ) 0.5 d d log n J I J I where l( ) is he maximised log likelihood d is he number of parameers and n is he sample size. S can hen be used o judge he srengh of evidence agains o M I wih respec M J. If S lies beween zero and wo Kass and Rafery sugges ha he evidence agains M I is no worh more han a bare menion. If S is beween wo and six he evidence agains In our case M I is srong and if i is greaer han en he evidence agains M J is Model and M I is very srong. M I is Model. Model has a slighly higher likelihood value bu i also has one more parameer han Model. The value of S is calculaed o be 0.33 which suggess ha he Bayes facor does no go agains Model wih respec o Model. 4. Two models excluding he real ineres rae differenial I is imporan o noe ha he ineres rae differenial did no play any role in he exchange rae equaion in Model. Therefore we removed his variable in Model 3 and insead conduced he esimaion using four non-saionary variables (he real exchange rae he erms of rade he produciviy differenial and he relaive governmen expendiure raio). The parameer esimaes of Model 3 are repored in Table 4. I can be seen ha excluding he real ineres rae differenial does no significanly aler he parameer esimaes of he oher variables in he model. {Table 4 abou here} Furhermore o obain a more parsimonious model han Model 3 we subsequenly resriced o be zero as i is insignifican in all of he previous esimaions conduced. Seing o zero indicaes ha he permanen componen of relaive produciviy has no impac on 9 The Bayes facor approach has also been used by Basisha and Sarz (008) and Basisha (009) o compare differen models used o esimae he oupu gap core inflaion and he non-acceleraing inflaion rae of unemploymen. 5

16 deermining he PEER. As a resul Model 4 canno be rejeced a he 0% level wih respec o Model 3. In addiion and 3 remain significanly posiive and negaive respecively as shown in Table 5. 0 {Table 5 abou here} 5 Tesing for coinegraion In he previous secion we found he sandard deviaion of he innovaion o he random walk process o be small. This may indicae ha he four non-saionary variables in he UC q η model are coinegraed. In his secion we perform hree coinegraion ess. The firs wo are based on he srucural ime series models proposed by Nyblom and Harvey (000). The hird es is Johansen s coinegraion es (Johansen and Juselius 990) which is a vecor auoregression based es used in he BEER/PEER approach o deermine he long-run relaionship beween he real exchange rae and he fundamenals. I is worh noing ha he difference beween Nyblom and Harvey (000) and Johansen s coinegraion es is analogous o he difference beween he KPSS and he ADF es. 5. Tesing for pre-specified coinegraing vecors In his subsecion we perform he firs coinegraion es proposed by Nyblom and Harvey (000) o deermine wheher q η = 0. This es can be regarded as esing he validiy of a se of pre-specified coinegraing vecors. I is based on a mulivariae local level model ha can be wrien as y IK 0 µ ε y = = + () y Θ I r µ ε where y is pariioned ino a K vecor y and an r vecor y wih r = N K. ε is also similarly pariioned. The K vecor random walk processes wih heir disurbance vecors µ and r vecor µ follow mulivariae η and η having posiive definie 0 Complee parameer esimaes of Model 4 are available upon reques. 6

17 covariance marices of coefficiens. Σ η and Σ η respecively. Finally Θ is an r K marix of If y is coinegraed wih y in equaion () here will be r linear combinaions of he observaions Ay ha are saionary. The rows of A consiue a se of r coinegraing vecors and can be pariioned as = ( ) A A A wih A being r r. Then Ay = A y + A y = (A + AΘ)µ + Aµ + Aε + Aε () where A + AΘ = 0 and Ση = 0. The r N marix A can be formed according o economic heory or hrough Θ. If we choose A such ha A + AΘ = 0 he es applied o Ay is he locally bes invarian (LBI) es of he null hypohesis ha Σ = 0 agains he alernaive ha ΘΣ Θ + ΘΣ + Σ Θ + Σ = 0 where he ii given by η Σ η is proporional o Σ s are he blocks of Σ ε. The es saisic is where - ( r; ) r ( ) η A = ASA ACA (3) and T j j C = T ( ) ( ) j= y - y = y y = T T = S = y y y y. ( )( ) The limiing disribuion of η ( r; A ) is he Cramér-von Mises CvM ( r ). In his subsecion we wish o es he null hypohesis ha q η = 0 agains he alernaive ha q η > 0. The coinegraing vecor A is formed such ha A + AΘ = 0 where 7

18 [ ] Θ = is esimaed from Models and 3. y = [ o pd gov q ] is pariioned ino 3 y = [ o pd gov ] and y = q. If he null ha = 0 canno be rejeced y is said o be coinegraed wih y q η. As he coefficien of he permanen componen of he produciviy differenial appears insignifican in all esimaions in Secion 4 we also es wheher coinegraion can be found beween y = [ o gov ] and y = q. In his case A is formed using = [ ] Θ 3 esimaed from Model 4 where is resriced o zero. Since he saionary componens in our UC models are no whie noise processes (i.e. he ransiory componens follow saionary AR() processes) he es saisic in equaion (3) is modified o allow for serial correlaion. The modificaion adaped here is in line wih he KPSS es which replaces S wih a consisen esimaor of he long-run variance τ = m ( m) = ω Γˆ ( τ ) S τ = m τ m where Γ ˆ ( τ ) is he sample auocovariance marix a lag τ ha is T ˆ Γ ( τ ) = T ( )( τ ) y y y y = τ + and ω is a weighing marix funcion such as ω τ ( ) τ m = + τ =.. m. Therefore as τ m m wih he KPSS es he selecion of lag lengh m may affec he conclusion reached. Alhough an increase in he lag lengh leads o a es closer o he desirable size i sacrifices he power of he es. Therefore we use m 4 for he es saisics presened in Table 6. The ess canno rejec he null hypohesis ha q η = 0 in Models and 3 when m implying y = [ o pd gov ] is coinegraed wih y = q. In addiion here is weaker evidence ha q η = 0 in Model 4 when he es saisics are adjused for serial correlaion wih m > 3. This suggess ha coinegraion may be found amongs he real exchange rae he erms of rade and he relaive governmen expendiure raio. We found ha excluding he real ineres rae differenial from Model o obain Model 3 does no aler he parameer esimaes in he res of he model. Therefore we only presen he coinegraion es using parameers esimaes from Model 3. 8

19 {Table 6 abou here} 5. Tesing for a specified number of common rends In his subsecion we performed a more general es oulined in Nyblom and Harvey (000) o cross-check he above coinegraion resuls. To do so we firs es wheher y = [ o pd gov ] is a se of variables ha are indeed no coinegraed hemselves. Second we examine wheher more coinegraion relaionships can be found in y = [ o pd gov q ]. In hese ess we do no pre-specify he marix A. Insead we es he null hypohesis ha rank ( Ση ) = K agains he alernaive ha rank ( η ) given by Σ > K for K < N. The es saisic is ξk N = λk λn (4) which is he sum of he ( N K ) smalles eigenvalues of S( m) - C. C and ( m) S are defined as in equaion (3) excep ha in his case hey are formed from he OLS residuals from regressing y on he vecors of consans and ime. The es saisics and criical values abulaed from Nyblom and Harvey (000) are presened in Table 7. The null hypohesis ha ( η ) ( η ) rank Σ = agains he alernaive ha rank Σ = 3 amongs y = [ o pd gov ] is srongly rejeced a he % level. This confirms he validiy of he firs coinegraion es. However evidence of a single coinegraing vecor amongs y = [ o pd gov q ] appears o be much weaker han he previous es suggesed as he null hypohesis ha ( η ) rank Σ = 3 canno be rejeced a he % level afer he es saisics have been adjused for serial correlaion wih m 4. {Table 7 abou here} 5.3 Johansen coinegraion es Given he second Nyblom and Harvey es offers only weak suppor for he presence of one coinegraing vecor among y = [ o pd gov q ] Johansen s coinegraion approach is 9

20 performed o cross-check wheher he variables are indeed coinegraed. If coinegraion is found using Johansen s es we can hen compare he coefficiens on he coinegraing vecor wih and 3 esimaed from he UC model. The saring poin of his approach is a VAR model of dimension four and order p which can be wrien in vecor-error correcion (VEC) form as follows p y = m + Π()y + Γ y + ε (5) i i i= where y = [ o pd gov q ] and Π() is he 4 4 long-run marix ha can be facored as ' αβ if Π() has reduced rank r < 4. Γ i represens he 4 4 marix of shor-run coefficiens m is a 4 vecor of consans and ε denoes a 4 vecor of whie noise residuals. If here are r linear combinaions of he variables in y ha yield saionary series y is said o have k = 4 r common rends. We se p equal o wo and he consan is resriced o he coinegraing space. The race es saisics repored in Table 8 indicae he presence of a single coinegraing vecor a he % level. This offers much sronger suppor for coinegraion han he second Nyblom and Harvey (000) es. Therefore we se he coinegraion rank equal o one and a sandard se of long-run exclusion and weak exogeneiy ess are conduced. The exclusion es indicaes ha he relaive produciviy variable can be excluded from he long-run relaionship given ha χ () = 0.3. This is broadly in line wih he resuls from he UC model which found ha his variable does no have a permanen impac on he real exchange rae. In addiion he relaive produciviy and erms of rade variables are also found o be weakly exogenous wih he join es saisic χ (3) =.. Consisen wih he resul from he UC models a significan and posiive long-run relaionship is found beween he erms of rade and he real exchange rae. However he posiive relaionship idenified beween he relaive governmen expendiure raio and he real exchange rae seems o conradic he conclusion drawn from UC models ha a higher governmen expendiure raio will evenually lead o a depreciaion of he real exchange rae. Neverheless his conradicion may be aribued o he UC model separaing he negaive and permanen effec of he relaive governmen expendiure raio We rea all non-saionary variables as I() processes in he VAR model as suggesed by he ADF ess. 0

21 from is poenially posiive ransiory impac on he real exchange rae while he long-run coefficien obained from he VECM jus reflecs he average of hese wo effecs. In addiion he adjusmen erm for he real exchange rae is negaive and significan. This implies ha he real dollar-euro exchange rae is one of he variables in he sysem ha adjuss o exogenous shocks. However he speed of adjusmen is found o be slow he halflife of deviaions from equilibrium being abou hree years. 3 {Table 8 abou here} Finally we calculae he PEER and he oal misalignmen using he Granger and Gonzalo (995) decomposiion. The resuls are ploed agains he corresponding componens obained from Model 3 in Figure 3. Alhough hese models use he same se of fundamenals o rack he long-run movemen of he dollar-euro exchange rae hey produce significanly differen resuls. The PEER esimaed from Model 3 indicaes ha he euro is closer o is fundamenal value han he coinegraion-based approach implied. Despie he differences observed boh models sugges ha he euro was undervalued agains he dollar during he mid-980s and early 000s. {Figure 3 abou here} 6 Ou-of-sample forecasing Ever since Meese and Rogoff (983) condiional ou-of-sample forecasing has become a sandard procedure for esing he validiy of exchange rae models. In his secion we compare he ou-of-sample forecasing abiliy of mulivariae UC models 3 and 4 wih he random walk process of he real exchange rae. 4 To conduc a ransparen comparison beween UC models wih he random walk process we firs compare he ou-of-sample forecasing abiliy of a univariae UC model of he real exchange rae specified as follows 3 The half-life is compued as log(0.5)/log(-α ) where α is he adjusmen erm in he equaion for he real exchange rae. 4 Model is no considered here as i has he same exchange rae equaion as Model 3 specified in equaion ().

22 q = q + q T C ( ) ( ) q = m + q + η η ~NID 0 η T T q q q q q = q + q + k k ~NID 0 C C C q q q q k q (6) T where he real exchange rae is decomposed ino a permanen ( q ) C componen ( q ) and a ransiory. If he univariae UC model is no beer han he random walk process in erms of forecasing while he mulivariae UC models are beer his suggess ha inclusion of he fundamenals helps o predic fuure real exchange raes. A rolling sample approach is used wih he full-sample period firs divided ino a pre-forecasing period from 975Q o 995Q4 and a forecasing period from 996Q o 008Q4. Alhough he choice of 996Q is ad hoc i provides a sufficienly large sample for iniial esimaion and for evaluaing ou-ofsample forecasing performances of he mulivariae UC models. The pre-forecasing sample moves forward quarer by quarer and he model s hyperparameers are re-esimaed a each sep unil he end of he sample is reached. In oal 53 one-quarer-ahead forecass and 4 welve-quarer-ahead forecass are calculaed. Table 9 repors he raios of roo-mean-squared errors (RMSE) of boh he univariae and mulivariae UC models relaive o he random walk process. One sriking resul revealed from Table 9 is ha he relaive RMSEs of he univariae UC model wih respec o he random walk process are very close o one and remain relaively consan across differen forecasing horizons. However for mulivariae UC models he longer he forecasing horizons he smaller he RMSE produced by hese models relaive o he random walk process. Diebold and Mariano s (995 DM henceforh) es of equal forecas accuracy is preformed o deermine wheher differences in forecasing errors beween a UC model and he random walk process are significan. The DM saisic is specified as DM = d d where d is he sample mean of a differenial loss funcion such ha n n j A j B j j= j= where A j d = n d = n ( e e ) e and e B j are he j h h sep-ahead forecas

23 errors obained from models A and B. heeroskedasic-auocorrelaion consisen (HAC) esimaor. d is he variance of d esimaed using he The values of he DM saisic wih he small sample modificaion proposed by Harvey Leybourne and Newbold (997 HLN henceforh) are also calculaed as: { ( ) } / * DM = DM n + H + H H n n and repored in Table 9 where n and H denoe he number of forecass and he forecas horizon respecively. Boh he DM and HLN saisics are calculaed under he null hypohesis ha he UC model is equivalen in forecasing accuracy o he random walk process. The alernaive hypohesis varies depending on he sign of he es saisic. If he sign on he es saisic is posiive he alernaive hypohesis is ha he mulivariae model is beer han he random walk process in erms of forecasing accuracy. If he es saisic has a negaive sign he alernaive hypohesis is invered. The calculaed saisics are compared o he criical values of he Suden s -disribuion wih n degree of freedom. The es saisics highlighed in bold indicae ha he null hypohesis of equivalen forecasing accuracy is rejeced. I can be seen ha none of he null hypoheses ha he univariae UC model is equivalen in forecasing accuracy o he random walk process can be rejeced. However Model (he mulivariae UC model) is significanly beer in erms of forecasing fuure exchange raes han he random walk process from eigh-quarer-ahead forecasing horizons onwards. This indicaes ha he mulivariae models which include he macroeconomic fundamenals will help o predic he long-run movemen of real exchange raes. Furhermore Model 3 which allows he ransiory componen of he real exchange rae o be modelled independenly o follow an AR() process improves he shor-run forecasing accuracy wih respec o Model. I is also ineresing o noe ha when he produciviy differenial is resriced o have zero impac on he real exchange rae in Model 4 he quadraic loss differenial sequence d j which is calculaed using he forecasing errors produced by his model and he random walk process presens a sronger auocorrelaion paern and in urn a larger value of d. This parly leads o less significan es saisics han hose of Model 3. 3

24 {Table 9 abou here} 7 Conclusions This paper proposes an aleraive approach o esimaing he PEER based on a UC model specificaion. We believe our approach offers a number of advanages over he convenional coinegraion-based PEER proposed by Clark and MacDonald (004). Firs we do no rely on he prerequisie ha coinegraion has o be found among he real exchange rae and macroeconomic fundamenals o obain non-spurious long-run relaionships and esimaes of he PEER. Insead in he UC model specificaions an unobserved random walk process is used o capure any missed variables from he coinegraion relaionship. This allows he esimaion of he long-run relaionships beween he inegraed variables using maximum likelihood and he use of he likelihood raio es o idenify he significance of hese longrun coefficiens even if coinegraion is rejeced. Second he impac ha he permanen and ransiory componens of he macroeconomic fundamenals have on he real exchange rae can be modelled separaely in he UC model. This is imporan for variables where he long and shor-run effecs may drive he real exchange rae in opposie direcions such as he relaive governmen expendiure raio. However he long-run coefficien on he coinegraing vecor esimaed using he VECM jus reflecs he average of hese wo effecs. This is illusraed in subsecion 5.3. In addiion alhough he UC model oulined above does no require coinegraion amongs he real exchange rae and he fundamenals as a prerequisie for obaining he PEER he UC model can also accommodae he coinegraion analysis as oulined in he firs wo subsecions of Secion 5. Following Nyblom and Harvey (000) using he pre-specified coinegraing vecor formed from he esimaed UC models we found ha he real exchange rae is coinegraed wih he macroeconomic fundamenals. Finally a forecasing exercise was conduced o es he validiy of our mulivariae UC models agains a random walk process of he real exchange rae. In general he longer he forecasing horizons he smaller he RMSE produced by he mulivariae UC models relaive o he random walk process. As suggesed by he DM and HLN saisics Model is significanly beer a forecasing fuure exchange raes han he random walk process from eigh-quarer-ahead periods onwards. This indicaes ha he mulivariae models which include he macroeconomic fundamenals will help o predic he long-run movemen of real exchange raes. However Model 3 which allows for he ransiory componen of he real 4

25 exchange rae o be modelled independenly o follow an AR() process considerably improves he shor-run forecasing accuracy wih respec o Model. In shor we demonsrae ha he mehod proposed in his paper can be a useful echnique for cenral banks o esimae he equilibrium exchange rae and o predic long-run exchange rae movemens. 5

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