Uncovering Yield Parity: A New Insight into the UIP Puzzle through the Stationarity of Long Maturity Forward Rates *

Transcription

1 Uncovering Yield Pariy: A New Insigh ino he UIP Puzzle hrough he Saionariy of Long Mauriy Forward Raes * Firs version: February 5, 4 This version: February 3, 6 Absrac Resuls and models of his paper are based on a srikingly new empirical observaion: long mauriy forward raes beween bilaeral currency pairs of he US, Germany, UK, and Swizerland are saionary. Based on his resul, we sugges a new explanaion for he UIP-puzzle mainaining raional expecaions and risk neuraliy. The model builds on he ineracion of foreign exchange and fixed income markes. Ex ane shor run and long run UIP and he EHTS is assumed. We show ha ex pos shocks o he erm srucure could explain he behavior of he nominal exchange rae including is volailiy and he failure of ex pos shor UIP regressions. We presen evidence on ex pos validiy of long run UIP and srikingly new evidence on he saionariy of he long forward exchange raes of major currencies. We se up, calibrae and simulae a sylized model ha well capures he observed properies of spo exchange raes and UIP regressions of major currencies. We define he noion of yield pariy and es is empirical performance for monhly series of major currencies wih favorable resuls. JEL Classificaion: E43, F3 Keywords: EHTS, forward discoun bias, saionariy of long mauriy forward raes, UIP, yield pariy * We are hankful for commens and suggesions o Andrew K. Rose, Péer Benczúr, Rober-Paul Berben, Philip R. Lane, Pierre L. Siklós, Timo Teräsvira, János Vincze, conference paricipans a he Applied Economerics Associaion conference on Exchange Rae Economerics, he h Annual Congress of he European Economic Associaion, he Inernaional Conference on Finance of he Universiy of Copenhagen, and seminar paricipans a he Cenral Bank of Hungary, De Nederlandsche Bank and he Sockholm School of Economics.

2 TABLE OF CONTENTS. Inroducion Uncovered Ineres Rae Pariy in he Long Run Empirical Evidence on he Saionariy of he Expeced Long Run Exchange Rae The Yield Pariy Approach The Basic Seup Dynamics of he Exchange Rae in he Shor Run Adjusmen in he Spo versus in he Expeced Long Run Exchange Rae A new explanaion for he failure of shor-run UIP: Exchange rae shocks are no orhogonal o he previous period shor ineres rae differenials Empirical Evidence on Yield Pariy for Major Exchange Raes Simulaion Evidence on he Yield Pariy Approach Summary References Daa Appendix Tables Figures... 46

3 LIST OF TABLES Table : Shor and long run UIP equaions Table : Uni roo and saionariy ess of ineres rae differenials Table 3: Uni roo and saionariy ess on spo and forward raes Table 4: Single equaion ECM for he change in spo exchange rae regressed on he previous period level of he long forward, and Dickey-Fuller equaions Table 5: Johansen unresriced coinegraion rank ess for spo exchange rae and shor and long ineres rae differenials Table 6: Variance of he logarihm of spo and forward raes Table 7: Yield pariy equaions: DEM/USD Table 8: Yield pariy equaions: GBP/USD... 4 Table 9: Yield pariy equaions: DEM/GBP... 4 Table : Simulaed momens and es saisics of he sylized model calibraed o monhly DEM/USD for 973M-3M... 4 Table : Simulaed momens and es saisics of he sylized model calibraed o monhly DEM/USD for 979M-3M Table : Simulaed momens and es saisics of he sylized model calibraed o monhly GBP/USD for 979M-3M Table 3: Simulaed momens and es saisics of he sylized model calibraed o monhly DEM/GBP for 979M-3M LIST OF FIGURES Figure : -year UIP regression β-esimaes wih confidence bands for non-overlapping samples, using differen monhs of he year Figure : Long run UIP: Yield differenials and acual fuure exchange rae changes Figure 3: Sabiliy of he long run expeced exchange rae: Spo and -year mauriy forward raes. 48 Figure 4: Variance of spo and long forward rae in 5-year long rolling samples Figure 5: The spo DEM/USD rae, and is hypoheical values if had absorbed all or none of EHTS shocks; he shor-run ineres rae differenial; and he generic foreign exchange shock, Figure 6: Monhly exchange rae changes and yield pariies Figure 7: Shor-run UIP β-coefficiens for rolling samples of, 5, and years long... 5 Figure 8: Yield pariy β-coefficiens for -year rolling samples Figure 9: Yield pariy β-coefficiens for 5-year rolling samples Figure : Yield pariy β-coefficiens for -year rolling samples

4 . Inroducion The connecion beween ineres raes and exchange raes is one of he mos inensively researched subjecs in inernaional macroeconomics. The uncovered ineres rae pariy (UIP) hypohesis, which claims he equalizaion of expeced yields inernaionally, is he criical building block of mos heoreical models, and a dismal empirical failure. (Flood-Rose,, p.5). As i is well know from hundreds of sudies and dozens of surveys, in shor horizons he exchange rae ends o move he opposie direcion han wha is prediced by UIP in flexible exchange rae sysems. Tradiional explanaions of his failure could be grouped ino wo main caegories: () Models emphasizing he way expecaions are formed. For example, he so-called peso-problem, learning, or bubbles could offer explanaions which are consisen wih raional expecaions, or even he rejecion of raional expecaions could be responsible. () Models rejecing risk-neuraliy and inroducing ime varying risk premium. However, in our reading of he lieraure, he consensus is ha none of hese aemps offers a saisfacory answer. For example, as Rogoff (, p.) makes i clear "Now, if here is a consensus resul in he empirical lieraure, i has o be ha nohing, bu nohing, can sysemaically explain exchange raes beween major currencies wih flexible exchange raes." This conclusion could be devasaing for furher research. However, here are some new approaches shedding new ligh on he apparen anomalies. Firs, a few papers already appeared arguing ha a longer mauriies UIP performs much beer ex pos as well. Since long run UIP will be a major building block of our approach, we will briefly summarize hese papers and presen new evidence in he nex secion. Second, here is a growing lieraure claiming ha in small macroeconomic models assuming ex ane UIP, he reacion of moneary policy could lead o ex pos failure of he hypohesis. Third, here are some papers sudying EHTS and UIP joinly, which papers usually do no direcly address he failure of UIP direcly, bu have achieved ineresing resuls in exchange rae forecasing. We briefly review hese papers as well since we also sudy EHTS and UIP joinly. In his pioneering work, McCallum (994) suggesed ha he reacion of moneary auhoriy aiming o smooh flucuaions of urbulence in financial markes could lead o negaive correlaion beween he ineres rae differenial and ex pos exchange rae changes. A major elemen of his models is ha he exchange rae appears in he moneary policy reacion funcion. Kugler () inroduces he spread beween he shor and long ends of he erm srucure ino he reacion funcion (keeping For an overview see, for insance, Taylor (995), Lewis (), and Engel (996). In a recen paper Benczúr (3) shows ha a model, mainaining UIP wih raional expecaion bu noise and parameer learning, could well describe he behavior of he nominal exchange rae a he beginning of disinflaion periods observed in many open economies. Our grouping of papers ino he second and hird caegories is somehow subjecive, since hese groups overlap. 4

5 he exchange rae in i as well). He shows ha he inensiy of moneary policy response o changes in he spread srongly affecs he resuls for UIP. Meredih-Chinn (998) exends McCallum's model o a small macroeconomic model in which hey drop he assumpion ha he exchange rae direcly appears in he reacion funcion. Insead, heir cenral bank follows a Taylor-rule and he feedback from he exchange rae could go hrough wo indirec channels, hrough inflaion and he oupu gap. They calibrae and simulae heir model o see wheher i can reproduce he ex pos failure of shor run UIP and he ex pos validiy of long run UIP. Their resuls are generally favorable, alhough hey need risk premium shocks in exreme size o fi o observed volailiy. Meredih-Ma () furher exend McCallum's model and derive he "correc" equaion for he shor run exchange rae movemen, which, ineresingly, does no include he ineres rae differenial. Hence, hey claim ha he simple UIP equaions suffer from he omied variable bias. Hence, hey conclude ha he negaive β-esimaes along he classic Fama (984) specificaion is no suiable o draw any conclusions regarding UIP. In our reading, his is he main message of he lieraure building on he reacion of moneary policy. Among he papers uilizing informaion from he erm srucure, Macdonald and Marsh (997) inegraed long run ineres rae differenial ino he PPP equaion. They se up a VAR model which performed beer in ou of sample forecasing han he random walk for horizons longer han a few monhs. Juselius-MacDonald (4) sudy several key pariy condiions beween he US and Japan and find, among ohers, ha he link is primarily from long-erm o shor-erm ineres raes, which resul suppors one of our key assumpions, namely, ha shocks o long erm ineres raes are imporan deerminans of he shor rae, and also possibly he spo exchange rae. Clarida e al. (3) se up a model, which explois informaion in he erm srucure, and find beer han random walk forecasing performance even for wihin-year predicions. Inci-Lu (3) also builds a model using erm srucure informaion which could replicae some basic properies (e.g. mean, variance) of major currencies. Alexius () sudies he yields of shor invesmen ino long bonds and finds regression resuls ha does no rejec he [α,β]=[,] hypohesis for many cases. Perhaps Bekaer and Hodrick () are he firs who explicily used he erm he expecaion hypoheses of he erm srucure of ineres raes and of he foreign exchange marke. Bekaer e. al. () es UIP and EHTS boh in he shor and he long run using VAR models, arguing ha sandard regression based ess have poor small-sample properies. Their saisical evidence agains UIP is mixed and is currency- bu no horizon- dependen. Their evidence agains EHTS is saisically more uniform, bu deviaions from EHTS are economically no imporan. While Bekaer e. al. () sudied parameer resricions in a VAR esimaed for he change in he exchange rae, level of nominal ineres raes, and he erm spreads, in his paper we base our 5

6 analysis on a srikingly new claim: saionary of long run nominal exchange rae expecaions. In our framework we ask he quesion: would i be possible ha, aking long run UIP as given, shocks o he erm srucure are responsible for he anomalies observed in he foreign exchange markes? We assume raionaliy and risk neuraliy; hence UIP for boh shor and long horizons and he EHTS are also valid ex ane. In his framework, which is raher simple compared o he highlighed papers above, we show ha ex pos failures of he EHTS does no deermine exacly he pah of he exchange rae. A shock o he EHTS can lead o exchange rae movemens ranging in wo corner soluions. One is he ex pos validiy of shor run UIP, in which case he expeced long-run exchange raes moves in accordance wih he shock o he EHTS. In he oher corner soluion he expeced long-run exchange raes says consan and he spo exchange rae bears he full adjusmen, hence, he acual pah of he spo exchange rae deviaes from wha was prediced by he UIP in he previous period. We define his second corner soluion as he ex pos pariy of oal yields, which we will phrase briefly as yield pariy (YP). Yield pariy is calculaed as he oneperiod ineres rae adjused wih he differenial of domesic and foreign unexpeced price changes of risk free discoun bonds. In suppor of our model we presen srikingly new evidence on he saionary of long forward raes, which correspond o long run exchange rae expecaions under our assumpions, based on boh simple uni roo ess and also on a vecor-error correcion models. We also es he empirical performance of yield pariy in explaining he shor run movemen of he exchange rae for monhly series of hree major currencies, he dollar, he mark/euro and he pound. The resuls indicae ha YP srongly ouperforms UIP and is properies (e.g. volailiy and sign changes) are similar o ha of shor run exchange rae flucuaions. We should clarify righ here wha our model is good for. I offers an explanaion of ex pos failure of shor run UIP when i is fulfilled ex ane. However, as we relae shocks o foreign exchange and fixed income markes, i canno be used as a direc forecasing ool. The res of he paper is organized as follows. Secion briefly surveys he lieraure on long run UIP and presens new evidence based on consisenly calculaed and publicly available high qualiy daases for hree major currencies. Secion 3 presens evidence on he saionariy of long forward raes. Secion 4 describes he heoreical model and inroduces he noion of yield pariy. Secion 5 offers a new explanaion for he failure of shor-run UIP regressions. The empirical resuls for yield pariy using monhly exchange raes of major currencies are presened in Secion 6. Secion 7 ses up a sylized model suiable for sochasic simulaion and sudies he simulaed properies of he spo exchange rae, UIP regressions and yield-pariy regressions. Secion 8 concludes. Daa is described in he Daa Appendix. 6

7 . Uncovered Ineres Rae Pariy in he Long Run The hypohesis of long run uncovered ineres rae pariy is essenial for our model. In his secion we briefly survey evidence presened in he lieraure and presen resuls using our daase. Perhaps he firs paper emphasizing he differences beween resuls for shor and long horizon UIP was Flood-Taylor (996). They sudied bilaeral USD exchange raes in a panel framework for he period One of heir resuls, for example, is ha calculaions for he 3-year horizon using governmen bond yields led o an esimaed value of.596 for β wih a.95 sandard error. The poin esimae is subsanially larger han usual esimaes for shor horizons, significanly posiive, and a he borderline of no being significanly differen from he heoreical value of one. Alexius () sudies he -year horizon using quarerly daa from he IFS for 3 OECD counries in he period As she also highlighs, here are wo key problems for he sudy of UIP inheren in he long governmen bond yields of he IFS. Firs, he yields do no refer o - year exacly, bu varies around years. Second, since he daa are yield o mauriy bu no holding period yields, ineres paymens disurb he resuls. She ries o circumven hese problems and concludes a he end ha UIP migh work much beer for long han for shor horizons. Meredih and Chinn (998) adops much beer daa by using, besides benchmark governmen bond yields, zero-coupon yields as well for 5 and -year horizons, which were available for hem for some G7 counries. They sudy bilaeral USD raes in using quarerly daa. Some of heir main resuls are ha () a longer horizons he esimaes for β are significanly posiive, () R are subsanially larger for long han for shor horizons, and (3) resuls using zero-coupon yields end o be beer and in some cases he esimae for β does no differ significanly from one. Chinn and Meredih () presen resuls for German mark based exchange raes and find less favorable resuls han for USD based raes, alhough hese are sill beer han resul for shor run UIP. They sugges ha liberalizaion of bond markes, which ook place laer in Germany and especially in Japan han in he US, could be responsible for he worse resuls. Finally, Chinn and Meredih (5) repor, using 5-year ineres rae differenials and daa for he US, Germany, Japan and Canada, ha long-horizon resuls are robus o he use of differen daa frequencies, sample periods, yield definiions, and base currencies. One of he main weaknesses of long run UIP esimaions is he shor sample. For example, in he -year case here are only hree independen observaions in he pos-breon-woods era (if we exended he analysis ill 3) and he inensiy of overlapping is very severe, leading o srong auocorrelaion of residuals. In heir pioneering work, Hansen-Hodrick (98) suggesed esimaing he parameers of overlapping UIP equaions by OLS bu correcing he covariance marix, which 7

8 became he sandard pracice in he lieraure. However, for long UIP equaions he equaion overlaps so much ha he usually suggesed order of auocovariance calculaion is unfeasible. Moreover, Darvas (998) have shown ha hypohesis ess based on Newey-Wes heeroskedasiciy and auocorrelaion consisen covariance (HAC) have subsanial size disorions when applied o overlapping samples, and Kirby (997) showed ha he sample R used o be seemingly larger for overlapping samples, even if in he rue daa generaing process R is low. Obviously, we canno circumven hese problems eiher. Our only advanage is he more reliable daa a a higher frequency. We use consan mauriy zero-coupon yields for he US, Germany, and UK. Our calculaions will be based on monhly frequency using end of monh daa. 3 Since 999 we subsiued he exchange rae of he mark wih he euro rae muliplied by he conversion rae, which is a sensible choice (see, e.g. Brüggemann and Lükepohl, 5). Hence, we esimaed he sandard UIP equaion: s + s = α + β n i i + ε +, (, + *(, + () n ( ) n where s denoes he log of he exchange rae (domesic currency price of a uni of foreign currency), (, + i and *(, + i are he n-monh domesic and foreign zero coupon yields, which are, similarly o he noaion used in he nex secion, are no annualized bu measured a he monhly level 4, and ε +n is he error erm. We aach hree idenifiers o he ineres rae: he subscrip denoes he dae of he quoe, while he wo values brackeed in he superscrip indicaes he beginning and he end of he period for which he ineres rae refers o. We will also add an F superscrip when he beginning of he period for which he ineres rae refers o will be laer hen dae of quoe o emphasize forward ineres raes. Table repors our resuls for -monh and for year periods, using monhly daa. For he DEM/USD rae we repor resuls for wo sample periods, he longes available period (973M- 3M) and he sample period of he oher wo relaions (979M-3M). The general feaures of all relaions are ha () he poin esimae of β is negaive for shorer and posiive for longer mauriies, wih he resuls improving wih he horizon, () here are some cases when he null hypohesis of α= & β = is no rejeced, (3) he R is very low for shorer mauriies bu subsanial for longer mauriies, (4) here is severe posiive auocorrelaion for all esimaions excep he non-overlapping -monh horizon. The wo differen sample periods for he DEM/USD rae led o differen poin esimaes of he parameers, bu he 95% confidence bands overlap for all mauriies, and he general endencies highlighed above are valid for boh cases. We may also noe 3 See he Daa Appendix for he full descripion of he daa. 8

9 ha our resuls for he DEM/GBP is also reasonable wih an R of.4 in he case of he -year horizon, which is in conras o resuls of Chinn-Meredih (), who repor an almos zero R, bu his resul could be also be he consequence of overlapping observaions (Kirby, 997). Posiive auocorrelaion in he regressions is due o he overlapping naure of he esimaes which could no be solved in any saisfacory way. As an illusraion, we calculaed he parameer esimaes in he case of he -year rae for non-overlapping samples. Since here are possible samples for his exercise, we calculaed all of hem and ploed he poin esimaes and confidence bands in Figure. The poin esimaes vary widely, for example, he difference beween he highes and lowes esimae is. for he DEM/USD and DEM/GBP rae and 3. for he GBP/USD rae, wih summer monhs being more favorable for UIP. (This resul should no have any reasonable explanaions). Indeed, he confidence bands are very wide and always includes zero and only in few cases include he heoreical value of one. There is a furher problem wih he esimaion of he UIP relaions in (). Namely, while he lef hand side, changes in he exchange rae, is saionary, he ineres rae differenials end o be nonsaionary, especially for longer horizons. Table repors he resuls of uni roo and saionariy ess. I has been argued ha sandard ess for uni roo, like he ess of Dickey and Fuller (979) and Phillips and Perron (988) have bad size and power properies; see, for example, Maddala and Kim (998) for an exensive survey, or Ng and Perron () for a more recen overview. For his reason we employ six oher uni roo ess and a saionariy es as well. Ellio e al. (996) proposed a family of es saisics ha are invarian o he rend parameers. They suggesed wo paricular ess, a modified version of he Dickey-Fuller -es, which is essenially based on a local GLS derending, and anoher feasible poin opimal es, boh having subsanially improved power when an unknown mean or rend is presen. Ng and Perron () exploied he findings of Ellio e al. (996), and applied he idea of GLS derending o modify exising ess and showed ha non-negligible size and power gains can be made when used in conjuncion wih an auoregressive specral densiy esimaor a frequency zero. They suggesed modificaions of hree es saisics sudied by Perron and Ng (996) and he feasible poin opimal es saisics of Ellio e al. (996). Furhermore, we also use he es developed by Kwiakowski e al. (99) o es he null hypohesis of saionariy agains he uni roo alernaive. 4 For example, a percen annualized ineres rae akes he value of./ per monh. 9

10 For all of hese ess we allow only he consan as a deerminisic componen, bu we do no allow a deerminisic rend, because in economic erms i would be difficul o raionalize a linear rend in ineres rae differenials. The general conclusion from Table is ha longer-mauriy ineres rae differenials are clearly found o be non-saionary, while resuls for he shor-run ineres rae differenials are somewha mixed. A uni roo in ineres rae differenial implies ha equaion () is unbalanced. To sum up, here are severe problems wih UIP regressions. Besides regression saisics, i is insrucive o simply plo yield differenials and acual fuure exchange rae changes, which are shown of Figure. For shorer horizons (i.e. - years) he variance of he exchange rae changes subsanially exceeds ha of he yield differenial, bu for longer horizons he wo variables have similar variance. Moreover, for longer horizons he parallel movemen of he variables is asonishing. Hence, in he long run UIP migh be a reasonable hypohesis. A possible raionale for long run UIP could be ha long yields predic fuure inflaion well, and in he long run exchange raes adjus o PPP. However, even if anoher mechanism was a work, from he poin of view of our model only long run UIP, bu no he mechanism leading o i, is imporan. 3. Empirical Evidence on he Saionariy of he Expeced Long Run Exchange Rae Since he saionariy of he long run exchange rae expecaions will be a key assumpion of our model, we firs presen new and surprising empirical evidence on his issue, using simple chars, uni roo ess, coinegraion analysis, and variance calculaions. Our assumpion of he ex ane UIP for boh shor and long horizons allows us o use he forward raes as expecaions, which could be calculaed using he ineres rae differenial assuming covered ineres rae pariy (CIP), which is a common pracice in he lieraure due o is widespread empirical suppor (see, for insance, Simpson e al., 5, for a recen evaluaio: () where s L ~ = s + n i, L L s denoes he long mauriy forward rae, which equals o he expeced long run exchange L rae under long run UIP, E [ s ] ~ L (, + *(, + =, i ( i i ) s + n denoes he long run ineres rae differenial wih a large n, measured a he monhly level as before, and n denoes he number of monhs ahead, e.g. in a en year horizon n= (number of monhs in years). Firs, Figure 3 simply plos he spo exchange rae and -year mauriy forward raes. A visual impression does indicae ha longer horizon forward raes are much sable han he spo exchange rae. For insance, he huge rise of he dollar in he firs half of eighies was signaled by bond

11 markes as parly emporary. Only in he recen low yield period follow he long forward raes closely he spo raes. Second, Table 3 shows he resuls of eigh uni roo ess and a saionariy es (KPSS) on he logarihm of spo and forward raes. The resuls clearly indicae in all cases ha he es saisics decline wih horizon. Tha is, for longer horizons, we generally can rejec he null hypohesis of uni roo bu canno rejec he null hypohesis of saionariy. These resuls are sriking in he ligh of various aemps o es for uni roos in real and nominal exchange raes. We do no need breaks or non-lineariy, which are frequenly adoped in he lieraure, o find saionariy of a measure of he exchange rae of major floaing currencies. We should also noe ha, by definiion, hese ess are no burdened wih he problem of overlapping observaions which was a quie severe problem in long-run UIP equaions. Sill, we do no wish o over-inerpre our findings. For he Japanese yen 5 we could no rejec uni roo in he -year forward raes, alhough he es saisics also declined wih horizon. We aribue his resul o he srong rending behaviour of he yen which is visible in abou half of our sample. Hence, our resuls are no applicable o all currencies of he world, bu applicable o a leas six major currency pairs, which consiue 5 percen of foreign exchange marke urnover according o he survey presened in BIS (5, p. ). We should also sress ha he same ess ha indicae non-saionariy of spo raes and shor mauriy forward raes indicae saionariy of long mauriy forward raes. See Darvas and Schepp (6) for more deails. Third, saionariy of he long forward rae has a clear coinegraion implicaion: he non-saionary spo exchange rae and he non-saionary long ineres rae differenial coinegrae. Hence, when is above is mean, i s expeced o come back o is mean. Adjusmen could be achieved by eiher he spo exchange rae or by he long ineres rae differenial. We esimaed he following error correcion model for he spo exchange rae: L (3) s + = α, + α, s + v +, where α, should be negaive when he shor rae adjuss (we did no subrac he mean of L s L s from L s since here is a consan in he regression anyway). For comparison, we also esimaed he simple Dickey-Fuller equaion: (4) s + = α, + α, s + v +, 5 Japanese governmen bond yield is available in he IMF: IFS daabase.

12 which should yield a non-significan (based on he Dickey-Fuller disribuio α, when he spo rae follows a random walk. In he nex paragraph we show resuls of a fully specified VECM model, bu here we briefly show he resuls of he simple ECM above o see he performance of his bivariae model. Table 4 indicaes ha α, is significanly negaive for almos all cases, while α, is no significan for he DF equaion. The R of he regressions end o be larger for he ECM han for he DF-equaion, and akes values which are no negligible considering ha we model fuure changes in nominal exchange raes. The able also shows resuls for sub-periods. The regression do no yield significan esimaes in , i.e. in he adjusmen period afer Breon-Woods, bu work nicely boh in and in Fourh, in addiion () here is anoher coinegraing relaionship among our variables, which is L S implied by he EHTS. The EHTS claims ha erm spreads wihin a counry is saionary, ( i i ) ~ I() and ( * L * i i S ) ~ I(), so rearranging, L S * L * S L * L S * S ~ L ~ S (5) ( i i ) ( i i ) = ( i i ) ( i i ) = i i = ~ p ~ I(), where ~ p denoes he erm premium differenial beween he wo counries. Hence, he long and shor ineres rae differenials, ~ ~ and S are coinegraed wih he vecor (,-). Consequenly, L i i we have hree I() variables and wo coinegraing vecors: s L n (6) ~ S s i = L p ~, wih n = ( years of monhly daa). ~ i We esed for coinegraion using he Johansen-es. Boh AIC and SIC indicaed lag as he opimal for all currency pairs. We resriced he consan o be in he coinegraion vecor only (o allow for, say, a erm premium differenial beween he wo counries.) Briish daa begin in 979, while German and US daa begin in 973 (or earlier). For he German-US relaion, we esed for coinegraion boh in he and in he samples, similarly o our oher calculaions. Table 5 shows he resuls of coinegraion ess. For he DEM/USD relaion only one coinegraion vecor is found, bu he rejecion of he second is no sound. For he GBP/USD relaion wo coinegraion vecors are found, alhough he join saionariy of all hree series can no be excluded. For he DEM/GBP relaion, however, no coinegraion is found. Again, he resul for he DEM/GBP rae is similar o findings of oher papers, and also o our oher findings in his paper, in ha USD-based exchange raes yield favorable resuls han he DEM/GBP cross rae. By and large,

13 we evaluae all hese resuls as evidence in favor for wo coinegraion vecors, and coninue wih esing he parameer resricion implied in (6). The unresriced (bu normalized) coinegraing vecors are he following (sandard errors are in brackes): DEM/USD, DEM/USD, GBP/USD, DEM/GBP, (6.3).7 (.8) 68.3 (3.5).98 (.7) 383. (8.5).67 (.35) 65.6 (39.5).64 (.4) LR =.73 (p =.69) LR =.98 (p =.37) LR= 4.89 (p =.87) LR =.6 (p =.59) Hence, for he DEM/USD and DEM/GBP relaions he LR es can no rejec he null hypohesis of he coefficien resricions, while for he GBP/USD relaion, rejecion can be made a 9%. To sum up, by and large, boh coinegraion vecors wih parameer resricions in (6) are suppored by he Johansen-es. 6 The resul has he implicaion ha a leas one of wo variables in he coinegraion vecor for he forward rae, ha is, he spo exchange rae and he long mauriy ineres rae differenial, could be forecased using he previous period long mauriy forward rae. Using ineres rae differenials up o one year Clarida e al. (3) have already shown ha he random walk model of exchange rae forecasing can be ouperformed. I remains o be analyzed wheher incorporaing informaion from long mauriy ineres rae can furher improve he forecasing accuracy. In his direcion, independenly of our work and wihou our saionary resul, Boudoukh e al. (5) have already shown ha forward raes up o 5-year mauriy do help in forecasing. The quesion, again, is wheher incorporaing our saionary resul helps forecasing furher. Finally, we calculae he variance of spo and forward raes. Our assumpion on he sabiliy of he long run exchange rae expecaions has he implicaion ha long forward raes should be less variable han spo exchange rae, and, our new resul on he saionariy of he long forward raes imply ha heir variance should be ime-invarian. Table 6 presens he variance of spo and forward raes. Our conjecure is rue for he DEM/USD rae and for he DEM/GBP rae, while for he GBP/USD rae he variance is pracically he same for spo and forward raes. In order o have a view on he sensiiviy o he sample period, Figure 4 show variances calculaed over 5-long sample periods. Again, resuls for he DEM/USD rae is he mos favorable. I is noable, for example, ha afer he urbulen period of he sevenies, he variance of he -year long forward rae is raher sable across differen sample period. 6 Resuls were similar using 7-year yields, for which univariae ess also indicaed saionariy of long forward raes. 3

14 Saionariy of he expeced long run exchange rae could be sriking from he poin of view of economic heory, since one would expec, if any, he real exchange rae o behave saionary. We will commen his anomaly in he heoreical secion, by arguing ha he nominal rae could be a good proxy for he real rae, a leas in he recen period of low inflaion. The high and volaile inflaion episode of he sevenies could be regarded as an adjusmen period oward he long run equilibrium. To sum up, evidence presened in his secion do sugges ha long run exchange rae expecaions are saionary and have smaller variance han spo exchange raes, which suppor our assumpion in he nex heoreical secion ha shocks o he yield curve could leave he long run expecaions relaively sable. 4. The Yield Pariy Approach In his secion we sugges a new explanaion of he forward-bias puzzle. We inegrae he expecaion hypoheses of he erm srucure (EHTS) and of he foreign exchange marke (UIP). Ex ane all of hese hypoheses, boh in he shor run and in he long run, are assumed. The shor run ex pos movemen of he exchange rae, however, is primarily deermined by capial gains/losses of long erm fixed income securiies creaed by shocks o he erm srucure. In he ex pos relaion of fixed income invesmens and he foreign exchange rae, we are going o define a concep ha we call as ex pos pariy of oal yields, or yield pariy o phrase i briefly. 4.. The Basic Seup We adop he sandard assumpions of perfec markes (free inernaional capial movemens, domesic and foreign invesmen opporuniies being perfec subsiues, governmen bonds are risk free, risk neural and raional invesors) and he assumpion of flexible exchange rae regime. We assume boh shor run and long run UIP and he EHTS wihou any risk or erm premium. Term premium is frequenly found by papers sudying he EHTS, however, we will always use he ineres rae differenial beween wo counries. If erm premiums in he US, Germany, and UK had similar magniude hen our assumpion of no erm premium was no resricive. We se up our model in discree ime. We sar wih he hypohesis ha he whole invesmen period conains n individual ime periods. The lengh of an individual period could be anyhing, e.g. one day, week, monh, ec. We assume ha n is larger han one and, depending on he frequency of he underlying individual ime period, could ake values represening, say, from one year o en years. For he exposiion of our model presened in his secion, he exac lengh of he invesmen horizon is no imporan (apar from he assumpion ha i is larger han one). In he empirical analysis 4

15 presened in he nex secion of his paper, we will assume, which we regard as a safe assumpion, ha usual invesmen horizons fall wihin he range of one o en years, for which we have consisenly calculaed available daa for zero coupon yield curves. We assume ha invesors inves ino he whole range of discoun bonds; hence we do no have o boher wih he issue of ineres paymens of fixed income securiies. However, o keep in line wih he wording of he lieraure, we will use he erm ineres raes, which we define, of course, he usual way. 7 As we have already said, he UIP and he EHTS are assumed ex ane in every ime period for boh shor and long mauriies. Shor run and long run UIP implies ha (, + ) *(, + ) (7) E ( s + ) = s + i i, (, + *(, + (8) E ( s ) s + n ( i i ) =, + n where E denoes he expecaions operaor based on informaion available a ime. UIP holds in he nex period as well, hence ( +, + *( +, + (9) E ( s ) s + ( n ) ( i i ) =. + + n + + Combining (7) and (9) and assuming EHTS wih no erm premium, we have (, + ) *(, + ) F ( +, + * F ( +, + () E ( s ) s + ( i i ) + ( n ) ( i i ) + =, + n where F ( +, + i denoes he forward ineres rae for period + o +n quoed a ime. A simple derivaion using he above equaions allows us o rewrie (7) as (, + *(, + F ( +, + * F ( +, + () E ( s ) s = n ( i i ) ( n ) ( i i ). + The empirically racable version of equaion () will serve as our es equaion. 4.. Dynamics of he Exchange Rae in he Shor Run Suppose ha here is an unexpeced shock o he erm srucure a ime poin +, ha is, he EHTS is violaed ex pos. In his case he ex pos movemen of he exchange rae from ime poin o + could be raher differen from he predicion of previous period UIP, even if UIP and EHTS held ex ane in ime period + as well. This resul, which we explore below, is he main conribuion of our model. The expeced long run exchange rae based on ime and + informaion can be expressed as 7 For he ease of exposiion of he model, in his secion we consider coninuous ineres compounding, bu his assumpion will be easily subsiued in he empirical secion wih he usually adoped mehod (simple linear ineres couning for wihin-year mauriies and compounded ineres couning for over he yearl period). 5

16 F ( +, + * F ( +, + () E ( s ) E ( s ) + ( n ) ( i i ) =, + n + ( +, + *( +, + (3) E ( s ) s + ( n ) ( i i ) =, + + n + + so he unexpeced change of he exchange rae is expressed as + ( +, + *( +, + F ( +, + * F ( +, + (4) s E ( s ) = [ E ( s ) E ( s )] ( n ) [( i i ) ( i i )] n + n + + Hence, whenever he EHTS is violaed ex pos, i.e. he second erm on he righ hand side of equaion (4) differs from zero, hen eiher he expeced long run exchange rae could change (i.e. he firs erm on he righ hand side), or he spo exchange rae could differ from is previous expecaions (or any combinaions of hem). Assume ha in he second period he long (he + o + ineres rae differenial is larger han expeced earlier (i.e. he second erm on he righ hand side is posiive, which is muliplied wih a negaive number). There are wo possible corner soluions. () The expeced long run exchange rae depreciaes o absorb he full adjusmen (he firs erm on he righ hand side), in which case he spo exchange rae equals o is previous period expecaion according o shor run UIP. () The expeced long run exchange rae says consan; hence he depreciaion of he spo rae will be less han prediced by UIP in he previous period. The firs corner soluion is consisen wih, besides ex ane UIP, ex pos shor UIP as well. The second corner soluion could be regarded as he ex pos pariy of oal yields. The erm ex pos pariy of oal yields refers o he case when he expeced value of oal yield from o +n based on informaion se is he same as he sum of he acual yield from o + and he expeced yield from + o +n based on + informaion se. The facor ha equalizes he expeced yields for he full invesmen period is he capial gain or loss assumed by he long mauriy bond price. Hence, we can express his pah as (, + ) (5) ( ~ *(, + ) ) ( ~ * s s = i + g i + g ), where ~ = g E [ g ] g is he unexpeced price change of risk free discoun bond. 8 We relae hese capial gains/losses o he exchange rae change over he same period, i.e. from o +, hence, heir values are no known ex ane bu only ex pos. Le us highligh again ha his oucome. 8 Since a discoun bond, by definiion, does no pay ineres, is one-period expeced change equals o he one-period (, + ) * ineres rae, i. Hence, we could simplify equaion (5) as s + = g+ g+. The reason for separaing he wo erms is o emphasize he known and he unexpeced elemens of he yields. 6

17 hinges on he assumpion ha shocks of he EHTS fully ransmi ino he spo exchange rae and o bond prices. 9 In empirical esing of his hypohesis, he righ hand side of equaion (5) can be equivalenly replaced wih he following formula, which is equals o our noion of yield pariy: (, + *(, + ( +, + *( +, + (, + ) (6) s s = n ( i i ) ( n ) ( i i ) YP Noe he similariy beween equaions () and (6). Equaion () was an ideniy in which boh he lef and he righ hand side variables are based on ime informaion se. Equaion (6), on he oher hand, is an expression for he acual change in he exchange rae from ime o + expressed as he funcion of boh ime and ime + informaion. Tha s why we subscriped yield pariy wih +. This poenially causes and endogeneiy problem if we are o regress ( s ) is, (, + *(, + ( +, + *( +, + (7) s+ s = + β [ n ( i i ) ( n ) ( i + i + )] + ε + α s on YP (, + ) + Noe ha when α= and β=, subracing equaion () from equaion (7) and rearranging for he error erm ( +, + *( +, + F ( +, + * F ( +, + (8) ( n ) [ ( i i ) ( i i )] + s E ( s ) ( ) ε. + = , ha Hence, he error erm is he sum of wo expecaion errors. Unforunaely, he expecaion error of ( +, + *( +, + he erm srucure is likely correlaed wih he regressor, namely wih ( i i ) + +, in which case esimaion of equaion (7) or is varians will have biased esimaes. This endogeneiy problem can no be handled in sandard ways. Hence, alhough we will esimae his regression and presen resuls in Secion 6, our main emphasis are he saionariy of long forwards showed already, and he simulaion evidence o be presened in Secion Adjusmen in he Spo versus in he Expeced Long Run Exchange Rae Before urning o esimaion and simulaion issues, we have o conclude he concepual secion by answering he key quesion: why he spo exchange rae should bear a leas par of he adjusmen, insead of he expeced long run exchange rae? We propose four possible reasons. 9 The second assumpion (i.e. ex pos failures of he EHTS ransmi ino bond prices) is he direc consequence of ex ane EHTS, since he repaymen value of he discoun bond a +n is known wih cerainy. Hence, wih ex ane pariies, while he exchange rae adjusmen could ake place eiher in is long run expecaion or in is spo value, in he case of bonds, only he spo price can adjus under our assumpion of risk-free bonds. 7

18 Firs, sylized facs are consisen wih his assumpion. Namely, he oher corner soluion (full adjusmen of he expeced long run exchange rae) implies he validiy of ex pos shor UIP and he failure of ex pos long UIP. Sylized facs, however, sugges he opposie. Second, we hink ha, in response o shocks, invesors could raher mainain heir long run expecaion and accep adjusmen in he spo exchange rae, han he reverse. An argumen in favor of sabiliy of longer run expecaions is he evidence presened in he previous secion on long run UIP. Why? Because, aking long UIP as given, he pricing of long bonds a is an unbiased predicor of +n nominal exchange rae, hence a + he impeus o revise long run expecaion could be lower, when he naure of he shock (i.e. wheher i is a pure noise or somehing fundamenal) could no be clearly recognized. This argumen has a esable implicaion ha he variance of he long run expeced exchange rae should be lower han ha of he spo rae, which we have already confirmed. Third, one may argue ha we are assuming he sabiliy of he long run expeced nominal exchange rae, alhough many economiss would favor he sabiliy of he long run real exchange rae. There is a sric correspondence beween expecaions on he nominal and real exchange rae sabiliy only if domesic and foreign moneary auhoriies follow credible price level argeing policies. However, we know from he buoying lieraure on moneary policy rules ha his is no he case in he counries we sudy. Hence, under inflaion argeing, floaing exchange rae regime, and real exchange rae sabiliy, he level of he long run nominal exchange rae is no pinned down, which quesions our assumpion. This is indeed a valid criique and we can only lis some argumens ha weaken is srengh. Namely, he curren moneary regimes led o low inflaion for many years, and one migh say ha inflaion is expeced o say a low levels in he fuure as well. In such an environmen invesors could expec ha shocks o domesic and foreign inflaion raes are low on he average on he one hand. On he oher hand, even if shocks did no sum o zero in a given counry, hey could have similar accumulaed values a home and abroad, resuling from, for example, and he globalized world economy. This argumen implies ha he nominal exchange rae is a good proxy for he real rae, which has been argued in numerous papers. Moreover, his argumen suggess ha our assumpion has more relevance in he recen hisory of low inflaion han in he uncerain periods of he sevenies and eighies. Fourh, empirical ess of he EHTS sugges ha i has more relevance for long run han for shor run changes in he erm srucure (Shiller 99, Campbell 995), suggesing again, ha invesors could give more credi o heir previous period long-erm expecaions. Needless o say ha we do no claim ha all shocks o financial markes leave he expeced long run exchange rae unchanged. In fac, a simple plo of, say, he -year ahead expecaions derived from he -year UIP shows 8

19 flucuaions, alhough no as wide as he spo rae, for major currencies, as we have already shown in Figure 3. Wha we do hink, however, is ha a significan fracion of shocks are shor run shocks which leave he expeced long run exchange rae relaively sable, and which is suppored by our srikingly new resul on he saionariy of he expeced long run exchange rae in Secion 3. Before our empirical analyses and simulaion sudy, we would like o emphasize ha our approach does no exclude he possibiliy of he feedback from he exchange rae o he erm srucure, due o, for example, he reacion of moneary policy. Shock o he erm srucure could be rooed in moneary policy response o various evens. Moreover, we assume perfec markes, bu he sandard imperfecions, like risk premium and erm premium, could be inegraed ino our approach as well. However, we wan o go wihou hese imperfecions o see how far we can reach in undersanding he forward discoun bias puzzle. 5. A new explanaion for he failure of shor-run UIP: Exchange rae shocks are no orhogonal o he previous period shor ineres rae differenials The concep of yield pariy defined in he previous secion could be also viewed as a measure of how much of he EHTS shocks are absorbed by he spo exchange rae. In his secion we calculae values of he spo exchange rae if i had absorbed all or none of he EHTS shocks. We use his decomposiion o offer a new explanaion for he shor-run UIP puzzle. Le us reurn o equaion (4). For simpliciy, denoe he shor-run UIP error wih UIP sl ε + s+ E ( s+ ), he change in he expeced long-run exchange rae as ε + E + ( s+ E ( s+ n ), EHTS ( +, + *( +, + F ( +, + * F ( +, + and he EHTS error as ε ( n ) [ ( i i ) ( i i )] equaion (4) can be rewrien as (4 ) ε UIP sl EHTS + = ε + ε Using his noaion, Equaion (4 ) is an ideniy: under our mainained assumpions (UIP in he shor and long run plus EHTS) he magniudes in (4 ) can be easily calculaed from he daa. Equaion (4 ), however, does no imply by iself any correlaion srucures. I could be possible, for example, ha a shock o he erm srucure is compleely offse by he opposie movemen of he expeced long run exchange rae, in which case he UIP error will be zero. I also could be possible ha shocks o shor run UIP and he expeced long run exchange rae are highly correlaed and he ETHS shock is independen of hem, adding only a small noise o he relaionship. Equaion (4 ) also does no imply by iself any causaliies, nor i implies he sources of shocks. For example, in he case of he euro/dollar rae, in December 4 here was a close o zero shock o he erm srucure, i s magniude was.8, while shocks o UIP and long run exchange rae were - 9

20 .4 and -.33, respecively, indicaing ha boh he spo and expeced long run exchange raes of he euro appreciaed by more han wo percen. Indeed, a simple graph showing he hree shocks UIP sl indicae a srong comovemen of ε and ε, which clearly indicaes he presence of generic s foreign exchange rae shocks. We will denoe his shock as ε + + _ CF +, where CF sands for common facor of spo and expeced long run exchange rae shocks. Hence, we aim o decompose he magniudes in (4 ) as: ε, + + ε + = ε + + ε + ε + s _ CF s _ S s _ CF s _ L EHTS (4 ) ( ) ( ) s _ S s _ L where ε and ε denoe he spo exchange rae specific and he expeced long-run exchange + + rae specific componens, which we will phrase briefly as spo-specific and long-specific componens; and he ilde indicaes unobserved shocks. Noe ha while equaion (4 ) is sill an ideniy, brackeed magniudes can no be calculaed from he daa wihou any furher assumpions. We idenify he common facor and spo and long-specific componens using a common facor model. Specifically, we assume ha he unobserved shocks, [ ] s _ CF s _ S s _ L mulivariae normal disribuion, ε ~ N( ) unobserved componens as ideniies: ε + = ε + ε + ε +, has a +,Ω, and ha he observed shocks are relaed o hese (9) ε ε UIP + s _ L + = ε = ε s _ CF + s _ CF + + ε + ε s _ S + s _ L + The Kalman-filer can be used o evaluae he likelihood funcion of he process and, having esimaed he parameers of he model by maximum likelihood, o infer he unobserved facors. For insance, coninuing he example above, he model indicaes ha he value of he common componen of he euro/dollar rae in December 4 was -.8, while he spo and long specific values were -.33 and -.5, respecively. s _ S s _ L s The general feaure of he esimaions were ha σ( ε ) < σ( ε ) < σ( ε + + _ CF + ), e.g. for he DEM/USD rae for he full period of he sandard deviaions are.8,.5, and.3, respecively. Noe, for comparison, ha he sandard deviaions of period are.33,.43, and.3, respecively. ε for he same UIP sl EHTS +, ε +, ε + When he componens are idenified, he magniudes in he rivial reducion of equaion (4 ) can be calculaed: s _ S s _ L EHTS (4 ) ε = ε ε

21 Hence, an EHTS shock is eiher absorbed by he shor or long-specific shocks, or by any combinaions of he wo. The idenificaion of foreign exchange rae shock componens allows us o ask he quesions: wha would have been he properies of he expeced long-run exchange rae, if (a) i absorbed all EHTS shocks, (b) i absorbed no EHTS shocks. For insance, when he expeced long run exchange rae s _ S s _ L absorbed all EHTS shocks hen ε + =, hence, ε + = ε EHTS +. When he spo rae absorbs all he s _ L EHTS shocks hen ε. Since ε sl is simply he change in he expeced long run exchange + = rae, hen he magniudes we are ineresed can be easily calculaed as + (.a) (.b) L _ all _ EHTS L _ all _ EHTS s _ CF EHTS s+ = s ε + s = s ε, L _ no _ EHTS L _ no _ EHTS s _ CF ε, where he recursion could be sared from any values (which only modifies he mean bu no he saisical properies of he consruced series). We sared he recursions in () from he acual vale of he expeced long run exchange rae, in order o be able o compare easily he acual and he consruced series. A similar exercise could be done for he spo rae as well. Recall ha s _ CF s _ S UIP ~ S ~ S s _ CF s _ S ε ε = = s E s ) = s s i, herefore, s = s + i + ε ε. When he ( ) ε + + ( s _ S EHTS spo rae absorbs all EHTS shocks, hen ε = ε from (4 ), hence: (.a) (.b) S _ full _ EHTS _ absorb S _ full _ EHTS _ absorb S s _ CF EHTS s + = s + i + + ε + s S _ no _ EHTS _ absorb S _ no _ EHTS _ absorb S s _ CF + = s + i + ε + ε, where we sared he recursion from he acual value of he spo exchange rae. Figure 5 shows he magniudes calculaed in (.a) and (.b), he shor-run ineres differenial ~ S ( ), and he generic foreign exchange rae shock ( s _ CF ε ), for he DEM/USD rae. The acual i exchange rae lies beween he wo hypoheical pahs. Recall from he previous secion ha when he long run expeced exchange rae absorbs all of he EHTS shocks and hence he spo exchange rae assumes none of hese shocks, hen shor run UIP holds in ex-pos daa as well. Hence, he arificial series in (.b) should fulfill ex pos UIP. Noe S _ no _ EHTS _ absorb S s _ CF also ha he daa generaing process for he difference is: ( ) Therefore, in a sandard regression ~ α, S _ no _ EHTS _ absorb S () ( s + ) = UIP + βuipi + v + s i ε. + = + +