The Forward Unbiasedness Hypothesis and the Forward Premium: A Nonlinear Analysis

Transcription

1 The Forward Unbiasedness Hypohesis and he Forward Premium: A Nonlinear Analysis Sofiane H Sekioua Docoral Programme Warwick Business School The Universiy of Warwick Covenry CV4 7AL sofiane.sekioua@wbs.ac.uk Sepember 003 This paper, which is par of he auhor s PhD hesis, has been prepared for he annual Money, Macro and Finance Conference in Cambridge in Sepember 003. I am graeful o Professor Lucio Sarno and paricipans a he finance seminar series a he Warwick Business School for helpful commens and feedback. Any errors and omissions are my own.

2 The Forward Unbiasedness Hypohesis and he Forward Premium: A Nonlinear Analysis Absrac This paper considers nonlinear aspecs of esing he mean-reversion of he forward premium. In conras o sandard linear mehods, we consider a novel approach ha allows for he join esing of nonlineariy and nonsaionariy. Wihin his approach, we employ nonlinear hreshold auoregressive (TAR) uni roo ess o invesigae wheher he and 3 monh forward premium series for seven indusrial counries are mean-revering. Overall, we are able o rejec he null hypoheses of lineariy and nonsaionariy indicaing nonlinear mean-reversion of he forward premium. Furher, large deviaions of he forward premium from is long-run equilibrium level are found o have faser speed of mean-reversion han small deviaions. JEL Classificaion: F30, F3, C3, C33. Keywords: Forward Premium, Panel Uni Roo Tess, Half-lives and Nonlinear Threshold Auoregressive Uni Roo Tess.

3 . Inroducion The relaionship beween spo and forward exchange raes has been sudied exensively in he inernaional finance lieraure. However, he lieraure has produced conflicing resuls relaed o a number of issues. One of hese conroversial issues concerns he saionariy of he forward premium and coinegraion beween forward and spo exchange raes (Kuan and Zhou, 00). The saionariy of he forward premium is imporan for a number of reasons. Firs, if he forward premium is a realisaion of a uni roo process i makes he commonly employed marke efficiency es of Fama (984) inappropriae since i suffers from he spurious regression ype criique (Granger and Newbold, 974; Baillie, 996). Second, since Hakkio and Rush (989), i has become common ha ess of he forward rae unbiasedness hypohesis (FRUH), which saes ha he forward exchange rae should be an unbiased predicor of he fuure spo rae, require ha he fuure spo rae s + and he forward rae f be coinegraed wih a coinegraing vecor of [,-]. I is also rue ha he FRUH requires he spo rae s and forward rae f o be coinegraed wih coinegraing vecor of [,-] or equivalenly ha he forward premium ( f s ) be saionary (Engel, 996; Zivo, 000). In a recen survey, Engel (996) summarized he evidence on ess of coinegraion beween spo and forward exchange raes and saionariy of he forward premium and forecas error / excess reurn as follows: Some have found (s - f ) is I (0); some have found i is I (); some have found i is fracionally inegraed. Some have found s + and f are coinegraed wih coinegraing vecor [,-]; some have found hey are coinegraing bu no wih coinegraing vecor [,-]; some have found hey are no coinegraed. These conflicing resuls hold on ess for he same se of currencies, (pp. 4). Engel (996) argues ha hese conflicing resuls may arise from differen sampling periods, bu more likely hey resul from differen properies of he various ess employed. Coinegraion beween he fuure spo rae s + and he forward rae f wih a coinegraing vecor of [,-] implies ha he foreign exchange forecas error / excess reurn (s + - f ) is saionary and vice versa (see definiion of coinegraion in Engle and Granger, 987). This also applies o he case of spo and forward exchange raes and he forward premium. 3

4 In his paper we focus on he properies of some of he ess used o invesigae he saionariy of he forward premium. We argue ha he major problem wih he sandard uni roo and coinegraion ess ha have been used so far o idenify he order of inegraion of he forward premium is ha hey implicily assume linear adjusmen. However, here are good reasons why, if spo and forward raes co-move and are linked by a coinegraing relaionship, adjusmen oward equilibrium may be nonlinear. For example, ransacion coss and marke fricions are ofen cied as he major sources of nonlineariies. According o Dumas (99) ransacion coss creae a band of inacion wihin which deviaions from he forward premium long run equilibrium level are lef uncorreced as hey are no large enough o be profiable. I is only deviaions ha are ouside he band ha are arbiraged by marke agens. In his framework, he forward premium follows a nonlinear process ha is mean-revering. Nonlineariies in foreign exchange markes are suppored by evidence of nonlineariies in nominal and real exchange raes (Obsfeld and Taylor, 997; Kilian and Taylor, 00; Taylor e al., 00) and in he adjusmen of he nominal exchange rae owards he long run equilibrium level suggesed by moneary fundamenals (Meese and Rose, 99; Taylor and Peel, 000). However, here is no clear way of differeniaing beween nonlineariy and nonsaionariy in he above lieraure. Indeed, hese sudies assume saionariy prior o fiing a nonlinear model and find evidence of fas nonlinear mean-reversion when hese nonlinear models are used. However, if saionariy is no valid and he variable under sudy has a uni roo, hen ess of lineariy versus a nonlinear hreshold alernaive will lead o incorrec inferences as hese ess will have non-sandard asympoic disribuions. Also, resuls found in he above sudies regarding nonlineariy and mean-reversion mus be inerpreed cauiously as long as a proper es of uni roo agains a saionary nonlinear (hreshold) alernaive has no been performed (Bec e al., 00). A recen paper by Taylor (00) has shown ha sandard uni roo ess have low power in he presence of nonlinear adjusmen, however. To circumven hese problems, we use a novel approach whose ess and disribuion heory have been recenly developed by Caner and Hansen (00) and which allows for he join esing of nonlineariy and nonsaionariy of he forward premium. This approach Sekioua (003a) uses his approach o model he deviaion of he nominal exchange rae from he level prediced by moneary fundamenals such as he money supply and income. Using nonlinear 4

5 uses a wo-regime symmeric hreshold auoregressive (TAR) model wih an auoregressive uni roo which allows for an inner no-arbirage band for small disequilibria and capures mean-reversion in response o shocks ouside he no-arbirage band. Wihin his model, we sudy Wald ess for nonlinear adjusmen and Wald and -ess for nonsaionariy. We also allow for general auoregressive orders and do no arificially resric coefficiens across regimes (Basci and Caner, 00). In his paper, we firs examine he univariae ime series properies of he forward premium for seven indusrial counries and for he recen floaing exchange rae period. Unsurprisingly, we find lile evidence of mean-reversion in he forward premium and his is explained by he low power of he univariae uni roo ess o rejec he uni roo null hypohesis especially when using shor daa samples 3. To overcome he power problems of he univariae ess we increase he number of observaions and es for he non-saionariy of he forward premium in a panel framework 4. However, despie he use of panel ess such as he Maddala and Wu (999) and Hadri (000) ess which are appropriaely sized and have excellen power agains close local alernaives, we were sill unable o rejec he null hypohesis of a uni roo in he forward premium. Given nonlinear exchange rae adjusmen, he dynamic relaionships implici in esing he saionariy of he forward premium series using hese linear univariae and panel ess are misspecified and i should come as no surprise ha we canno deec any mean-reversion in he forward premium. Allowing for he possibiliy of nonlinear adjusmen and using nonlinear TAR uni roo and boosrap uni roo ess, he auhor uncovers evidence of nonlinear mean reversion of he deviaion series. This evidence reinforces hose of sudies which deeced nonlineariies in exchange rae adjusmen. However, he resuls are more robus since one of he drawbacks of previous sudies which was he failure o es for uni roo was deal wih. 3 The failure o esablish mean reversion in he forward premium may be due o lack of power of convenional uni roo ess. Taylor (995) argued ha uni roo ess are biased downward in he sense hey are more likely o rejec saionariy when he underlying process driving he daa is in fac saionary. This bias is due o he small size of available daa samples. 4 The use of panel daa allows us far greaer flexibiliy and his has also been proven quie saisfacory in improving he power of uni roo ess. The addiional cross secional dimension in he panel leads o beer power properies of he panel ess as compared o he lower power of he sandard individual uni roo es agains near uni roo alernaives especially when using shor daa samples. 5

6 likelihood ess, we firs obain srong evidence of nonlineariies in he forward premium. Second, using Wald and -ess we were able o rejec he uni roo null hypohesis in a leas one regime (he regimes being inside and ouside he ransacion cos band). Specifically, we find ha he forward premium displays uni roo behavior when i is wihin he ransacion coss band and when here is a large shock i is mean-revering. Finally, he compued halflives of deviaions indicae ha he speed of adjusmen ouside he band is faser han han inside he ransacion cos band. Finally, he evidence of nonlinear mean-reversion of he forward premium, and he forecas error / excess reurn, uncovered in his paper provides suppor for he forward rae unbiasedness hypohesis (FRUH) and he uncovered ineres rae pariy (UIP) hypohesis which requires a saionary forecas error. This evidence also indicaes ha he ineres rae differenial, which is linked o he forward premium hrough he covered ineres rae pariy (CIP) condiion, is also a saionary TAR process. This paper is srucured as follows: Secion provides a descripion of he workhorse of our analysis, namely he forward rae unbiasedness hypohesis (FRUH). The empirical mehodology is described in Secion 3. Daa and empirical resuls are provided in Secion 4. The las secion concludes.. Forward Rae Unbiasedness Hypohesis (FRUH) and he Forward Premium The forward rae unbiasedness hypohesis (FRUH), which saes ha he forward exchange rae is an unbiased predicor of he fuure spo exchange rae, has been sudied exensively in he empirical lieraure of inernaional finance (Engel, 996). This hypohesis was developed as a corollary o he efficien marke hypohesis. Assuming raional expecaions we ge: E (s + /I )=f () where f is he log forward rae for a or 3-monh conrac a ime, s + is he corresponding log spo rae a ime +, and E (.) is he expecaions operaor given informaion a ime, I. Given he assumpion of raional expecaions, we ge: s + =E (s + )+u + () where u +, is a zero mean, whie noise process. Now, subsiuing () ino () yields: 6

7 s + =f +u + (3) Equaion (3) delivers he FRUH. This hypohesis is esed empirically using he following regression: s (4) + = η + βf + u + where FRUH holds ifη = 0, β = and E [u + ]=0. Naka and Whiney (995) and Hai e al. (997) refer o esing η = 0, β = as esing he forward rae unbiasedness hypohesis. Tesing he orhogonaliy condiion E [u + ]=0, condiional on no rejecing FRUH, is hen referred o as esing marke efficiency under raional expecaions and risk neuraliy (Zivo, 000). Several researchers over he years have examined regression (4) wih various improvemens in economeric echniques and several early resuls are supporive of he unbiasedness hypohesis (Cornell, 977; Frenkel, 977). However, subsequen research on esing for uni roo drew doub as o he appropriaeness of (4), as exchange raes display uni roo behaviour. Concerns over spurious regressions promped researchers o look for saionary alernaives o (4). This resuled in he following specificaion: s (5) ' ' + s = η + β (f s ) + ε + Equaion (5) shows ha he forward premium ( f s ) is an unbiased predicor of he ' fuure currency depreciaion ( s + s ) if = ' η 0, β = and E [ ] = ε + 0. If he spo rae is difference saionary hen for regression (5) o be balanced and he FRUH hypohesis o hold, ( ' ' f s ) mus be saionary, assuming η 0, β = and E [ ε ] = 0. This can be = + shown by applying he hypohesis ha he risk premium ( rp f E [ s ] expecaions are raional ( E [ s + ] = s + u + s = E [ s ] s = s s u ). We can hen wrie: = + ) is null and ha f (6) whereu + is he forecas error, which mus be saionary under raional expecaions. Hence, if ( s + s ) and u + are saionary, he forward premium ( f s ) mus also be saionary. The laer condiion implies ha he spo and forward raes have o be coinegraed wih a coinegraing vecor of [,-]. Hakkio and Rush (989) and Engel (996) sressed ha esing he unbiasedness hypohesis from (4) or (5) involves coinegraion beween eiher s + and f or f and s wih he same coinegraing vecor of [,-]. I is clear from 7

8 s f = s + (f s ) (7) + ha if he change in he spo exchange rae is saionary hen coinegraion beween he fuure spo rae and he forward rae wih a coinegraing vecor of [,-] involves coinegraion wih he coinegraing vecor [,-] beween he spo and forward exchange rae. Essenially, saionariy of he forecas error / excess reurn implies saionariy of he forward premium and vice versa. In his paper we focus on he relaionship beween he forward and spo rae only, i.e. he forward premium, since his relaionship has implicaions for regression ess based on (5) and he unbiasedness hypohesis. Surveying he empirical evidence on he saionariy of he forward premium and coinegraion beween spo and forward exchange raes we find ha he resuls are somewha mixed. Clarida and Taylor (997), Hai e al. (997) and Barkoulas e al. (003) suppor he saionariy of he forward premium. Ohers obain evidence supporive of uni roo behaviour of he forward premium (Crowder, 994; Evans and Lewis, 995; Horvah and Wason, 995; Luinel and Paudyal, 998). In general, he resuls depend on he esing procedure, he daa frequency and ime period. Furher, he high persisence and heerogeneiy of he forward premium reduce he power of uni roo ess and disor he size of saionariy ess (Engel, 996). These problems have led some researchers o consider non-sandard models of uni roo and coinegraion esing. For example, Baillie and Bollerslev (994) and Byers and Peel (996) concur ha he forward premium is fracionally inegraed, whereas Baillie (994) considers fracional coinegraion beween spo and forward exchange raes. Kuan and Zhou (00), on he oher hand, use rolling coinegraion echniques and find ha he relaionship beween spo and forward rae broke down in he 980s. Alhough, hey became coinegraed again in he 990s, hey no longer co-moved proporionally, however. The main problem wih previous sudies of he saionariy of he forward premium is ha hey implicily assume ha exchange rae adjusmen is linear. Nonlineariies in he relaionship beween spo and forward exchange raes can be due o a variable speed of adjusmen owards a long-run equilibrium. This may arise because small deviaions are no considered imporan by he marke and he auhoriies, whereas for larger deviaions, he pressure from he marke o reurn he exchange rae near is equilibrium value becomes larger (Taylor and Allen, 99; Taylor and Peel, 000). The same reasoning applies in he presence of ransacion coss (Dumas, 99). Transacion coss keep arbirage on hold unil 8

9 he forward premium beween wo currencies depars sufficienly above a hreshold of inacion. Neverheless, in spie of he evidence supporive of nonlinear exchange rae adjusmen, here are only a few sudies of FRUH which ake nonlineariies ino accoun. Coakley and Fueres (00 a, b) address he forward premium puzzle 5 in a nonlinear framework and argue ha he insignifican coefficiens from recen daa spans can be explained by an unbalanced regression problems caused by asymmeries and nonlineariies in spo reurns. 3. Empirical Mehodology 3.. The Maddala and Wu (999) Panel Uni Roo Tes Maddala and Wu (999) proposed a panel uni roo es ha is based on he p-values of N independen ess of a hypohesis. This es, herefore, can be used wih any kind of uni roo or saionariy es. The significance levels p i (i =,, 3,, N) are independen uniform (0, ) variables and lnp i has a propery of he χ disribuion wih wo degrees of freedom. Using he addiive χ variables, he following es (wih a χ disribuion and N degrees of freedom) can be derived: N MW = ln (8) i = p i This es can be used wih any kind of uni roo or saionariy es. Indeed, he es s null hypohesis is he same as ha of he univariae es used o obain he significance levels p i. This is is bigges advanage when compared wih oher panel uni roo daa ess, such as hose proposed by Im e al. (997) and Pedroni (998). Furher his es does no require a balanced panel and is non-parameric. 5 In a regression of he change in he spo exchange rae on he forward premium, he slope coefficien is no only differen from one bu i is also negaive. This is he forward premium puzzle. 9

10 3.. The Hadri (000) Panel Saionariy Tes The Hadri panel saionariy es is an exension of he univariae KPSS es o he panel framework wih N cross-secions. Following single ime series saionary esing procedures proposed by Kwiakowski e al. (99), consider he following wo models: fp i = r i + ε i, i=,,t (9) or fp i = r i + β i + ε i (0) where r i is a random walk defined hrough r i = r i- + u i wih fixed iniial values r i0. To es he null of saionariy, consider he following hypohesis: H o : λ = 0 agains H a : λ > 0 () where λ = σ u / σ ε and σ u = 0 under he null. The panel can be presened as: fp i = X i B i + e i () ' ' where fp = [fp...fp ], e = [e...e ] and X i is a T uni () vecor. The Hadri LM es is: i i it i i it LM = N N i = T σ T = ε,i S i (3) where S and are he parial sum of residuals and he variance from each individual series. i σ ε,i Hadri (000) suggess he following semi-parameric correcion for serial correlaion o be applied o each variance erm in he panel: T * σ i (x) γ = 0 + s = κ(x)γ (4) s where σ * T i = γ0, he bandwidh x=s/l+, l is he lag runcaion and γs = e e T i = s + number of choices are available for he Kernelκ(x). Hadri has suggesed ha he Quadraic- Specral (QS) kernel migh be opimal, bu for comparison Barle (BT) and Tukey-Hanning (TH) kernels are also included. The following finie sample correcion o he Hadri LM es is used: i s. A 0

11 Z µ N(LM ξ µ ) = (5) ς µ where = / 6, ς / 45 when an inercep is included and ξ µ µ = ξ µ µ = = / 5, ς / 6300 when an inercep and a rend are included The Caner and Hansen (00) Threshold Auoregressive (TAR) Uni Roo Tes The model suggesed by Caner and Hansen (00) is a hreshold auoregressive (TAR) process of he form: ' ' = θ x {Z λ} + θ x {Z λ} + e < (6) ' ' =,, T, where x = (fp r... k ), {.} is he indicaor funcion, e is an iid error, Z = fp fp m for some m and r is a vecor of deerminisic componens including an inercep and possibly a linear ime rend. The hreshold λ is unknown and akes on values in he inerval λ Λ = [λ,λ ] where λ and λ are picked so ha P(Z λ ) = π > 0 and P(Z λ ) = π < 6. I is convenien o show he componens ofθ and θ as: µ µ θ = β and θ = β (7) ρ ρ The TAR model is esimaed by leas squares (LS): Le ˆ ' ˆ ' θ ) ) ˆ (λ x {Z < λ} + θ(λ x {Z λ} e(λ ) = + T σˆ (λ ) = T eˆ (λ ) (9) (8) 6 Since only he magniude of he change in he forward premium ha maers and no he sign, we consider he absolue value of he change as he swiching variable. We, herefore, reain a symmeric hreshold λ = λ = λ. This makes our TAR a wo-regime symmeric model.

12 be he OLS esimae of σ for fixed λ. The leas squares esimae of he hreshold λ is found by minimizing σ (λ ): ˆ λ = arg min σˆ (λ ) (0) λ Λ In model (5), an imporan issue is wheher here is a hreshold effec. The hreshold effec disappears under he hypohesis ha: H = = () : µ µ, ρ ρ 0 This hypohesis is esed using a sandard Wald es W T. This is wrien as: W T = T( σˆ σˆ ) () 0 where ˆσ is he residual variance from (8), and ˆσ 0 is he residual variance from OLS esimaion of he null linear model. Le (3) denoe he Wald of hypohesis () for fixed hreshold. W T (λ ) = T( σˆ σˆ 0 (λ ) ) (3) Then since W (λ) is a decreasing funcion of σ ˆ (λ ), we find ha he Wald is: W T T = W (λ) ˆ = supw (λ ) (4) T λ Λ T In model (8), he parameers ρ and ρ conrol he saionariy of he process fp. A leading case is when fp is a uni roo process such ha: H : ρ = ρ 0 (5) 0 = The sandard es for (0) agains he unresriced alernaive ρ or ρ 0 is he Wald : 0 R T = + (6) where and are raios for ρˆ ˆ and ρ from he OLS regression of he TAR model. While i is unclear how o form an opimal one-sided Wald es, Caner and Hansen (00) recommend focusing on negaive values of ρˆ ˆ and ρ o end up wih a simple one-sided Wald es : = (7) R T {ρˆ 0} < + {ρ ˆ < 0} which is esing he uni roo null hypohesis agains he one-sided alernaive ρ < 0 or ρ < 0. Generally, Caner and Hansen sugges examining he individual s

13 and such ha an insignifican provides evidence in favour of he presence of a uni roo in he TAR process. While he disribuions of R T and RT have asympoic approximaions, improved finie sample inference may be conduced using a boosrap disribuion. In his paper, we obain he exac p-values of he R T and RT s using 0000 boosrap simulaions consisen wih Caner and Hansen (00). 4. Daa and Empirical Resuls 4.. Daa Monhly spo and forward exchange raes daa for seven indusrial counries, namely, Ausralia, Canada, France, Germany, Ialy, Japan, Swizerland and he USA, agains he UK pound serling are used in he empirical analysis for his paper. The daa is for he period spanning from January 979 o December 998 and is obained from DaaSream. We use hese currencies because hey are heavily uilized in earlier sudies. By using monhly daa and - and 3-monh forward conracs, overlapping daa effecs are avoided. The empirical analysis in his paper is performed using version 3.. of he GAUSS programming language. 4.. Univariae and Panel Uni Roo Tess Mean-reversion of he forward premium series would provide evidence of coinegraion beween spo and forward exchange raes. Table presens he values of he Augmened Dickey Fuller (ADF) es performed on he - and 3-monh forward premium series. The ADF es on he log levels of he -monh forward premium series shows ha he uni-roo null hypohesis can only be rejeced for Swizerland and Japan a he 5% significance level and for Canada a he 0% level. Resuls of he ADF es on he 3-monh forward premium series are quaniaively similar o hose on he -monh forward premium series. However, he inabiliy o rejec he uni roo null hypohesis using he ADF es should no be regarded as prima facie evidence agains he saionariy of he forward premium. Indeed, resuls of he ADF es have o be inerpreed wih care since a major criicism of he ADF uni roo esing procedure is ha i canno disinguish beween uni roo and near uni roo processes 3

14 due o is low power in he presence of such a persisen variable as he forward premium (Engel, 996). Given he conflicing findings of he univariae ADF uni roo es, we now apply panel uni roo and saionariy ess o sysems of forward premium series. By exploiing crossequaion dependencies and increasing he span of he daa by joinly esing for a uni roo and saionariy across a number of series, panel ess are likely o lead o subsanial gains in es power. In able we implemen correlaion analysis o idenify cross-secional dependencies among he sysem variables. As repored in able, here are subsanial correlaions among he panel forward premium series, wih he noable excepion of he Swiss / USA and Swiss / Ialy forward premium series. Overall, hese resuls sugges ha implemenaion of panel uni roo and saionariy ess would lead o subsanial power gains. In able repors he individual ADF -s and heir corresponding p-values for each one of he forward premium series. These values are used o compue he Maddala and Wu (999) es. The resuls of he Maddala and Wu (999) show ha here is suppor for he rejecion of he uni roo null hypohesis as he Maddala and Wu is greaer han he criical value a he % significance level. However, i is possible ha his es is influenced by ouliers in he panels, and ha, in he curren applicaion esing is of limied value due o heerogeneiy of he daa of differen counries, some being saionary and ohers inegraed of order. To illusrae his poin, reconsider he Maddala and Wu (999) es wihou he forward premium series whose ADF s are significan a he 5% level. If we removed hese series and considered a panel of 4 forward premium series he calculaed Maddala and Wu (999) es falls from o for - monh forward premium series, and now no significan even a he 0% significance level. We reach a similar conclusion for he 3-monh forward premium. Given he mixed resuls obained wih he Maddala and Wu (999) es, we now urn our aenion o esing he saionariy of he forward premium series using a es ha does no rely upon he -s of he ADF es. For his panel es, we use he nonparameric correcion o he saionariy es due o Hadri (000) o ake accoun of heerogeneous serial dependence across our panel ha is made up of 7 forward premium series. The es considers he null of saionariy and for a sample wih more han 50 observaions Hadri (000) shows ha his es has high power agains close local alernaives 4

15 (Huner and Simpson, 00). The resuls of he Hadri (000) are repored in able 3. Ordering he ess by heir specificaion, he es s rejec he null hypohesis of saionariy for he -monh forward exchange rae premium. As for he 3-monh forward premium he evidence is mixed. The es s correced for serial correlaion accep he null of saionariy whereas he null hypohesis is rejeced when here is no correcion for serial correlaion. Therefore, despie he fac ha his es appears suied o es wheher he forward premium series are saionary he resuls are inconclusive Threshold Auoregressive (TAR) Boosrap Tess The major problem wih he ess ha we have used so far is ha hey implicily assume linear adjusmen. Nonlineariies in he relaionship beween spo and forward exchange raes can be due o a variable speed of adjusmen owards a long-run equilibrium. This may arise because small deviaions are no considered imporan by he marke and he auhoriies, whereas for larger deviaions, he pressure from he marke o reurn he exchange rae near is equilibrium value becomes larger (Taylor and Allen, 99; Taylor and Peel, 000). Nonlineariies can also arise as a consequence of ransacion coss and marke fricions (Dumas, 99). To explore he forward premium for poenial fricions capured by nonlineariies; we employ he nonlinear hreshold auoregressive (TAR) framework of Caner and Hansen (00). The model used is a wo-regime symmeric TAR model wih an auoregressive roo ha is local-o-uniy. The firs sep in our analysis involves esing for lineariy and he appropriae es for his is a Wald es. In ables 4 o 7 we repor he Wald es, % boosrap criical values and boosrap p-values for hreshold variables of he form Z = fp fp m for delay parameers m from o. However, he opimal delay parameer is chosen so ha i minimises he residual variance of he TAR model of each forward premium series. The delay parameers for he - (3- ) monh forward premium series are 4 (7) for he USA, (8) for Swizerland, 7 (4) for Japan, () for Canada, 5 (5) for France, (8) for Germany and 6 (7) for Ialy. From ables 4 and 6, i is clear ha each corresponding o he opimal delay parameer is highly significan and easily rejecs he null hypohesis of lineariy in favour of he nonlinear hreshold model. 5

16 In ables 8 and 9 we repor he hreshold uni roo and s for each delay parameer m from o, and heir p-values obained using 0000 boosrap simulaions. However, he -s of ineres are hose ha correspond o he delay parameers ha minimise he residual variances. These delay parameers are he same as hose used o es for lineariy. For hese delay parameers he boosrap p-values for for he - (3- ) monh forward premium series are (0.043) for he UK, (0.0849) for Swizerland, (0.880) for Japan, (0.9530) for Canada, (0.033) for France, (0.0) for Germany and (0.0894) for Ialy. The significan s for -monh forward premium series for he UK, Japan and Ialy indicae ha we are able o rejec he uni roo null hypohesis in favour of ρ < 0 in he firs regime. For he 3-monh forward premium we are able o rejec he null hypohesis for he UK, Swizerland, France and Ialy. The boosrap p-values for he for he - (3- ) monh forward premium series are (0.300) for he UK, (0.540) for Swizerland, 0.80 (0.0047) for Japan, (0.0) for Canada, (0.5550) for France, (0.440) for Germany and (0.6550) for Ialy. In his case, p-values indicae ha we are able o rejec he uni roo null hypohesis in he second regime for - and 3-monh forward premium series for Canada and -monh forward premium for Japan only. However, we are able o sele he uni roo quesion using he R T and R T s which are repored in ables 9 and. The one-sided Wald es R T, which ess uni roo agains a wo-regime saionary nonlinear model, is rejeced in ou of 7 cases for he - monh forward premium. For he 3-monh forward premium, he uni roo null hypohesis is rejeced in 5 ou of 7 cases. However, on closer inspecion we find ha he resuls for he one-sided uni roo es are due o he fac ha he firs regime is mean-revering whereas he second regime has a uni roo and his regime is dominan. Taking his reasoning ino consideraion and following Basci and Caner (00) we rejec he uni roo null in favour of a saionary process for 4 ou of 7 -monh forward premium series. For he 3-monh forward premium series, we rejec he uni roo null for 6 ou of he 7 cases considered. This is an imporan resul since i indicaes ha once allowance is made for nonlineariies we are able o rejec he uni roo null hypohesis for he forward premium. This would explain why we were unable o rejec he uni roo null hypohesis using linear univariae and panel uni roo esing echniques. 6

17 For he chosen delay parameer we repor in ables 3 and 4 he leas squares esimaes of he TAR models for each forward premium series. For he US 3-monh forward premium, in paricular, he poin esimae of he hreshold is This value indicaes ha he TAR splis he regression funcion depending on wheher he change in he forward premium lies above he hreshold of in absolue erms. If he change in he forward premium in he pas 7 monhs is above he hreshold in absolue erms we swich o he second regime. The firs regime (ouside he band) behaves as a saionary process and has 8.9% of observaions, whereas he second regime (inside he band) is a random walk wih a drif and has 8.% of observaions 7. Overall, he US 3-monh forward premium is shown o be globally saionary since we are able o rejec he uni roo null for he R T wih a 5.78% boosrap p-value Compued Half-lives of Deviaions 8 In able 4 we repor he half-lives of deviaions from he relaionship beween spo and forward exchange raes. The half-lives are defined as he number of monhs i akes for deviaions o subside permanenly below 50% in response o a uni shock in he level of he series and are compued because hey essenially provide a measure of he degree of meanreversion. In he firs wo columns we repor he half-lives esimaed using he linear auoregressive model and for he wo mauriy periods i.e. and 3 monhs. I is clear from hese wo columns ha he speed of mean-reversion is very slow. The esimaes of he halflives vary from a low of 4.6 monhs o a high of 5.77 monhs. In he remaining columns we repor he half-lives esimaed for he nonlinear TAR models. The half-lives are compued for he firs regime (ouside he band) and he second regime (inside he band). The values of he half-lives indicae ha deviaions ouside he band of inacion are 7 Given ha mos observaions lie wihin he uni roo regime, his would explain why sandard univariae and panel uni roo ess fail o rejec he null hypohesis. 8 Alhough i would have been useful o compare our resuls wih oher sudies, we could no find any previous research which compued he half-life of deviaions from he relaionship beween forward and spo exchange raes using eiher linear or nonlinear models. Therefore, ou resuls represen an imporan conribuion. 7

18 correced or die ou very quickly and he speed of adjusmen is as low as 0.80 monhs. The esimaes for he half-lives ouside he band vary from 0.80 monhs o 4 monhs and his is much faser han he values we obained using he linear model. As for he inner regime, adjusmen is exremely slow. However, his is no surprising since he inner regime behaves essenially as a uni roo process and is roo is insignificanly differen from zero in mos cases. In general, i is only when he forward premium is in he neighbourhood of is long run equilibrium level do he nonlinear TAR models yield very slow speeds of adjusmen. For large deviaions, arbirage kicks in and adjusmen is almos insananeous. Indeed, hese half-lives are very fas when compared o he kind of esimaes found in linear exchange rae lieraure (for example he PPP lieraure). 8

19 5. Conclusion This paper addresses he issue of conflicing resuls associaed wih sudies rying o deec mean-reversion in he forward premium. I is argued ha he negaive resuls are due o nonlineariies in he adjusmen owards he long run. These sem from marke fricions such as ransacion coss. However, unlike previous sudies, which esed for nonlineariies only, we propose an approach ha allows he join esing of nonlineariy and nonsaionariy. Using univariae hreshold auoregressive (TAR) uni roo and boosrap ess, which allow for he possibiliy of nonlinear adjusmen, we find evidence o suppor he hypohesis ha he forward premium is a saionary TAR process. Specifically, he forward premium follows a nonlinear process and is found o display uni roo behaviour when i is inside he ransacion cos band whereas ouside his band he forward premium is mean-revering. The speed of adjusmen ouside he band is much faser han inside he band. These resuls indicae ha once we allow for he possibiliy of nonlinear adjusmen we are able o deec rapid mean-reversion of he forward premium. Overall, here are wo significan conclusions ha we can draw from he empirical analysis in his paper. Firs, since he forward premium has been found o be nonlinearly mean-revering a corollary of his is ha he forward marke forecas error / excess reurn is also a nonlinear mean-revering process. This provides evidence in favour of he forward rae unbiasedness hypohesis (FRUH) and also confirms he validiy of he uncovered ineres rae pariy (UIP) hypohesis which requires a saionary excess reurn. Second, he domesic and foreign nominal ineres rae differenials are saionary TAR processes, given he covered ineres rae pariy (CIP) condiion. 9

20 6. References Baillie, R.T. (994), The Long Memory of he Forward Premium, Journal of Inernaional Money and Finance,, pp Baillie, R.T. (996), Long Memory Processes and Fracional Inegraion in Economerics, Journal of Economerics, 73, pp Baillie, R.T and T. Bollerslev (994), The Long Memory of he Forward Premium, Journal of Inernaional Money and Finance, 3, pp Baillie, R.T and T. Bollerslev (000), The Forward Premium Anomaly is no as Bad as you Think, Journal of Inernaional Money and Finance, 9, pp Barkoulas, J., C.F. Baum and A. Chakrabory (003), Forward Premiums and Marke Efficiency: Panel Uni-roo Evidence from he Term Srucure of Forward Premiums, Journal of Macroeconomics, 5, pp Basci, E and M. Caner (00), Are Real Exchange Raes Nonlinear or Nonsaionary? Evidence from a New Threshold Uni Roo es, Bilken Universiy. Baum, C.F., J.T. Barkoulas and M. Caglayan (00), Non-linear Adjusmen o Purchasing Power Pariy in he pos-breon Woods Era, Deparmen of Economics, Boson College. Bec, F., M. Ben Salem and R. MacDonald (00), Real Exchange Raes and Real Ineres Raes: A Nonlinear Perspecive, CREST- ENSAE. Bilson, J.F.O. (98), The Speculaive Efficiency Hypohesis, Journal of Business, 54, pp Byers, J.D and D.A. Peel (996), Long Memory Risk Premia in Exchange Raes, Mancheser School, 64, pp Caner, M and B.E. Hansen (00), Threshold Auoregression wih a Uni Roo, Economerica, 69, pp Clarida, R.H and M.P. Taylor (997), The Term Srucure Of Forward Exchange Premiums and he Forecasabiliy of Spo Exchange Raes: Correcing he Errors, The Review of Economics & Saisics, 79, pp Clemens, M.P and J. Smih (00), Evaluaing Forecass from SETAR models of Exchange Raes, Journal of Inernaional Money and Finance, 0, pp

21 Coakley, J and A.M. Fueres (00), Exchange Rae Overshooing and he Forward Premium Puzzle, Ciy Universiy Business School. Coakley, J and A.M. Fueres (00), Rehinking he Forward Premium Puzzle in a Nonlinear Framework, Ciy Universiy Business School. Cornell, B. (977), Spo Raes, Forward Raes and Exchange Raes Efficiency, Journal of Financial Economics, 5, pp Crowder, W.J. (994), Foreign Exchange Marke Efficiency and Common Sochasic Trends, Journal of Inernaional Money and Finance, 3, pp Darba, G and U.R. Pael (00), Nonlinear Adjusmen in Real Exchange Raes and Long Run Purchasing Power Pariy- Furher Evidence, NSEI, India. Delcoure, N., J. Barkoulas., C.F. Baum and A. Chakrabory (000), The Forward Rae Unbiasedness Hypohesis Revisied: Evidence from a New Tes, forhcoming, Global Finance Journal. Dumas, B. (99), Dynamic Equilibrium and he Real Exchange Rae in a Spaially Separaed World, Review of Financial Sudies, 5, pp Enders, W and S. Dibooglu (00), Long Run Purchasing Power Pariy wih Asymmeric Adjusmen, Souhern Economic Journal, 68, pp Engel, C. (996), The Forward Discoun Anomaly and he Risk Premium: A Survey of Recen Evidence, Journal of Empirical Finance 3, pp Engle, R.F. and C.W.J. Granger (987), Co-inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, 55, pp Evans, M.D and K.K. Lewis (995), Do Long-Term Swings in he Dollar affec esimaed of he risk premia?, Review of Financial Sudies, 8, pp Fama, E.F. (984), Forward and Spo Exchange Raes, Journal of Moneary Economics, 4, pp Frenkel, J. (977), The Forward Exchange Rae, Expecaions and he Demand for Money- The German Hyperinflaion: Reply, American Economic Review, 70, pp Fueres, A-M and J. Coakley (00), A Nonlinear Analysis of Excess Foreign Exchange Reurns, The Mancheser School, 69, pp Granger, C.W.J and P. Newbold (974), Spurious Regressions in Economerics, Journal of Economerics,, pp. -0.

22 Grossman, S and R. Shiller (98), The Deerminans of he Variabiliy of Sock Marke Prices, American Economic Review, 7, pp Hadri, K. (000), Tesing for Saionariy in Heerogeneous Panel Daa, Economerics Journal, 3:, pp Hai, W., N.C. Mark and W. Yangru (997), Undersanding Spo and Forward Exchange Rae Regressions, Journal of Applied Economerics, pp Hakkio, C.S and M. Rush (989), Marke Efficiency and Coinegraion: An Applicaion o he Serling and Deuschemark Exchange Raes, Journal of Inernaional Money and Finance, 8, pp Hansen, L.P and R.J. Hodrick (983), Risk Averse Speculaion in he Forward Foreign Exchange Marke: A Economeric Analysis of Linear Models, In Jacob A. Frenkel, ed. Exchange Raes and Inernaional Macroeconomics (Universiy of Chicago Press, Chicago, IL). Horvah, M.T.K and M.W. Wason (995), Tesing for Coinegraion When Some of he Coinegraing Vecors are Prespecified, Economeric Theory,, pp Huner, J and M. Simpson (00), A Panel Tes of Purchasing Power Pariy under he Null of Saionariy, Deparmen of Economics, Brunel Universiy. Im, K. S., M.H. Pesaran and Y. Shin (997), Tesing for Uni Roos in Heerogeneous Panels, mimeo. Kilian, L and M.P. Taylor (00), Why is i so Difficul o Bea he Random Walk Forecas of Exchange Raes?, mimeo, Universiy of Michigan and Universiy of Warwick. Kuan, A.M and S. Zhou (00), Has he Link Beween he Spo and Forward Exchange Raes Broken Down? Evidence from Rolling Coinegraion Tess, Deparmen of Economics and Finance, Souhern Illinois Universiy. Lohian, J.R and M.P. Taylor (997), Real Exchange Rae Behaviour: The Problem of Power and Sample Size, Journal of Inernaional Money and Finance, 6, pp Luinel, K.B and K. Paudyal (998), Common Sochasic Trends Beween Forward and Spo Exchange Raes, Journal of Inernaional Money and Finance, 7, pp Maddala, G. S and S. Wu (999), A Comparaive Sudy of Uni Roo Tess wih Panel Daa and a New Simple Tes, Oxford Bulleing of Economics and Saisics, Special Issue, 6, pp Meese, R.A and A.K. Rose (99), An Empirical Assessmen of Non-lineariies in Models of Exchange Rae Deerminaion, Review of Economic Sudies, 58, pp

23 Naka, A and G. Whiney (995), The Unbiased Forward Rae Hypohesis Re-examined, Journal of Inernaional Money and Finance, 4, pp Obsfeld, M and A.M. Taylor (997), Nonlinear Aspecs of Goods-Marke Arbirage and Adjusmen: Heckscher's Commodiy Poins Revisied, Cenre for Inernaional and Developmen Economics Research (CIDER) Working Papers C97-088, Universiy of California a Berkeley. Pedroni, P. (999), Criical Values for Coinegraion Tess in Heerogeneous Panels wih Muliple Regressors, Oxford Bulleing of Economics and Saisics, Special Issue, 6, pp Sekioua, S.H. (003a), The Nominal Exchange Rae and Fundamenals: New Evidence from Threshold Uni Roo Tess, forhcoming in he Finance Leers. Sarno, L and M.P. Taylor (998), Real Exchange Raes under he Recen Floa: Unequivocal Evidence of Mean Reversion, Economics Leers, 60, pp Taylor, A. (00), Poenial Pifalls for he PPP Puzzle? Sampling and Specificaion Biases in Mean Reversion Tess of he LOOP, Economerica, 69, pp Taylor, M.P. (995), The Economics of Exchange Raes, Journal of Economic Lieraure, 33, Taylor, M.P and H. Allen (99), The Use of Technical Analysis in he Foreign Exchange Marke, Journal of Inernaional Money and Finance,, pp Taylor, M.P and D.A. Peel (000), Nonlinear Adjusmen, Long-run Equilibrium and Exchange Rae Fundamenals, Journal of Inernaional Money and Finance, 9, pp Taylor, M.P., D.A. Peel and L. Sarno (00), Nonlinear Mean Reversion in Real Exchange Raes: Towards a Soluion o he Purchasing Power Puzzles, Inernaional Economics Review, 4, pp Taylor, M.P and L. Sarno (998), The Behaviour of Real Exchange Raes During he Pos-Breon Woods Period, Journal of Inernaional Economics, 46, pp Zivo, E. (000), Coinegraion and Forward and Spo Exchange Rae Regressions, Journal of Inernaional Money and Finance 9, pp

24 7. Empirical Resuls Table Descripive s for forward exchange rae premium series. USA Swizerland Japan Canada France Germany Ialy USA Swizerland Japan Canada France Germany Ialy k 3 N Mean Maximum Minimum S.D Skewness Kurosis Jarque-Bera ** (0.0344) 8.980** (0.06) 6.345** (0.0465).4640* (0.0000) * (0.0000) ** (0.0303) * (0.0000).66 (0.696) * (0.0079) (0.3) * (0.0000) * (0.0000) ADF: Tes (0.3535) -.896** (0.0478) ** (0.000) *** (0.0637) -.30 (0.7) -.93 (0.754) -.57 (0.495) (0.46) *** (0.0789) * (0.0064) (0.054) (0.753) (0.866) (0.4565) ρ (0.037) (0.06) (0.043) (0.0389) (0.048) (0.037) (0.045) (0.077) (0.03) (0.0397) (0.038) (0.037) (0.090) (0.09) Half-life Correlaions USA Swizerland Japan Canada France Germany Ialy Noe: ρ is he roo of he auoregressive (AR) model. Criical values for he ADF es wih an inercep are (%), (5%) and (0%). *, ** and *** indicae significance a %, 5% and 0% ** (0.083) * (0.0000) 4

25 Table Maddala and Wu (999) uni roo es based on he p-values of he ADF uni roo es. Level ADF - p-value ADF - p-value ADF - p-value ADF - p-value k 3 USA Swizerland -.896** *** Japan ** * Canada *** France Germany Ialy Maddala and Wu es 3.838* (0.005) (0.36) 8.944** (0.006) (0.843) Degrees of Freedom Noe: Criical values for he ADF es wih an inercep are (%), (5%) and (0%). The asympoic disribuion of he Maddala and Wu (MW) es is χ (N). *, ** and *** indicae significance a %, 5% and 0%. Criical Values d.f 0.% %.5% 5% 0%

26 Table 3 Hadri (000) panel saionariy es for heerogeneous panels. Tes, k= Tes, k=3 Inercep: LM ().660* (0.0043) (0.09) Inercep: LM () * (0.0000) 8.337* (0.0000) Trend and Inercep: LM (3).6883** (0.0457) (0.606) Trend and Inercep: LM (4) * (0.0000) * (0.0000) Noe: LM () denoes he value of he one-sided es LM es, esing for saionariy around an inercep wih serially correced errors. LM () denoes he value of he one-sided es LM es, esing for saionariy around an inercep. LM (3) denoes he value of he one-sided es LM es, esing for saionariy around an inercep and a ime rend wih serially correced errors. LM (4) denoes he value of he one-sided es LM es, esing for saionariy around an inercep and a ime rend. *, ** and *** indicae rejecion of he saionariy null hypohesis a %, 5% and 0%. 6

27 Table 4 Threshold ess wih fixed delay parameer, m. -monh forward exchange rae premium series. k= USA Swizerland Japan Canada France Germany Ialy Delay Order Wald Saisic Saisic p-value Wald Saisic Saisic p- value Wald Saisic Saisic p- value Wald Saisic Saisic p- value Wald Saisic Saisic p-value Wald Saisic Saisic p- value Wald Saisic Saisic p- value * * * * * * * * * * ** * * ** ** * * * * * * * * ** * * ** ** * * * * * ** * * ** * * * * * * * * *** ** * * ** * * *** ** * * * ** * * ** * * * * *** * * * * * ** * *** * * * Table 5 % Boosrap criical values USA Swizerland Japan Canada France Germany Ialy m % C.V. % C.V. % C.V. % C.V. % C.V. % C.V. % C.V Noe: Boosrap criical values calculaed from 0000 replicaions. 7

28 Table 6 Threshold ess wih fixed delay parameer, m. 3-monh forward exchange rae premium series. k=3 USA Swizerland Japan Canada France Germany Ialy Delay Order Wald Saisic Saisic p- value Wald Saisic Saisic p- value Wald Saisic Saisic p- value Wald Saisic Saisic p- value Wald Saisic Saisic p-value Wald Saisic Saisic p- value Wald Saisic Saisic p- value * *** ** * * * * * ** * * * ** * * * * *** * ** * * * * ** * * * ** ** *** * ** * * * * * * * * * ** *** * * * * * *** ** * * * * * ** ** * * * * * *** * * * * ** ** *** * * * ** * *** * * * Table 7 % Boosrap criical values USA Swizerland Japan Canada France Germany Ialy m % C.V. % C.V. % C.V. % C.V. % C.V. % C.V. % C.V Noe: Boosrap criical values calculaed from 0000 replicaions. 8