I S T H E A U S T R A L I A N F O R E X M A R K E T E F F I C I E N T? A T E S T O F T H E F O R W A R D R A T E

Transcription

1 I S T H E A U S T R A L I A N F O R E X M A R K E T E F F I C I E N T? A T E S T O F T H E F O R W A R D R A T E U N B I A S E D N E S S H Y P O T H E S I S By D.E. Allen and Paul Taco School o Accouning, Finance and Economic and FEMARC Working Paper Serie Edih Cowan Univeriy June 2007 Working Paper 0706 Correpondence auhor: David E. Allen School o Accouning, Finance and Economic Faculy o Buine and Law Edih Cowan Univeriy Joondalup, WA 6027 Auralia Phone: Fax: d.allen@ecu.edu.au_ 1 School o Accouning, Finance and Economic, Edih Cowan Univeriy, Joondalup Drive, Joondalup. Tel , Fax , d.allen@ecu.edu.au

2 Abrac Thi paper eaure a e o he orward rae unbiaedne hypohei (FRUH), uing he Auralian dollar wih he Unied Sae and Japanee currencie uing daily requencie. We evaluae he FRUH on he 1-monh orward rae, or boh currencie, and he 3-monh and 6-monh orward rae or he US dollar only. We adop a coinegraion ramework or aeing he FRUH applying a coinegraing VAR model involving Johanen ML approach. Our reul indicae ha in all cae he po and orward rae are inegraed o order 1. Furhermore here i evidence o coinegraion and in all bu one cae he coinegraing vecor i (1, -1). The error correcion erm in all cae i aiically igniican and ha he correc ign. JEL Code: F31 G13 Keyword: Inere pariy, exchange rae, marke eiciency, coinegraion 2

3 1. Inroducion In hi paper we e he heory o he Forward Rae Unbiaedne Hypohei (FRUH); a heory ha underpin much o he work done on he oreign exchange (FOREX) marke. Thee marke coninually grow in igniicance; he Reerve Bank o Auralia (RBA) repored ha he ize o he oreign exchange in Auralia had grown o he evenh large in he world by Some udie o exchange rae by Huang (1981), Vander Kraa and Booh (1983) and Wadhwani (1987) have uggeed ha exchange rae are oo volaile wih repec o he behaviour o heir underlying deerminan. The iue o exchange rae volailiy can be viewed a an eicien marke iue. I exchange rae are oo volaile wih repec o a reaonable benchmark in an eicien marke, hen here may be ground or hrowing and in he gear o currency marke, o a o low down change in exchange rae and keep hem in line wih hoe o heir undamenal (Eichengreen, Tobin e al. 1995). Converely, i exchange rae volailiy i largely conien wih ha prediced by convenional model, hen one may imply have o wallow he rae abrup wing a he poenially eicien repone o underlying hock (Barolini and Giorgianni 2001). Conequenly hi i a criical iue or policy maker, a exchange rae volailiy ha been linked o growing rade imbalance, increaed inancial marke volailiy and le eecive domeic macroeconomic policie (Kahn 2000). In hi udy we analye he inere rae and exchange rae relaionhip, and explore how i aec he orward and po rae o Auralian denominaed currency. We e wheher he Auralian Foreign Exchange i an eicien marke in he conex o a coinegraing relaionhip beween orward rae and po rae. Prior work in hi 3

4 area include Hakkio and Ruh (1989), Felmingham and Leong (2003) and Zivo (1998). The paper i divided ino ix ecion; a review o pariy condiion ollow in ecion 2, ecion hree exend he heoreical ramework and ecion our inroduce he empirical mehod adoped and he daa e. Secion ive review he reul and a brie concluion ollow in ecion ix. 2. Inere pariy condiion Economic agen have a choice beween holding domeic-currency ae, which yield he own rae o inere r d, or ae denominaed in oreign currency, which yield he own rae o inere. To he exen ha inveor can accumulae eiher (1+ r d ) or (1+ r )/ uni o domeic currency wih cerainy, arbirageur in purui o aured proi will move und in whaever amoun are required o eliminae any dicrepancie beween hee inere acor. Thu an inere-rae pariy condiion i creaed, hi condiion aer an ae-marke equilibrium, where (1+ r d ) = (1+ r )/, which implie ( ) = ( + r ) ( 1+ r ) 1 = ( r rd ) ( 1+ rd ) r rd 1...(1) In oher word, he percenage orward premium on domeic currency i.e. he percenage by which he orward price o domeic currency exceed he po price will equilibrae o he exce o he oreign inere rae over he domeic inere rae (Iard, 1978). Arbirageur hould preven large dicrepancie beween orward exchange rae and po rae ha are expeced o prevail on he dae on which orward conrac maured. 4

5 Covered inere pariy An alernaive way o expreing (1.1) i uing a covered inere pariy analyi, accouning or oreign exchange rik. Thi rik i ha he uure po exchange rae may no equal he expeced uure po rae. By purchaing a orward conrac, he individual guaranee a rae o exchange a mauriy, hu eliminaing all oreign exchange uncerainy. Agen, who cover oreign exchange rik wih orward conrac, hould be indieren beween wo dieren counrie inancial inrumen. Thu: ( )( ) 1 + r = 1+ d r ( ) ( ) + ( ) = + ( ) = 1, rewriing [ 1+ ( ) ]( ) 1 + r = 1+, impliying d r 1 + r = 1+ d ( ) + r + r ( ) becaue r and ( ) are boh ypically mall racion, heir produc i approx. 0, hu impliying r d r Again equaion 1.1 i reproduced, wih an imporan implicaion ha inroduce he parnering inere rae condiion o Uncovered inere pariy (UIP). Uncovered inere pariy 5

6 Conrary o CIP, here i no aemp o hedge oreign exchanger rik uing orward conrac, leaving oreign ranacion uncovered and hinging on expecaion o he uure po rae. Thi creae he uncovered inere pariy (UIP), (Daniel and VanHooe 2001). Thi can be ummaried a: r d r e +1.(2) Furhermore, becaue he ranacion i uncovered, here i no eec on he orward ( e ) marke, he eec on he nex period po exchange rae 1 +, would be he ame a on he orward marke in CIP. I i eviden ha he wo pariy condiion are linked hrough inere rae dierenial, uppoe ha boh condiion are aiied, hu: rd e r (3) Thee condiion imply ha he orward premium i equal o he change in he po rae, and impliying we obain he ollowing condiion: = e (4) Thi ignal ha he orward exchange rae hould equal he po rae expeced o prevail a he ime o he elemen o he orward conrac. I he orward rae yemaically dier rom he po rae, hen in he abence o a rik premium, a proi opporuniy exi. The implicaion o combining an eicien marke wih condiion (4) i ha he orward exchange rae hould on average equal he expeced uure po exchange rae. In oher word, he orward exchange rae i an unbiaed predicor o he 6

7 expeced uure po rae. In he ollowing analyi I he orward exchange rae or +1 i an unbiaed raional expecaion o he rae a +1, coinegraion o he orward and uure po rae ollow imply (Dwyer and Wallace 1992). 3. The heoreical ramework Thi udy analye he orward rae unbiaedne hypohei uing he Auralian dollar wih he Unied Sae dollar and he Japanee yen uing a coinegraion ramework. Japan i currenly Auralia large rading parner and he Unied-Sae i Auralia hird large rading parner. We echewed he ue o China given ha i currency i managed o a degree and no ubjec o he ame marke orce. There are numerou previou Sudie eing he FRUH; Engle (1996) provide a review, neverhele only a ew have analyzed he relaionhip beween po and orward rae uing Auralian daa. The mo recen udy done wa by Felmingham and Leong (2003) who ued Auralian and US daily daa or 90 and 180-day orward marke or he period o 1985 o Felmingham and Leong (2003) indicae ha orward rae aced a an unbiaed predicor o he po rae or he period uing AUD/USD daa bu only in he 90-day marke. However, when hey conider he cah rae imulaneouly, he orward rae in he 180-day marke i alo unbiaed. Thu hey conclude ha he predicabiliy o he po exchange rae hrough he orward rae ha improved over ime. In hi udy, we will examine boh exchange rae uing Auralia a he denominaed currency. Our daa will be daily and he mo recen, or example he AUD/USD pan he la hree year, rom Jan 03 o Dec 06. Furhermore, due o he growh in coinegraing VAR analyi and he ramework provided by Zivo (1998) we have new mehod or eing he FRUH, which will be applied in hi paper. 7

8 Hakkio and Ruh (1989) eablih ha hiorically here have been wo general approache o eing he orward rae unbiaedne hypohei (FRUH); he ir i he level regreion. In hi regreion, reearche regre he uure por rae, + 1, on he orward rae,,, a in equaion (5). = a + b + e +1.. (5) Early reearche like Frenkel (1976, 1979) ound ha b eimae were very cloe o 1 and hence uppored he FRUH. However ubequen auhor uch a Meee and Singleon (1982), have criicied hi approach becaue o he poenial nonaionariy o he po and orward rae, where by boh erie ollow uni roo procee and hi implie ha po and orward rae are coinegraion. Thi implie poenial puriou regreion problem a decribed in Granger and Newbold (1974). The econd approach o eing eiciency i he dierence equaion, hi mehod regree he rae o depreciaion on he orward premium, a in equaion (6) [ ] u +1 = a + b + (6) Empirical udie uch a Bilon (1981), Hanen and Hodrick (1980) and Huang (1981) have ound ha under hi approach he FRUH wa overwhelmingly rejeced. Furhermore ypical reul o he eimae b acro a wide range o currencie and ampling requencie are ound o be igniicanly negaive. Thee reul are hereore reerred o a he orward dicoun anomaly, orward dicoun bia or orward dicoun puzzle and eem o conradic he reul baed on he level regreion. 8

9 Hakkio (1981) and Baillie, Lippen, and McMahon (1983) in he ligh o hee deiciencie and new developmen in he heory o coinegraion by Engle and Granger (1987) ormed a hird approach. Marke eiciency implie ha even i he po and orward rae are non-aionary, hey never dri apar o ha hey will be coinegraed (Hakkio and Ruh 1989). The coinegraing relaionhip beween po and orward rae, in eiher he level or dierence approach, ha been examined by many including Hakkio and Ruh (1989), Barhar and Szakmary (1991), Naka and Whiney (1995), Hai, Mark and Yu (1997), and Clarida and Taylor (1997). However reul rom hee udie were mixed and rongly dependen on he how coinegraion relaionhip wa modelled. An imporan inigh rom all he coinegraion udie wa he recogniion ha he FRUH require ha + 1 and or and be coinegraed wih a coinegraing vecor o (1,-1). Moreover Zivo (1997) poin ou ha ince: + 1 = + 1 ( ) I i rivial o ee under he aumpion ha and are I(1) ha (i) i and are coinegraed wih coinegraing vecor (1,-1) hen + 1 and mu be coinegraed wih coinegraing vecor (1,-1); and (ii) i + 1 and are coinegraed wih coinegraed vecor (1,-1) hen and mu be coinegraed wih coinegraing vecor (1,-1) (Zivo 1997). However imple ir order vecor error correcion model ha ue +, ) mu be ued wih cauion, a hey mi ome imporan dynamic in ( 1 monhly daa and a a reul indicae ha he FRUH appear o hold. 4 The Reearch Mehod and daa e 9

10 We commence by underaking uni roo e o deermine wheher he erie are nonaionary uing he Augmened Dickey-Fuller (ADF) e recommended by Engle and Granger (1987). We hen proceed o capure dynamic uing a imple y ( ) coinegraed VAR(1) model or =, and hu inerence rom hi model are rue and reliable. We ollow Zivo (1998) wriing a bivariae VAR(1) model or y, which i: y = + Φy 1 µ +..(7) y = A +.(8) g where y i an m 1 vecor o joinly deermined (endogenou) variable, and i an m 1 vecor o unoberved diurbance aumed o aiy he ollowing aumpion: Having eablihed VAR(1) model, we can alo wrie he yem o equaion in (7) a a SURE model (8) wih all he equaion having he ame e o regreor, g = y, y, K, y ), in common. Thi i done o calculae he Maximum ( 1 2 p Likelihood (ML) eimaor o he unknown coeicien when regreing y on g and urher o calculae he VAR order elecion. Firly we mu wrie (8) in marix noaion o have Y G = n m n A +..(9) m n m where: = mp + 2; 10

11 Y n m = ( y, y, ) 2, 1 K y n ; A = m ( µ, Φ) ; g = n ( mp + 2) ( ι,, Y, Y,, Y ) n n 1 2 K p vecor (1,1,,1) and (1,2,,n), repecively., where ι n and n are he n-dimenional The ML eimaor o A and Σ are given by A ˆ 1 = ( G G) G Z..(10) and ~ 1 = n ( Y GAˆ) ( Y GAˆ)..(11) Thee equaion will be ued when elecing he VAR order elecion crierion, in paricular when uing he Akaike Inormaion Crierion (AIC) and he Schwarz Bayeian Crierion (SBC); Reurning o he our VAR(1) model, and uing he analyi rom (Zivo, 1998) we calculae he coinegraing VAR model, by rewriing equaion (7) a y = µ + y 1 +.(12) where = Φ I, we are able o ummarize he long-run inormaion in y by he long-run impac marix, ; i i he rank o hi marix ha deermine he number o coinegraing vecor. In our cae he rank i 1 and here exi

12 vecor α and β uch ha = α β. Uing he normalizaion β ( 1 β ), (2.7) = become he vecor error correcion model (VECM): = µ + α ( 1 β 1 ) + (12a) = µ + α ( 1 β 1 ) +.(12b) Since po and orward rae do no exhibi a yemaic endency o dri up or down i may be more appropriae o reric he inercep in (12) o he error correcion erm. Tha i, µ α = µ c and µ α = µ c. Under hi rericion and are I(1) wihou dri and he coinegraing reidual, β, i allowed o have a µ nonzero mean c. Wih he inercep in (12) rericed o he error correcion erm, he VECM can be olved o give a imple AR(1) model or he coinegraing reidual β y µ c = β µ c. Pre-muliplying (9) by β and rearranging give: ( β 1 µ c ) β µ c = Φ 1 + η (13) Φ = I + α I ( α β α ) where β = + η = β = β and. Since (13) i imply an AR (1) model, he coinegraing reidual i able and aionary i Φ = 1+ ( α β α ) I(1) hu and are no coinegraed. < 1. Noice ha i a = β a hen he coinegraing reidual i Anoher imporan iue wih hi model i he exogeneiy au o po and orward rae wih regard o he coinegraing parameer α and β. Thi wa he 12

13 ocu o many auhor, who were concerned wih exogeneiy iue in error correcion model, uch a Johanen (1992, 1995) and Zivo (1998). In hi model, exogeneiy o po and orward rae place rericion on he parameer o he VECM (12), in paricular i i weakly exogenou wih repec o ( ) α β. I hi i he cae hen α = 0 and eicien eimaion o he coinegraing parameer can be made rom he ingle equaion condiional error correcion model, ( β 1 ) + γ ν = µ + α 1 +..(13a) where γ 1 = σ σ and ν i uncorrelaed wih. Thi i alo he cae i i weakly exogenou wih repec o ( ) α hen α = 0 and eicien eimaion o he coinegraing parameer can be made rom he ingle equaion condiional error correcion model, β ( β 1 ) + γ j ν = µ + α 1 +..(13b) where γ 1 = σ σ and ν i uncorrelaed wih. I β = 1 hen he orward premium i I(0) and ollow an AR(1) proce and he VECM (12) become, ( 1 1 ) + = µ + α,..(14a) ( 1 1 ) + = µ + α (14b) Noice ha (14b) i imply he andard dierence regreion ued o e he FRUH. Furhermore, i α j and α are o he ame ign and magniude hen he 13

14 implied value o Φ in (13) i cloe o 1 and hi correpond o he ylized ac decribed above, in paricular where he orward premium i aionary bu very highly auocorrelaed. Alo, he implied variance o he orward premium rom (12) i σ ηη and 1 ( σ σ ) 2 = σ + σ ρ and will be very mall relaive o he variance o 2 given he ylized ac ha σ σ and ρ 1. The FRUH place eable rericion on he VECM (). Neceary condiion or he FRUH o hold are (i) and are coinegraed (ii) β = 1 and (iii) µ = 0. c In addiion, he FRUH require ha α = 1 in order or he oreca error o have condiional mean zero. Togeher hee wo rericion limi boh he long-run and hor-run behaviour o po and orward rae. Applying hee rericion, (12), one period ahead, become, ( ) +, 1 α.(15a) + 1 = + ( ) +, = +..(15b) Noice ha he FRUH require ha he expeced change po rae i equal o he orward premium or, equivalenly, ha he adjumen o long-run equilibrium occur in one period. The change in he orward rae, on he oher hand, i direcly relaed o he perience o inere rae dierenial now meaured by ( α I ) = α α ince Φ = I +. Sabiliy o he VECM under he FRUH require ha α < 1. Thu, he FRUH i conien wih a highly perien orward premium. The repreenaion in (15) how ha weak exogeneiy o po rae wih repec o he coinegraing parameer i inconien wih he FRUH becaue i po 14

15 rae are weakly exogenou hen α = 0 and he FRUH canno hold. In addiion i he FRUH i rue and orward rae are weakly exogenou hen (15) canno capure he dynamic o ypical daa. To ee hi, uppoe ha orward rae are weakly 2 exogenou o ha α = 0. Since σ σ σ = σ i ollow ha γ γ 1. I µ = 0, α = 1 and β = 1, hen (13) become = ν = + ν which imply ae ha he curren po rae i equal o he curren orward rae plu a whie noie error. We proceed by perorming he ADF e, hen e he hypohei o wheher he realized po and orward rae are coinegraed uing a vecor auoregreion (VAR) model. We ir run an unrericed VAR on Spo and orward rae, in order o elec and deermine he order o VAR. Uing he Akaike Inormaion Crierion (AIC) and Schwarz Bayeian Crierion (SBC), and hen acerain he number o coinegraing relaionhip; Our prior i one coinegraing relaion, and we can impoe i on he model, by peciying r (he number o coinegraing vecor). Thi hen allow u o ar our long-run rucural modelling analyi, Pearan and Smih (1996), in paricular he eing o he coinegraing vecor o (1,-1), which ha been exenively documened in previou udie. 15

16 Daa e We ue daily daa rom January 2003 o December The daily daa or AUD/JPY will be rom July 06 o December 06. The orward marke ued or AUD/USD analyi wa he 1-monh orward, 3-monh orward, and 6-monh orward rae, whil he AUD/JPY analyi ued on he 1-monh orward rae. All AUD/JPY daa wa obained rom Daaream, whil all AUD/USD daa wa obained rom he Reerve Bank o Auralia (RBA). 5. Reul We conduced ADF e, o examine wheher he ollowing po or orward rae are aionary or non-aionary. From he reul in Table 1 i i eviden ha he AUD/USD Spo and 1-monh orward are non-aionary, indeed all variable conirm he neceary condiion or coinegraion analyi. We conduced e o elec he order o VAR. In Table 2, where we evaluae he AUD/USD Spo and 1-monh orward rae, we can ee he highe value or AIC i and or SBC i which boh correpond o a VAR o he order o 1. Thi i alo conirmed in able 3 and 4, where we conider he AUD/USD 3-monh and 6-monh orward rae. Evaluaing Table 5, he AUD/JPY Spo and 1-monh orward rae, we alo come o he ame concluion where he AIC and SBC boh elec a VAR order o 1. Thi mean ha our model will be VAR(1) model, which i conien wih work by Zivo (1997). 16

17 Table 1: Augmened Dickey Fuller (ADF) Te or AUD/USD Spo and All Forward Rae Variable ADF Te Saiic ADF 95% Criical Signiican a 5% level Acual Exchange Rae Spo Rae No 1-Monh Forward Rae 3-Monh Forward Rae 6-Monh Forward Rae No No No 1 Dierence Exchange Rae Spo Rae Ye 1-Monh Forward Rae 3-Monh Forward Rae 6-Monh Forward Rae Ye Ye Ye Noe: Thi able diplay he ADF e aiic in abolue value, where he number o lag or each e i 1. Furhermore he 95% criical value, he null hypohei, o a uni roo, again he alernaive, o no uni roo. I a variable ha a non-igniican e reul a a 5% level o igniicance, i conain a uni roo and hu i non-aionary or inegraed. Table 8: ADF Te or AUD/JPY Spo and 1-Monh Forward Rae Variable ADF Te Saiic ADF 95% Criical Signiican a 5% level Acual Exchange Rae Spo Rae No 1-Monh Forward Rae No 1 Dierence Exchange Rae Spo Rae Ye 1-Monh Forward Rae Ye 17

18 Table 2: Unrericed VAR analyi AUD/USD Spo Rae and 1-Monh Forward Rae Order LL AIC SBC LR Te CHSQ (4) = [.138] CHSQ (8) = [.277] * * CHSQ (12) = [.154] CHSQ (16) = [.000] AIC=Akaike Inormaion Crierion SBC=Schwarz Bayeian Crierion Noe: Thee able are ued o ae he lag lengh in hi coinegraing VAR model; irly a hypoheied lag order o 1 o 4 i eleced. The reuling AIC and SBC model elecion crieria are compued or each correponding lag lengh. The order o VAR [VAR(p)] eleced correpond wih he highe value o he SBC and AIC. The highe AIC and SBC i denoed by *. Table 3: Unrericed VAR analyi AUD/USD Spo Rae and 3-Monh Forward Rae Order LL AIC SBC LR Te CHSQ (4) = [.164] CHSQ (8) = [.463] * * CHSQ (12) = [.567] CHSQ (16) = [.000] AIC=Akaike Inormaion Crierion SBC=Schwarz Bayeian Crierion Table 4: Unrericed VAR Analyi AUD/USD Spo Rae and 6-Monh Forward Rae Order LL AIC SBC LR Te CHSQ (4) = [.198] CHSQ (8) = [.446] * * CHSQ (12) = [.647] CHSQ (16) = [.000] AIC=Akaike Inormaion Crierion SBC=Schwarz Bayeian Crierion Table 5: Unrericed VAR Analyi AUD/JPY Spo and 1-Monh Forward Rae Order LL AIC SBC LR Te CHSQ (4) = [.321] CHSQ (8) = [.381] * * CHSQ (12) = [.587] CHSQ (16) = [.000] AIC=Akaike Inormaion Crierion SBC=Schwarz Bayeian Crierion 18

19 Coinegraion o po and orward rae. Once he order o VAR i e o 1, we can hen deermine how many coinegraing relaionhip are preen beween Spo and he orward rae. Logic ugge ha i i unlikely ha a rend exi in he coinegraing relaionhip beween po and orward rae, hu he rericed inercep and no rend opion wa choen or hi model. The reul in able 6 how ha boh he Maximal eigenvalue and Trace aiic rongly rejec he hypohei ha here exi no coinegraion. However boh e airm ha here exi more han one coinegraion relaionhip. Converely looking a he Model Selecion Crieria only Schwarz Bayeian Crierion, ugge ha here exi 1 coinegraing relaionhip, and he oher ugge more han one. The reul in able 7 or he hree monh orward rae are imilar, wih only he SBC uggeing one coinegraing relaionhip whil he oher e all ugge wo or more. In Table 8, he Spo and 6-monh orward rae are he only AUD/USD e ha airm one coinegraing relaionhip, however no collecively. A only he Trace aiic rejec r = 0 (no coinegraion) and airm r = 1 (one coinegraion relaion), bu he Maximal Eigenvalue doe no rejec r = 0 and hu ugge no coinegraion exi beween he Spo and 6-monh orward rae. Now viewing Model Selecion Crieria, we again ee ha only he SBC uppor he Trace aiic inding, whil he oher crieria uppor eiher no-coinegraion or wo or more coinegraing relaion. Finally in able 9, he AUD/JPY Spo and 1-monh orward rae, how inconcluive reul. Boh he Maximal Eigenvalue and Trace Saiic how ha here exi no coinegraion (r = 0) a boh he 90% and 95% level o igniicance. Furhermore only he SBC in he Model Selecion Crieria uppor he hypohei o r = 1. 19

20 Table 6: Coinegraion o AUD/USD Spo and 1-Monh Forward Rae Coinegraion LR Te baed on Maximal Eigenvalue Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Coinegraion LR Te baed on Trace Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Choice o he Number o Coinegraing Relaion Uing Model Selecion Crieria Rank Maximized LL AIC SBC HQC r = r = * r = * * * AIC = Akaike Inormaion Crierion SBC = Schwarz Bayeian Crierion HQC = Hannan-Quinn Crierion Noe 1 : Thi able diplay he ollowing e and model elecion crieria ha i ued o deermine he appropriae number o coinegraing relaion ha are likely o exi among he I(1) variable. Noe 2 : The e compued are he Trace and Maximum Eigenvalue aiic, hey are ued or eing he rank o he long-run marix, Πy, a 0, 1, or 2 ogeher wih he relevan 90% and 95% criical value. Any igniican value a a 90% level will be denoe a *, and any igniican value a a 95% level will be denoed a **. Noe 3 : Thi able urher preen he maximized value o he log-likelihood uncion o he coinegraing VAR model, Akaike, Schwarz, and Hannan and Quinn model elecion crieria, or he dieren value o r, he rank o he long run marix, Πy. The highe value or each crieria i diplayed a *. Table 7: Coinegraion o AUD/USD Spo and 3-Monh Forward Rae Coinegraion LR Te baed on Maximal Eigenvalue Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Coinegraion LR Te baed on Trace Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Choice o he Number o Coinegraing Relaion Uing Model Selecion Crieria Rank Maximized LL AIC SBC HQC r = r = * r = * * * AIC = Akaike Inormaion Crierion SBC = Schwarz Bayeian Crierion HQC = Hannan-Quinn Crierion 20

21 Table 8: Coinegraion o AUD/USD Spo and 6-Monh Forward Rae Coinegraion LR Te baed on Maximal Eigenvalue Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Coinegraion LR Te baed on Trace Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = * Choice o he Number o Coinegraing Relaion Uing Model Selecion Crieria Rank Maximized LL AIC SBC HQC r = * * r = r = * * AIC = Akaike Inormaion Crierion SBC = Schwarz Bayeian Crierion HQC = Hannan-Quinn Crierion Table 9: Coinegraion o AUD/JPY Spo and 1-Monh Forward Rae Coinegraion LR Te baed on Maximal Eigenvalue Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Coinegraion LR Te baed on Trace Null Alernaive Saiic 95% Criical 90% Criical r = 0 r = r <= 1 r = Choice o he Number o Coinegraing Relaion Uing Model Selecion Crieria Rank Maximized LL AIC SBC HQC r = * * r = r = * * AIC = Akaike Inormaion Crierion SBC = Schwarz Bayeian Crierion HQC = Hannan-Quinn Crierion Thee reul are puzzling and diicul o quare wih exiing lieraure on FRUH. Moreover, here i a general conenu ha here exi 1 coinegraing relaionhip beween po and orward rae, combining hi wih he mixed reul rom hor daa ample, we will impoe r = 1 on he model. The ollowing eimae o he coinegraing coeicien are normalized on he coeicien o he po exchange rae. Thi i achieved by placing he rericion o A1 = 1 o he model, which i he Spo 21

22 rae or he ir variable in he coinegraing VAR. In he ollowing able we diplay he eimae o he orward rae and heir aympoic andard error in Panel A. Looking a able 10, we can ee ha he 1-monh orward rae i negaive and very cloe o one, which according o preceding lieraure and economic heory i preciely wha i hould be, hu impoing he urher rericion o [A2 = -1] hould only renghen hi relaionhip. Panel B in able 11 diplay hi urher rericion, and looking a he log-likelihood raio aiic or eing hi rericion, which i [0.943], conirm ha hi rericion canno be rejeced, a i i no aiically igniican and hu airm hi relaionhip. Table 11 panel A how ha he 3-monh orward rae i negaive and ju over one, which again i he highly repored (1, -1) coinegraing relaionhip. By impoing he urher exac rericion in panel B, we obain he ollowing loglikelihood raio aiic (0.353), which again i no igniican and hu hi relaionhip canno be rejeced. Table 12: panel (A), diplay he 6-monh orward rae a a very mall negaive, which i he only AUD/USD rae no cloe o one. Impoing he urher rericion diplayed in panel B, how he ollowing log-likelihood raio aiic (.026), which i aiically igniican a a 95% level, and hu provide evidence again he validiy o he coinegraing relaion (1, -1). Table 13: panel (A), how he 1-monh orward rae a poiive however no large number, again conradicory o he coinegraing relaion (1, -1). Impoing he over-ideniying rericion o A2=-1, diplayed in panel B, we obain a non-igniican log-likelihood raio aiic. Thi airm he coinegraion relaion beween po and orward rae. 22

23 Table 10: Rericion on AUD/USD Spo and 1-Monh Forward Rae Panel A A1 = -1 Spo (*None*) 1-Monh Forward ( ) Panel B A1 = -1; A2 = 1 Spo 1-Monh Forward LL Subjec o exacly ideniying LR e o Rericion (*None*) (*None*) CHSQ (1) = [0.943] Noe 1 : Panel A diplay he eimae o he VAR model ubjec o he general rericion on he conegraing coeicien, in hi cae he coeicien o he Spo rae i normalized o 1, he ir variable in he conegraing VAR. Furher he eimae o he coinegraing coeicien (1-Monh Forward rae) i diplayed along wih i aympoic andard error. Noe2: Panel B diplay he impoed over-ideniying rericion o A2=-1, where A2 and or he coeicien o he 1-Monh Forward rae. Thi rericion i eed uing a log-likelihood raio aiic denoed by CHSQ (1). Table 11: Rericion on AUD/USD Spo and 3-Monh Forward Rae Panel A A1 = -1 Spo (*None*) 1-Monh Forward (2.0072) Panel B A1 = -1; A2 = 1 Spo 1-Monh Forward LL Subjec o exacly ideniying LR e o Rericion (*None*) (*None*) CHSQ (1) = [.353] Table 12: Rericion on AUD/USD Spo and 6-Monh Forward Rae Panel A A1 = -1 Spo (*None*) 1-Monh Forward ( ) Panel B A1 = -1; A2 = 1 Spo 1-Monh Forward LL Subjec o exacly ideniying LR e o Rericion (*None*) (*None*) CHSQ (1) = [.026] 23

24 Table 13: Rericion on AUD/JPY Spo and 1-Monh Forward Rae Panel A A1 = -1 Spo (*None*) 1-Monh Forward ( ) Panel B A1 = -1; A2 = 1 Spo 1-Monh Forward LL Subjec o exacly ideniying LR e o Rericion (*None*) (*None*) CHSQ (1) = [.319] Error Correcion Model (ECM) eimaed by OLS baed on Coinegraing VAR (1) Model In hi ecion, we ae he error correcion orm o he relaion in he coinegraing VAR model. The reul diplayed in able 14, how he error correcion equaion and heir correponding e aiic or each AUD/USD 1-monh, 3-monh and 6- monh orward rae. In panel A, we can ee ha he error-correcion erm, ( ) i aiically igniican a he 95% level, hu reinorcing he exience o a long-run relaionhip beween Spo and 1-monh orward rae. The EC erm ha he correc negaive ign, bu i very mall, uggeing ha i converge very lowly o equilibrium, wih only 5.03% o he dicrepancy correced or each period. Furher he goodne o i aiic i , indicaing ha he model doe no explain he relaionhip well. From panel B, we ee ha he error-correcion erm, ( ) i again aiically igniican, and again uppor he exience o a long-run relaionhip beween he Spo and 3-monh orward rae. The EC erm ha he correc ign, and again i very mall. Finally rom panel C, we ee ha he errorcorrecion erm, ( ), i igniican, again upporing he long-run 24

25 relaionhip beween he Spo and 6-monh orward rae. The EC erm ha he correc ign, bu again i exremely mall. Similarly in Table 15, we ee ha he error-correcion erm, ( ), i igniican a he 95% level, uggeing a long-run relaionhip beween he AUD/JPY Spo and 1-monh orward rae and he EC erm ha he correc ign, bu again i very mall. Table 14: Error Correcion Model or AUD/USD Spo and Forward Rae Panel A The Error correcion model Sp = β Fwd + βec The E. Model Spo = -1(Forward Rae) (EC -1 ) Coeicien Sandard Error T-Raio [Prob.] [.000] R-Squared R-Bar-Squared DW Saiic Diagnoic Te Te aiic [p-value] A: Serial Correlaion CHSQ (1) = [.712] B: Funcional Form CHSQ (1) = E-4 [.994] C: Normaliy CHSQ (2) = [.000] D: Heer-ocedaiciy CHSQ (1) = [.002] Noe 1 : The diagnoic reul uilize he ollowing e: A: Lagrange muliplier e o reidual erial correlaion B: Ramey RESET e uing he quare o he ied value C: Baed on a e o kewne and kuroi o reidual D: Baed on he regreion o quared reidual on quared ied value Panel B The Error correcion model Sp = β Fwd + βec The E. Model Spo = -1(Forward Rae) (EC -1 ) Coeicien Sandard Error T-Raio [Prob.] [.000] R-Squared R-Bar-Squared DW Saiic Diagnoic Te Te aiic [p-value] A: Serial Correlaion CHSQ (1) = [.711] B: Funcional Form CHSQ (1) = [.739] C: Normaliy CHSQ (2) = [.000] D: Heer-ocedaiciy CHSQ (1) = [.594]

26 Panel C The Error correcion model Sp = β Fwd + βec The E. Model Spo = -1(Forward Rae) (EC -1 ) Coeicien Sandard Error T-Raio [Prob.] [.005] R-Squared R-Bar-Squared DW Saiic Diagnoic Te Te aiic [p-value] A: Serial Correlaion CHSQ (1) = [.700] B: Funcional Form CHSQ (1) = [.205] C: Normaliy CHSQ (2) = [.000] D: Heer-ocedaiciy CHSQ (1) = [.585] 1 Table 15: Error Correcion Model or AUD/JPY Spo and 1-Monh Forward Rae The Error correcion model Sp = β Fwd + βec The E. Model Spo = -1(Forward Rae) (EC -1 ) Coeicien Sandard Error T-Raio [Prob.] [0.008] R-Squared R-Bar-Squared DW Saiic Diagnoic Te Te aiic [p-value] A: Serial Correlaion CHSQ (1) = [0.349] B: Funcional Form CHSQ (1) = [0.668] C: Normaliy CHSQ (2) = [0.348] D: Heer-ocedaiciy CHSQ (1) = [0.959] 1 Noe1: The diagnoic reul uilize he ollowing e: A: Lagrange muliplier e o reidual erial correlaion B: Ramey' RESET e uing he quare o he ied value C: Baed on a e o kewne and kuroi o reidual D: Baed on he regreion o quared reidual on quared ied value 6. Concluion On balance our reul eem o uppor he orward rae unbiaedne hypohei (FRUH). A boh exchange rae, he AUD/USD and he AUD/JPY, exhibi a coinegraing relaionhip beween po and orward rae, and urhermore our longrun rucural modelling e do no rejec he coinegraing vecor being (1, -1). 26

27 We began by eing he aionariy o he po and orward rae or boh exchange rae and he reul ugge he po and orward rae are inegraed o order 1, I(1). We hen aeed he order o VAR or hi model, unanimouly conirming he ue o VAR(1) model. Nex, we eed or he number o coinegraing relaion preen. Thee were omewha inconien wih dieren e uggeing dieren number o coinegraing relaion; ranging rom no-coinegraion preen, o 1 coinegraing relaion, and mo commonly 2 or more. However we appealed o wih economic heory and impoed 1 coinegraing relaionhip on he model. Thu, we hen impoed rericion on he model and appealed o he daa e o ee i i rejeced he rericion o a coinegraing relaionhip o he vecor (1, -1). The reul rom he rericion applied on boh exchange rae were in avour o he hi rericion and did no rejec i. However in one cae, he AUD/USD po and 6- monh orward rae, he e yielded a aiically igniican reul, providing evidence again he rericing coinegraing vecor. Finally, we evaluaed he error correcion model or boh exchange rae and commonly ound ha he error correcion erm where aiically igniican, were o he correc ign bu were o an exremely mall naure. Thi mean ha convergence o equilibrium would be very low, wih only 5.03%, 1.55%, 0.67%, and 4.13% o dicrepancie correced or each period. Furhermore diagnoically he model uered rom non-normaliy and in one cae heerocedaiciy. To conclude, our reul were broadly conien wih he FRUH and marke eiciency. 27

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