ECONOMETRICS OF THE FORWARD PREMIUM PUZZLE

Transcription

1 ECONOMETRICS OF THE FORWARD PREMIUM PUZZLE Avik Chakrabory Universiy of Tennessee Sephen E. Haynes Universiy of Oregon Ocober 5, 2005 ABSTRACT This paper explores from a new perspecive he forward premium puzzle, i.e., why a regression of he change in he fuure spo exchange rae on he forward premium paradoxically yields a coefficien ha is frequenly negaive. This radiional specificaion is compared heoreically and empirically o a "level" regression of he fuure spo rae on he curren forward rae, which does no display he puzzle. We explore boh non-raionaliy and risk premium explanaions. The general conclusion is ha, wih non-raionaliy, any modes deviaion from uniy in he level coefficien becomes grealy magnified in he forward premium coefficien because of he saionary/nonsaionary properies of he relevan variables, hereby generaing he puzzle. JEL Classificaion: F30, F3 Keywords: Forward premium puzzle, Spo and forward exchange raes, Foreign exchange marke efficiency, Non-raionaliy in foreign exchange markes

2 ECONOMETRICS OF THE FORWARD PREMIUM PUZZLE Avik Chakrabory and Sephen Haynes Ocober 5, Inroducion The Forward Premium Puzzle is an empirical paradox in he foreign exchange marke ha coninues o pose a challenge o inernaional economiss. A regression of he fuure change in he log of he spo exchange rae on he forward premium (he log of he forward exchange rae minus he log of he spo exchange rae) is expeced wih efficien markes o yield a coefficien of uniy. Insead, regression esimaes of his "forward premium" specificaion yield a coefficien ha is significanly less han uniy and frequenly negaive! Much of he burgeoning lieraure aemping o solve he puzzle has focused on explanaions involving a risk premium in he forward exchange marke, wih quie mixed findings. A second specificaion involving spo and forward exchange raes, referred herein as he "level" specificaion, was pursued early in his lieraure -- a regression of he log of he fuure spo exchange rae on he log of he curren forward exchange rae. 2 Alhough no wihou economeric concerns, his regression ypically yields a coefficien close o uniy, a finding which seems consisen wih efficien markes. Comparing esimaes from hese wo similar specificaions suggess a relaed puzzle -- how can a small insignifican deviaion of he coefficien from uniy in he level specificaion For discussion abou he forward premium puzzle, see Froo and Thaler (990) and Obsfeld and Rogoff (997, pp ). For surveys of he research, wih focus on he risk premium approach, see Lewis (995) and Engel (996). 2 For some early papers, see Cornell (977), Levich (979), and Frenkel (980); for a recen discussion, see Zivo (2000).

3 become so grealy magnified ha i causes a sign reversal in he forward premium specificaion? The focus of his paper, which we believe is novel in he lieraure, is o find a saisfacory answer o his relaed puzzle. In addiion, he analysis sheds ligh on a probable explanaion why here is a deviaion in he coefficien from uniy in eiher specificaion relaing spo and forward exchange raes. Is he level form or he forward premium form more appropriae o evaluae marke efficiency? The mos obvious choice would seem o be he level form since i is a direc approach. However, he variables in he level form (he fuure spo and curren forward exchange raes) are non-saionary I() variables, which implies ha regressing one of hem on he oher may lead o inconsisency given he well-known uni roo problem in ime series regression. 3 The forward premium form involves saionary I(0) variables (he fuure change in he spo exchange rae and he forward premium), so he resuling regression coefficien is consisen, which explains he lieraure's almos universal reliance on his specificaion. 4 More recenly, Evans and Lewis (993) demonsrae ha he variables in he level specificaion, he fuure spo and he curren forward exchange raes, are coinegraed, implying ha he level regression is in fac super consisen 5. If so, he level form is indeed legiimae o evaluae marke efficiency and one need no focus only on he radiional forward premium specificaion. 3 For an early saemen, see Granger and Newbold (974). For a full reamen of he uni roo problem, see Hamilon (994, ). 4 The forward premium regression is a popular es for marke efficiency. The conenion is ha if he marke is efficien hen he agens have full informaion and make unbiased predicions abou he fuure exchange rae. Hence, wih an efficien marke he forward premium regression would resul in a slope coefficien of uniy. 5 See Engle and Granger (987) and Hamilon (994, ) for general developmen on coinegraion and super consisency. For relaed applicaions of coinegraion o spo and forward exchange raes, see Hakkio and Rush (989), Hai, Mark, and Wu (997), and Zivo (2000). 2

4 To explore why he forward premium specificaion grealy magnifies (wih a frequen sign reversal) any coefficien deviaion from uniy in he level specificaion, we decompose he slope coefficiens in boh specificaions ino a combinaion of risk premium and non-raionaliy erms, and hen express he coefficiens as variances and covariances of he relevan variables. The heoreical analysis leads o sark empirical predicions, which are hen esed using daa on spo and forward exchange raes beween he US dollar and hree oher major currencies. The general conclusion is ha he dramaic magnificaion in he coefficien deviaion from uniy and possible sign reversal shifing from he level o he forward premium specificaion can be explained by he variance-covariance properies of he relevan I(0) and I() variables in he wo specificaions, i.e., he fac ha he variables are saionary in he forward premium form and non-saionary in he level form. The paper also concludes ha he key reason he coefficiens in eiher form deviae from uniy is non-raionaliy of agens in he foreign exchange marke. This finding does no rule ou he possibiliy of he exisence of a risk premium, bu does indicae ha he puzzle is no solely a consequence of a risk premium. The nex secion develops he heoreical decomposiion of he coefficiens as variances and covariances of he relevan variables in a combined model of risk aversion and non-raionaliy. Secion 3 presens esimaion resuls, and secion 4 concludes. 2. Level and Forward Premium Models The "level" specificaion of he relaionship beween he forward exchange rae F and he fuure spo exchange rae s +, where boh exchange raes are defined as he dollar price of foreign exchange and expressed in logarihms, is he following: s δ γf ψ + = () 3

5 where δ is he inercep, γ is he slope coefficien, and ψ is a random error erm. The key null hypohesis for marke efficiency is ha he slope coefficien γ is uniy. The ordinary leas squares (OLS) esimaor of γ is ˆ γ : Cov( s, F) ˆ = (2) V( F) + γ where V and Cov are he sample variance and covariances, respecively. The radiional "forward premium" specificaion is α β ( ) s+ = + F s + u + (3) where α is he inercep, β is he slope coefficien, and µ is a random error erm. The null hypohesis for marke efficiency in his form is ha he slope coefficien β is uniy. Similarly, he OLS esimae of β is ˆ β : ˆ Cov( s, F s ) = V( F s ) + β (4) The forward premium puzzle is ha ˆ β is significanly less han uniy, and in he majoriy of sudies is negaive. General Model Suppose firs ha agens are risk averse. In his case, he forward rae is heir expeced value of he fuure spo rae minus a premium hey are willing o forego in order o eliminae foreign exchange risk. Thus, E[ s+ ] = F + RP (5) where E[ s + ] is he expeced value in period of he spo rae in period +, and he risk premium in period. Nex, suppose ha agens are no raional and make sysemaic forecas errors in period +, denoed e +. As a consequence, RP is 4

6 s+ = E[ s+ ] + e+ (6) Combining eq. (5) and eq. (6) we obain s F RP e = + + (7) + + Nex, subrac s from boh sides of eq. (7): s+ = ( F s) + RP + e + (8) Combining eq. (2) and eq. (7) yields CovRPF (, ) CovF (, e ) ˆ = + + (9) V( F) V( F) γ + Similarly, combining eq. (4) and eq. (8) yields ˆ Cov( RP, F s) Cov( F s, e+ β ) = + + V( F s ) V( F s ) (0) Thus, in general, risk aversion and/or non-raionaliy offer plausible explanaions why OLS esimaes of γ and β may differ from uniy. To see his, consider he special case of risk neuraliy and raional expecaions. Wih risk neuraliy, RP = 0, and he second erm on he righ-hand side of eqs. (9) and (0) becomes zero. In addiion, if agens possess raional expecaions, hen he forecas error e + is uncorrelaed wih he informaion se in period (including F s ), which implies ha he hird erm in eqs. (9) and (0) is also zero. Thus, wih risk neuraliy and raional expecaions, eqs. (9) and (0) collapse o ˆ γ = ˆ β =. Nex, consider he condiions required o generae he forward premium puzzle, i.e., ˆ β less han uniy and ofen negaive. For ˆ β o be less han uniy, i follows from eq. (0) ha 5

7 Cov( RP, F s) Cov( F s, e+ ) + < 0 V( F s ) V( F s ) () which implies ha a leas one of he wo erms on he lef-hand side of eq. () mus be negaive, and heir sum mus be negaive. For ˆ β o be negaive, i follows from eq. (0) ha Cov( RP, F s) Cov( F s, e+ ) + <. (2) V( F s ) V( F s ) Risk Premium Wihin he forward premium specificaion, firs suppose ha agens have raional expecaions, i.e., Cov( F s, e + ) = 0, bu ha agens possess a risk premium, i.e., Cov( RP, F s ) 0. In order for ˆ β <, i follows from eq. (0) in his case ha Cov( RP, F s) <0. Thus, when he forward premium F s is posiive, he risk premium erm RP mus be negaive (and vice-versa) in he majoriy of he cases. However, when F s is posiive, here is a posiive reurn on purchasing foreign currency. If agens are risk-averse, hey would be willing o accep a smaller sure reurn. In his case, he forward premium mus be less han he change in he exchange rae when boh are posiive. Since on average s = ( F s ) + R, hen he + + P risk premium on average mus be posiive when he forward premium is posiive. And he converse is rue when he forward premium is negaive. As a consequence, if agens are risk averse, hen i mus follow ha Cov( RP, F s ) > 0 6. Hence, he risk-premium 6 This resul is no new. Fama (984) and Engel (996) demonsrae he same resul bu analyically differenly. 6

8 approach alone does no seem sufficien o explain he puzzle of ˆ β <, which requires Cov( RP, F s ) <0. Thus, in his paper we focus on a non-raionaliy approach. Non-raionaliy Alernaively, assume ha agens are risk neural, i.e., Cov( RP, F s ) = 0, bu ha expecaions are no raional, i.e., Cov( F s, e + ) 0. Thus, he forecas error in he nex period is correlaed wih informaion his period, and agens make sysemaic errors in predicion of he spo exchange rae. In order for ˆ β <, i follows from eq. (0) in his case ha Cov( F s, e + ) < 0. Since i is no possible a priori o predic he sign of his covariance, non-raionaliy can poenially explain he puzzle of ˆ β <. If here also exiss a risk premium, hen from he previous subsecion he effec of non-raionaliy mus dominae he effec of he risk premium in order for he ne bias in ˆ β o remain negaive. Comparing he Level o he Forward Premium Specificaion Suppose ha agens are risk-neural, and ha non-raionaliy is he only source of bias in eqs. 9 and 0. Thus, from eq. 9 he bias in ˆ γ in he level specificaion is Cov( F, e+ ), and from eq. 0 he bias in ˆ β in he forward premium specificaion is V( F) Cov( F s, e V( F s ) + ). Evidence discussed in he inroducion suggess ha, paradoxically, he bias in he level specificaion is minimal, ye he bias in he forward premium specificaion is srongly negaive, causing a frequen sign reversal in he coefficien 7

9 esimae ˆ β. A plausible resoluion o his paradox can be found by exploring he saionary-nonsaionary properies of he relevan variables in he wo bias erms. Cov( F, e+ ) Firs, consider he bias erm in he level specificaion,. Assume V( F) ha he forward exchange rae F is a non-saionary variable and he forecas error e + is saionary, conjecures suppored by empirical evidence presened below. Given hese saisical properies of F and e +, i can be shown ha plim[ Cov( F, e + )] = 0, plim[ V( F )] =, p Cov( F e ), + lim[ ] 0 V( F) =, and p lim ˆ γ =. Thus, he bias erm in he Cov( F, e+ ) level specificaion is likely o be relaively "small" for samples of a leas V( F) moderae size. Furhermore, since i depends on he sample size, he bias erm should decline in absolue value moving from quarerly daa o monhly daa for a fixed number of years in he sample. These implicaions of he level model are esed below. Cov( F s, e V( F s ) Nex, consider he bias erm in he forward premium specificaion: + ). Since esimaes of ˆ β are significanly less han uniy and ofen negaive, his bias erm is expeced o be relaively "large" in magniude and negaive, Cov( F s, e+ ) and in he majoriy of cases we should find ha <. Assume ha he V( F s ) variables F s and are saionary, conjecures also suppored by our daa. Given e + hese saisical properies of F s and e +, hen i can be shown ha Cov( F s, e+ ) p lim[ Cov( F s, e+ )] = k, p lim[ V( F s)] = k2, plim[ ] = k3, and V( F s ) 8

10 plim ˆ β = + k3, where k, k 2, and k 3 are finie numbers. Thus, he bias erm in he forward premium form Cov( F s, e V( F s ) + ) may have any finie magniude and is no sysemaically relaed o sample size, implicaions also esed below. However, he sign of he covariance erm is ambiguous wihou placing resricions on he source of he nonraionaliy. To summarize, he heoreical analysis shows ha, assuming non-raionaliy, he bias erm in he level specificaion is small in magniude and declines as he sample size increases given he non-saionary properies of F and s +. However, he bias erm in he forward premium specificaion becomes grealy magnified given he saionary properies of is variables. Thus, non-raionaliy combined wih he saionary/nonsaionary properies of he relevan variables offers a poenial explanaion for he apparen puzzle of lile or no bias in he level specificaion beween he spo and forward exchange raes, ye a dramaic negaive bias wih frequen sign reversal in he radiional forward premium specificaion. 3. Evidence The sample is hree exchange raes -- he US dollar price of he UK poundserling, he French Franc and he Japanese Yen (he daa are from Harris Bank's Weekly Review). The daa are boh monhly and quarerly, and include spo exchange raes, and one and hree-monh forward raes. Monhly daa are from March 973 o Augus 992, and quarerly daa are from 973-quarer o 992-quarer 2 for he French Franc, and 973-quarer o 994-quarer for he UK pound-serling and Japanese Yen. The daa 9

11 are drawn from he las Fridays of he calendar monh for monhly daa, and he calendar quarer for quarerly daa. 7 Coinegraion of he Level Specificaion Valid esimaion of he level specificaion requires coinegraion beween he fuure spo and curren forward exchange raes. Wih coinegraion, he uni roos in he variables will no lead o inconsisen parameer esimaes; in fac, regression esimaes will be super consisen. To es his coinegraion requiremen, Johansen's es is applied o he hree exchange raes using boh monhly and quarerly daa in our sample. The fuure spo rae is proxied by he spo rae on he las Friday one-monh ahead for monhly daa, and hree-monhs ahead for quarerly daa. Tes resuls are presened in Table for monhly daa and Table 2 for quarerly daa. In Tables and 2, race saisics indicae ha coinegraing relaions exis a he 5% level of significance beween he fuure spo rae and he curren forward rae for all hree exchange raes wih boh monhly and quarerly daa. In he case of he UK poundserling, wo coinegraing relaions exis a he 5% level for boh monhly and quarerly daa. Thus, he esimaes in Tables and 2 clearly indicae ha he non-saionariy of he variables in he level specificaion does no lead o inconsisency in OLS esimaion, bu raher super consisency. As a consequence, a side-by-side comparison of boh he level 7 We hank Nelson Mark for providing his daa (originally from Harris Bank's Weekly Review), which he examined in Mark and Wu (998). However, here are caveas wih his daa. The sample ignores he ransacions coss of rading currencies due o he exisence of bid-ask spreads and he delivery srucure. Also, daa are recorded on Fridays. When maching a forward rae wih a corresponding spo rae in quarerly daa, he delivery dae for he forward ransacion should be exacly hree monhs from ha day. By aking he las Friday of every monh, his required delivery srucure could be los. These limiaions wih he daa may inroduce bias in he esimaes. Forunaely, Bekaer and Hodrick (993) argue ha hese daa limiaions seem unimporan in explaining he exchange rae saisics examined in his paper. 0

12 and forward premium coefficien esimaes is feasible in order o deermine why hey are so dramaically differen. Esimaes of he Level and Forward Premium Specificaions We nex es he heoreical implicaions derived in secion 2 assuming nonraionaliy bu risk neuraliy using he same daa. Esimaes of he ˆ γ coefficien from he level specificaion are presened in Table 3. The evidence seems very consisen wih he heoreical predicions of secion 2. Firs, ˆ γ esimaes in all six cases are no saisically differen from uniy, and are numerically close o uniy. 8 This evidence suggess ha he forward rae may be a reliable predicor of he fuure spo rae, and ha here may be no significan deviaion from raionaliy. In paricular, he evidence does no sugges a paern of coefficiens ha are significanly less han uniy and even negaive, as in he forward premium puzzle. Anoher implicaion from Table 3 is ha for all hree exchange raes he coefficiens increase in magniude oward uniy shifing from quarerly o monhly daa. Cov( F, e+ ) This is consisen wih he predicion ha he bias erm decreases as he V( F) sample size increases, hence ha he coefficien ˆ γ ends o approach uniy as he sample size increases. Esimaion resuls for ˆ β from he forward premium specificaion are summarized in Table 4. This evidence also suppors he heoreical predicions from secion 2. Firs, he ˆ β coefficiens are significanly less han uniy a he % level in five of he six cases, and in fac are numerically negaive in all six cases, replicaing he 8 However, a cavea is ha he power of he es is low.

13 forward premium puzzle. In combinaion wih evidence from Table 3, his evidence demonsraes ha he modes deviaion from uniy in level form is sufficienly magnified o generae a negaive ˆ β in he forward premium form in all six cases. Anoher observaion is ha he ˆ β esimaes do no change sysemaically wih a change in he sample size, unlike in he level specificaion. The analysis in secion 2 suggess ha his is primarily a consequence of he saionariy propery of he forward premium. The variance in he denominaor should no explode wih increasing sample Cov( F, size, hence he bias erm s e V( F s ) + ) should no change sysemaically. Direc Esimaes of he Bias Finally, we explore a more direc mehod of esing he heoreical predicions in Secion 2. Table 5 presens, for he level model, esimaes of he numeraor and denominaor of he bias erm, i.e., esimaes of he covariance beween he forecas error and he forward rae, and he variance of he forward rae. Table 6 presens analogous esimaes for he forward premium model. This evidence is consisen wih ha presened in Tables 3 and 4. The forward rae has a much larger variance in Table 5 compared o any oher variance or covariance esimaes in Tables 5 and 6, as prediced given is non-saionariy. Also, he covariance beween he forward rae and he forecas error in Table 5 is very small, as prediced. Thus, he bias erm in he level specificaion Cov( F, e+ ) is small in magniude. The numeraor and denominaor erms for he V( F) forward premium specificaion in Table 6 are also small and are roughly of he same Cov( F, order of magniude. Consequenly, he bias erm s e V( F s ) + ) is sufficienly large 2

14 and negaive o drive he ˆ β coefficien o become negaive, creaing he forward premium puzzle. 9 In sum, he empirical evidence in Tables 3 hrough 6 is srongly consisen wih he heoreical analysis relaing he presence of a modes bias in he slope coefficien in he level specificaion o a subsanial bias and sign reversal in he slope coefficien in he forward premium specificaion. 4. Conclusion This paper explores he economerics behind he forward premium puzzle from a novel perspecive. The perspecive is o explain why any small deviaion from uniy in he coefficien of a regression of he fuure spo exchange rae on he curren forward exchange rae (he level specificaion) becomes so magnified in he radiional forward premium specificaion o frequenly yield a negaive regression coefficien, i.e., he forward premium puzzle. This paper demonsraes, we believe, ha he relaionship beween spo and forward exchange raes can be beer undersood by examining heir link using boh he level and forward premium specificaions joinly raher han focusing solely on he radiional forward premium specificaion. We decompose he OLS regression coefficiens in he level and forward premium models, permiing boh a risk premium and non-raionaliy. The heoreical decomposiion and subsequen empirical analysis leads o wo findings. Firs, for he forward premium model, he downward bias in he slope coefficien, wih frequen sign reversal, can be explained by non-raionaliy bu no a risk premium, as he bias sems 9 As noed above, non-raionaliy alone does no predic he sign of he bias in ˆ β. As described in foonoe 0, however, non-raionaliy semming from recursive leas squares learning does imply a negaive bias since i implies a negaive covariance beween he forecas error and he forward premium. 3

15 from a negaive covariance beween he forecas error and he forward premium. Second, wih non-raionaliy, any modes deviaion from uniy in he level coefficien becomes grealy magnified in he radiional forward premium coefficien because of he saionary-nonsaionary properies of he relevan variables. In his paper, we make no conjecure abou he source of he non-raionaliy ha may generae he negaive covariance beween he forecas error and he forward premium. 0 Also, we do no explore wheher or no marginal non-raionaliy in he level form is evidence of marke inefficiency. I would seem ha a key o undersanding he implicaions for marke efficiency lies in he naure of he non-raionaliy ha can explain his negaive covariance, and if he covariance is of such a magniude as o yield a negaive ˆ β in a majoriy of cases. Noneheless, our heory and evidence clearly suggess ha only a modes deviaion from raionaliy in he level specificaion is sufficien o cause a sign reversal in he forward premium regression, i.e., he forward premium puzzle. 0 See Chakrabory (2005) for a plausible explanaion of his non-raionaliy and negaive covariance in erms of recursive leas squares learning. The key assumpion is ha risk neural agens do no have perfec knowledge abou he foreign exchange marke, bu aemp o learn he parameers of he sochasic process generaing he exchange rae using consan-gain recursive leas squares. Crucially, his approach predics a negaive covariance beween he forecas error and he forward premium, which is necessary o explain he forward premium puzzle. 4

16 Table : Johansen s es for coinegraion beween he fuure spo exchange rae s + and he curren one- period ahead forward rae Yen (monhly daa) F for he US dollar price of he UK Pound-serling, French Franc and Japanese The null hypohesis in Johansen s unresriced coinegraion rank es is ha here exiss no coinegraion beween he variables. Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY No. of observaions No. of coinegraing Relaions None Trace saisic 39.83** 28.23** 7.85* 5% Criical value % Criical value A mos Trace saisic 4.93* % Criical value % Criical value Normalized coefficiens s F ( ) ( ) ( ) Noes: * and ** denoe saisical significance (wo-ailed es) a 5% and % level, respecively. Numbers in parenheses are sandard errors. Sources: Monhly daa are aken for he period March 973 o Augus 992. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar monh. 5

17 Table 2: Johansen s es for coinegraion beween he fuure spo exchange rae s + and he curren one- period ahead forward rae Yen (quarerly daa) F for he US dollar price of he UK Pound-serling, French Franc and Japanese The null hypohesis in Johansen s unresriced coinegraion rank es is ha here exiss no coinegraion beween he variables. Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY No. of observaions No. of coinegraing Relaions None Trace saisic 5.74* 5.92* 24.43** 5% Criical value % Criical value A mos Trace saisic 4.98* % Criical value % Criical value Normalized coefficiens s F (0.0356) ( ) (0.0089) Noes: * and ** denoe saisical significance (wo-ailed es) a 5% and % level, respecively. Numbers in parenheses are sandard errors. Sources: Quarerly daa are aken for he period 973-quarer o 992-quarer 2 for French Franc and 973-quarer o 994-quarer for UK pound-serling and Japanese Yen. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar quarer. 6

18 δ γ ψ Table 3: Esimaes from he "Level" regression equaion s+ = + F + + for USD price of UK Pound-serling, French Franc and Japanese Yen (monhly and quarerly daa) The dependen variable is fuure spo rae s +. Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY Monhly daa Quarerly daa No. of observaions ˆ γ (0.06) (0.0094) (0.0074) 2 R No. of observaions ˆ γ (0.0357) (0.03) (0.023) 2 R Noes: * and ** denoe saisical significance (wo-ailed es) a 5% and % level, respecively, for H γ =. Numbers in parenheses are sandard errors. 0 : Sources: Monhly daa are aken for he period March 973 o Augus 992 and he quarerly daa span he period 973-quarer o 992-quarer 2 for French Franc and 973-quarer o 994-quarer for UK pound-serling and Japanese Yen. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar monh for monhly daa and calendar quarer for quarerly daa. 7

19 + = α + β + u+ for Table 4: Esimaes from he "Forward Premium" regression equaion s ( F s ) USD price of UK Pound-serling, French Franc and Japanese Yen (monhly and quarerly daa) The dependen variable is he change in he fuure spo rae s +. Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY Monhly daa Quarerly daa No. of observaions ˆβ -0.73** -0.96** -0.53** (0.606) (0.659) (0.397) 2 R No. of observaions ˆβ -.323** ** (0.79) (0.874) (0.47) 2 R Noes: * and ** denoe saisical significance (wo-ailed es) a 5% and % level, respecively, for H β =. Numbers in parenheses are sandard errors. 0 : Sources: Monhly daa are aken for he period March 973 o Augus 992 and he quarerly daa span he period 973-quarer o 992-quarer 2 for French Franc and 973-quarer o 994-quarer for UK pound-serling and Japanese Yen. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar monh for monhly daa and calendar quarer for quarerly daa. 8

20 Table 5: "Level" Specificaion: Variance and covariance erms for he forward rae and forecas error for USD price of UK Pound-serling, French Franc and Japanese Yen (monhly and quarerly daa) Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY Monhly daa: No. of observaions Cov( F, e + ) V( F ) Cov( F, e+ ) V( F) Quarerly daa No. of observaions Cov( F, e + ) ( ) V F Cov( F, e+ ) V( F) Sources: Monhly daa are aken for he period March 973 o Augus 992 and he quarerly daa span he period 973-quarer o 992-quarer 2 for French Franc and 973-quarer o 994-quarer for UK pound-serling and Japanese Yen. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar monh for monhly daa and calendar quarer for quarerly daa. 9

21 Table 6: "Forward Premium" Specificaion: Variance and covariance erms for he forward premium and he forecas error for USD price of UK Pound-serling, French Franc and Japanese Yen (monhly and quarerly daa) Exchange rae (US dollar price of foreign currency) USD/UKP USD/FRF USD/JPY Monhly daa: No. of observaions Cov( F s, e + ) V F s ) ( Cov( F s, e V( F s ) + ) Quarerly daa No. of observaions Cov( F s, e + ) V F s ) ( Cov( F s, e V( F s ) + ) Sources: Monhly daa are aken for he period March 973 o Augus 992 and he quarerly daa span he period 973-quarer o 992-quarer 2 for French Franc and 973-quarer o 994-quarer for UK pound-serling and Japanese Yen. The spo exchange rae, monh forward rae and 3 monh forward rae daa are aken from Harris Bank s Weekly Review. They are drawn from he Fridays occurring neares o he end of he calendar monh for monhly daa and calendar quarer for quarerly daa. 20

22 References Bekaer, G. and Hodrick, R On biases in he measuremen of foreign exchange risk premiums. Journal of Inernaional Money and Finance 2, Chakrabory, A, Learning, he forward premium puzzle, and marke efficiency. Universiy of Tennessee Working Paper. Cornell, B., 977. Spo raes, forward raes and exchange marke efficiency. Journal of Financial Economics 5, Engel, C., 996. The forward discoun anomaly and he risk premium: A survey of recen evidence. Journal of Empirical Finance 3, Engel, R.F. and Granger, C.W.J., 987. Co-Inegraion and Error Correcion: Represenaion, esimaion, and Tesing. Economerica 55, Evans, F. and Lewis, K.K., 993. Trends in Excess Reurns in Currency and Bond Markes. European Economic Review 37, Fama, E., 984. Forward and spo exchange raes. Journal of Moneary Economics. 4, Frenkel, J.A., 980. Exchange raes, prices and money: lessons from he 920s. American Economic Review 70, Froo, K.A. and Thaler, R.H Anomalies: Foreign Exchange. Journal of Economic Perspecives 4, Granger, C.W.J. and Newbold, P Spurious regressions in economerics. Journal of Economerics 2, -20. Hai, W., Mark, N., and Wu, Y Undersanding spo and forward exchange rae regressions. Journal of Applied Economerics 2, Hakkio, C.S. and Rush, M Marke efficiency and coinegraion: an applicaion o he serling and Deuschmark exchange markes. Journal of Inernaional Money and Finance 8, Hamilon, J Time Series Analysis. Princeon Universiy Press, Princeon, NJ. Levich, R., 979. On he efficiency of markes of foreign exchange. In: Dornbusch, R., Frenkel, J. (Eds.), Inernaional Economic Policy heory and Evidence. John Hopkins Press,

23 Lewis, K.K Puzzles in inernaional financial markes. Handbook of Inernaional Economics, Vol. 3, Rogoff, K. and Grossman, G. (eds), Norh Holland, Amserdam, Mark, N.C. and Wu, Y., 998. Rehinking deviaions from uncovered ineres pariy: he role of covariance risk and noise. The Economic Journal 08, Obsfeld, M. and Rogoff, K., 997. Foundaions of Inernaional Macroeconomics. MIT Press, Cambridge, Mass. Zivo, E., Coinegraion and forward and spo exchange rae regressions. Journal of Inernaional Money and Finance 9,