Marke Overreacion and Under-reacion for Currency Fuures Prices Sephen J. Larson and Sephen E. Wilcox* Minnesoa Sae Universiy, Mankao January 2008 Sephen J. Larson, Ph.D., CFP Minnesoa Sae Universiy, Mankao Deparmen of Finance 150 Morris Hall Mankao, Minnesoa 56001 (507)-389-2324 Sephen.Larson@mnsu.edu *Conac Auhor Sephen E. Wilcox, Ph. D., CFA Minnesoa Sae Universiy, Mankao Deparmen of Finance 150 Morris Hall Mankao, Minnesoa 56001 (507)-389-5344 Sephen.Wilcox@mnsu.edu
Marke Overreacion and Under-reacion for Currency Fuures Prices Absrac Research has documened overreacion and under-reacion in many markes including he sock marke and he foreign currency spo marke. This paper addresses marke over- and under-reacion for foreign currency fuures conracs. Our daa se consiss of daily observaions of fuures prices, spo exchange raes, and Eurocurrency LIBOR for he Briish Pound, Japanese Yen and Swiss Franc from January 2, 1991 o December 31, 2006. Using a 5-year moving window mehod and he foreign currency fuures pricing model of Amin and Jarrow (1991), we find repeaed evidence of coinegraion among he fuures price, he spo exchange rae, and ineres raes over en differen esimaion periods. An error-correcion model is hen used o develop a series of prediced fuures price changes. We assess wheher he marke overreaced or under-reaced o new informaion by comparing he acual fuures price change o he change prediced by he error correcion model. For each even (exreme, one-day price change), Lexis-Nexis is accessed o deermine if news services offered an explanaion. An informed even (winner or loser) refers o an exreme currency fuures price change ha corresponds wih an explanaion in Lexis/Nexis. The informed winner and informed loser samples are each broken down according o wheher he announcemen is economic or poliical in naure. Uninformed winners and uninformed losers are exreme, oneday fuures price changes ha do no correspond o Lexis/Nexis news announcemens. Our resuls sugges he ype of underlying announcemen is useful in pinpoining when he marke over- and under-reacs o new informaion peraining o foreign currency fuures prices. Specifically, for winners i appears he marke overreacs o poliical news and news ha is no widely publicized (uninformed evens). For losers here is evidence suggesing he marke under-reacs o poliical announcemens and here is some evidence of overreacion for he sample of uninformed losers. Overall, here are wo major findings. Firs, i appears he marke is oo opimisic when favorable and unfavorable poliical news is released peraining o he Briish Pound, Japanese Yen and Swiss Franc. Second, i appears he marke overreacs o new informaion when ha informaion is no widely publicized.
Inroducion and Lieraure Review This paper addresses he behavior of currency fuures prices upon he release of new informaion. We examine exreme, one-day changes in fuures prices where a loser is defined as an exreme, one-day decline in he fuures price and a winner is defined as an exreme, one-day increase in he fuures price. The underlying informaion a he ime of each exreme price change (even) is gahered from Lexis/Nexis and he ype of informaion is caegorized. An informed even is an exreme price change ha corresponds o a release of news while an uninformed even does no correspond o publicly released informaion. The sample of informed evens is caegorized furher ino wo groups. The firs group perains o he release of economic news and he second group perains o he release of poliical news. Our resuls sugges conrolling for he ype of underlying announcemen is useful in pinpoining currency fuures marke over-and under-reacion. The full sample of winners and he full sample of losers are no associaed wih over- or under-reacion. However, for winners he poliical sub-sample and uninformed sub-sample are each associaed wih overreacion. There is also some evidence ha economic announcemens are associaed wih under-reacion. For losers, he poliical sub-sample is associaed wih under-reacion and here is some evidence of overreacion for he uninformed sub-sample. In heir influenial paper, DeBond and Thaler (1985) were he firs o formally sudy overreacion in he sock marke. Using a hree year period, DeBond and Thaler formed en porfolios of socks based on performance. During he subsequen hree year period he lowes decile of socks ou-performs he highes decile of socks by 24.6% (saisically significan). Many sudies follow such as Akins and Dyl (1990) and Bremer and Sweeney (1991) where evidence suggess he sock marke overreacs o news a he ime i is released and subsequenly correcs iself over he nex few rading days. Larson and Madura (2003) examine sock price overreacion for winners and losers while conrolling for he underlying informaion releases. They conrol for he informaion ha was released a he ime of he exreme price changes. Informed evens (losers and winners) are associaed wih underlying informaion releases while uninformed evens are no. Their resuls sugges conrolling for he underlying informaion is useful when aemping o pinpoin when he marke over- and under-reacs. Specifically, heir uninformed winners are associaed wih an overreacion phenomenon, bu heir informed winners are no. This suggess he marke overreacs o informaion when rading of privae informaion, bu efficienly reacs o public informaion. This may be relaed o invesor self-aribuion bias as discussed by Daniel, Hirshleifer, and Subrahmanyam (1998); hey posulae ha marke paricipans are more prone o overreac when rading on non-publicized informaion. Using an even-sudy mehod Larson and Madura (2001) also examine overreacion and under-reacion for spo currency exchange raes. For emerging 1
currencies hese auhors find evidence suggesing he marke overreacs, bu for indusrial currencies hese auhors find evidence suggesing he marke underreacs. These auhors also conrol for he underlying informaion releases and find evidence ha he degree of overreacion is condiioned upon he underlying informaion released. In his paper, we examine he response of currency fuures prices o underlying informaion releases using he heory of coinegraed processes. We also conrol for he underlying informaion releases o help pinpoin when he marke is prone o over- and under-reac. Hypoheses I seems reasonable o assume he currency fuures marke will under-reac o informaion causing exreme, one-day price adjusmens for currency fuures prices. Larson and Madura (2001) examine spo exchange raes and heir resuls sugges he marke under-reacs o news abou indusrial currency. Since spo and fuures prices almos always move in phase we expec o find under-reacion for he currency fuures prices in our sample. Our firs hypohesis is formally saed below: Hypohesis 1: Exreme, one-day price changes in currency fuures prices will be followed by price changes in he same direcion. Exreme fuures price changes are expeced o be associaed wih he release of public informaion. In ligh of findings in he lieraure, i seems reasonable o presuppose he marke will respond differenly o differen ypes of informaion. Larson and Madura (2001) find evidence pursuan o indusrial spo exchange raes ha suggess he marke is more likely o overreac when exreme currency price changes are no associaed wih underlying informaion releases. In heir heoreical paper, Daniel, Hirshleifer, and Subrahmanyam (1998) sugges marke paricipans overreac more when rading on privae informaion. These auhors sugges marke paricipans possess self aribuion bias, which suggess hey are over confiden when hey hold informaion ha has no been delegaed o he general public. For his reason we believe our samples of uninformed winners and uninformed losers will be associaed wih overreacion. Our second hypohesis is formally saed below: Hypohesis 2: Exreme, one-day price changes in foreign currency fuures prices will be associaed wih reversals (overreacion) when no news corresponds o he price change. A cross-secional analysis is conduced and he main purpose is o examine wheher he marke reacs differenly o differen ypes of announcemens while conrolling for oher facors. The oher facors are he iniial degree of mispricing 2
on day 0 and he degree of mispricing (leakage) on day -1. We offer wo hypoheses based on he exising lieraure wih regard o hese wo facors. Firs, Brown and Harlow (1988); and Akhigbe, Gosnell and Harikumar (1998) find evidence ha larger abnormal reurns on day zero are associaed wih a higher degree of overreacion. These auhors reason ha larger evens are associaed wih larger degrees of uncerainy and herefore larger degrees of overreacion. Hypohesis hree is saed formally below: Hypohesis 3: Larger price movemens on he even day will be associaed wih larger degrees of overreacion. Second, Daniel, Hirshleifer, and Subrahmanyam (1998) posulae ha sock prices will overreac o privae informaion signals. They reason ha invesors overweigh privae signals and herefore overreac. They aribue his o invesor self-aribuion bias, or overconfidence. If heir heory is correc, higher degrees of pre-even leakage will be associaed wih a higher degree of overreacion. Hypohesis four is saed formally below: Hypohesis 4: Larger price movemens on day -1 will be associaed wih higher degrees of overreacion. Model Developmen In heir seminal paper, Engle and Granger (1987) inroduced he heory of coinegraed processes as a means of esing long-run heories among nonsaionary variables. Because many financial ime series conain sochasic rends, much aenion in he financial lieraure has been devoed o he possibiliy of wo or more asses being coinegraed, ha is, sharing a common sochasic rend. Examples of coinegraion among equiies can be found in Bossaers (1988), Cerchi and Havenner (1998), and Kasa (1992). Coinegraion has also frequenly been found among foreign exchange raes and Baillie and Bollerslev (1989) find ha seven currencies are coinegraed. Coinegraion has been found in many commodiy markes. For example, Bachmeier and Griffin (2006) use a coinegraion mehod o evaluae he inegraion beween and wihin he markes for coal, naural gas and crude oil. They find ha he markes for crude oil are highly inegraed and can be viewed as a single global marke. In conras, while he US coal indusry is coinegraed across regions, i shows less inegraion han he oil indusry as indicaed by he slow speeds of adjusmen in he error correcion represenaion. Finally, he marke for naural gas is only weakly relaed o he oher wo sources of energy. Based on heir analysis, he auhors conclude ha a single energy complex does no exis. Warrel (2006) uses he Engle-Granger wo sage coinegraion mehod o analyze inernaional inegraion in he coal indusry. She finds coinegraion 3
and inerpres his as evidence of a global coal marke. She concludes ha marke concenraion concerns for mergers in he coal indusry may be exaggeraed because he relevan reference region is he enire global marke. In he case of fuures markes, raders agree o receive or deliver a given spo marke commodiy a a cerain ime in he fuure for a price ha is deermined oday. In such circumsances, i is no surprising ha a long-run relaionship beween fuures prices and spo prices may prevail. Coinegraion in fuures markes does no necessarily occur in every insance, bu, under many circumsances, coinegraion beween spo and fuures prices would be expeced on heoreical grounds and has been documened empirically. The heoreical argumens for coinegraion beween spo and fuures prices are ypically based on marke efficiency, price convergence, and/or he saionariy of he cos of carry. Hakkio and Rush (1989) and Shen and Wang (1990) demonsrae ha coinegraion beween spo and fuures prices is a necessary condiion for marke efficiency if here is no risk premium. Chowdhury (1991) and Lai and Lai (1991) argue ha price convergence a mauriy will lead o coinegraion beween he spo and fuures prices. Lien and Luo (1993) discuss he relaionship beween coinegraion and he cos of carry and argue ha a saionary cos of carry should exis for shor mauriy conracs, paricularly if ineres raes are low. For longer mauriy conracs, coinegraion may sill apply if he cos of carry is near zero due o a rade-off beween he convenience yield and sorage coss. Empirically, Chowdhury (1991) finds evidence of coinegraion beween cerain spo and fuures prices in meal markes. Wahab and Lashgari (1993) and Ghosh (1993) find coinegraion beween he S&P 500 spo index and fuures conracs. Quan (1992) and Sereis and Banack (1990) discover coinegraion in crude oil markes. In currency markes, Baillie and Bollerslev (1989), Hakkio and Rush (1989), Kroner and Sulan (1993), Ghosh (1993), and Lien and Luo (1993) find coinegraion beween foreign exchange fuures and spo markes. More recenly, error-correcion mehods have been used o invesigae marke inegraion and o forecas commodiy prices, paricularly in he energy complex. In he elecriciy and naural gas fuures markes, Emery and Liu (2002) find ha he mean reversion in heir rading rule simulaion is boh saisically and economically significan. Girma and Paulson (1999) find ha risk arbirage opporuniies exis in he crack spread (crude oil, heaing oil and unleaded gasoline) complex for he period 1983 o 1994. Lanza e al. (2005) build an error-correcion model for he dynamics of en grades of crude oil and foureen differen refinery producs. They compare he ou-of-sample forecasing performance of he error-correcion models wih a naïve auoregressive model which lacks he coinegraion consrains. They find ha imposing he coinegraing consrain marginally improves some of heir forecass. Ewing e al. (2006) apply a varian of an error-correcion model, he momenum-hreshold auoregression (M-TAR), o he gasoline, heaing oil and 4
crude oil markes. Their model is beer able o accommodae asymmeric responses o shocks in hese markes. They emphasize ha modeling he ineracions beween he spo and fuures markes is imporan for proper hedging and forecasing. In he foreign currency markes, Sequeira, e al (1999) find coinegraing relaionships beween he Ausralian dollar spo and fuures prices, and U.S. and Ausralian risk-free raes of ineres. These coinegraing relaionships sugges an error-correcion represenaion for he cos-of-carry model which, wih zero resricions, yields he error-correcion formulaion for he unbiased expecaions hypohesis. The auhors find he cos-of-carry model o be empirically superior o he unbiased expecaions hypohesis for he four sample ses considered. In his effor, we make use of he foreign currency fuures pricing model derived by Amin and Jarrow (1991) wihin he framework of Heah, e al (1992) as indicaed in equaion (1). F,T is he fuures exchange rae beween a domesic currency, d, and a foreign currency, f, a ime for a fuures conrac wih mauriy T, S is he spo exchange rae beween domesic currency d and foreign currency d f f a ime, e is Euler s number, and are he domesic and foreign T-period i,t i,t ineres raes a ime, respecively, and θ,t is an adjusmen erm for he markedo-marke feaure of a fuures marke conrac. F d f ( i, T i, T ) T θ = S e, T (1), T e Assuming T =1 and applying he properies of he naural logarihm o boh sides of equaion (1) resuls in equaion (2). and are he naural logarihms F,T of and, respecively. S f,t s f = + (2) d f, T s + i,t i,t θ,t The marked-o-marke adjusmen erm is no direcly observable, bu i is reasonable o assume ha his erm is covariance saionary and can reaed as an inercep erm in an empirical model. Accordingly, he model we es for coinegraion is presened in equaion (3) where he esimaed coefficiens for β 3 and 2 are prediced o be posiive and he esimaed coefficien for β is prediced o be negaive. f = + (3) d f, T β 0 + β1s + β 2i,T + β3i,t ε β 1 5
If coinegraion beween he daa series in (3) is presen, hen here exiss an error-correcion model ha predics changes in based on pas changes in, d i,t f, and, and deviaions in any exising coinegraing relaionships. i,t f,t s Daa We make use of daily closing prices for he Briish pound (BP), Japanese yen (JY), and Swiss franc (SF) fuures conracs ha rade via open oucry on he Chicago Mercanile Exchange. We assume coninuous conrac pricing, so he fuures price used is ha for he nearby fuures conrac. We also make use of daily BP, JY, and SF spo exchange raes, expressed as American quoes o mach he pricing convenion of he fuures conracs. Finally, we make use of daily observaions of he 3-monh BP, JY, SF, and U.S. dollar ($) LIBOR raes for ineres rae daa. The daa se begins in 1991 and runs hrough year-end 2005. Following Norbin, e al (1997), we employ a 5-year rolling window mehodology. The rolling regressions help o validae he sabiliy of he relaionship and allow us o evaluae wheher he forecasing abiliy is robus o varying ime periods. Our analysis makes use of en in-sample 5-year esimaion periods from January 2, 1991 o December 31, 2004. The error-correcion model for each of hese en 5- year esimaion periods is hen used o predic changes in he currency fuures exchange rae in he following 1-year ou-of-sample ime period. Table 1 illusraes he en in-sample esimaion periods as well as he en ou-of-sample esing periods. -INSERT TABLE 1 HERE- Uni Roo Tess The firs condiion for a se of series o be coinegraed is ha each series mus be inegraed of he same order. The augmened Dickey-Fuller (ADF) [see Dickey and Fuller (1979)] and Phillips-Perron (PP) [see Phillips and Perron (1988)] uni roo ess can be used o es he values of all he specified daa series for nonsaionariy. The saring poin for a uni roo in ime series x is o firs consider a firs-order auoregressive process [AR(1)] such as ha in equaion (4). x µ + ρx + ε = (4) 1 µ and ρ are parameers and he error erm, ε, is assumed o be whie noise. Tess are carried ou by esimaing equaion (5) where x 1 is subraced from boh sides of equaion (4) where α = ρ 1. 6
x µ + αx + ε = (5) 1 The null hypohesis of a non-saionary series [an I(1) series or series wih a uni roo] can be evaluaed by esing wheher he value of he esimaed coefficien for α, αˆ, is zero. Because a α greaer han zero implies an explosive series ha makes lile economic sense, he hypohesis is esed agains he onesided alernaive ha α ˆ is less han zero. The simple uni roo es described above is valid only if he series is an AR(1) process. If he series is correlaed a higher order lags, he assumpion of whie noise disurbances is violaed. The ADF and PP ess use differen mehods o conrol for higher-order serial correlaion in he series. The ADF approach conrols for higher-order correlaion by adding lagged difference erms of he dependen variable, x, o he righ-hand side of he regression, resuling in 1 equaion (6). (6) x = µ + αx 1 L + β x + ε i i= 1 i The lag lengh, L, is chosen o render he error erm ε whie noise. 2 If he ADF - saisic for αˆ is negaive and significan, he null hypohesis of a uni roo is rejeced and he series canno be considered non-saionary. If he null hypohesis is no rejeced, hen here is no evidence ha he series is saionary and he series is assumed o be non-saionary. The PP es makes use of a non-parameric mehod of conrolling for higherorder serial correlaion. The es regression for he PP es is he AR(1) process presened in equaion (6) above. The PP es makes a correcion o he es saisic of he α coefficien o accoun for serial correlaion in ε. The correcion is non-parameric because he procedure makes use of an esimae of he specrum of ε ha is robus o heereoskedasiciy and auocorrelaion of unknown form. The asympoic disribuion of he PP -saisic is he same as he ADF -saisic and es resuls are inerpreed in he same manner. The Kwiakowski, Phillips, Schmid, and Shin (KPSS) es (1992) differs from he ADF and PP ess in ha he series x is assumed o be saionary under he null hypohesis. The KPSS es saisic makes use of he residuals from he OLS 1 There are acually hree possible variaions of he ADF and PP uni roo ess: (1) esimaion wih a consan and a rend erm, (2) esimaion wih a consan [see equaion (8)], and (3) esimaion wih neiher a consan nor rend erm. We have included a consan erm, µ, in our analysis because he mean change in some of he daa series was saisically differen from zero and all daa series exhibied some skewness. We did no include a deerminisic rend erm because economic heory predics none will exis in he daa series we use for his paper. 2 We choose a lag lengh ha minimizes he Schwarz Informaion Crierion for he opimal lag, L, in our ADF ess. 7
regression of x on he exogenous variables. The (Lagrange muliplier) KPSS es saisic is based on a cumulaive residual funcion and an esimae of he residual specrum a frequency zero. Criical values are based upon he 3 asympoic resuls presened in KPSS. Table 2A presens he resuls of he uni roo ess for he naural logarihm of $/BP he Briish pound fuures exchange rae, f, he naural logarihm of he Briish $/BP pound spo exchange rae, s, he 3-monh U.S. dollar LIBOR, i $, and he 3- BP monh Briish pound LIBOR, i. For he ADF and PP ess, he null hypohesis ha he series is non-saionary is rejeced in only wo of he en 5-year ime $/BP $/BP BP periods esed for f, s, and i. The null hypohesis of non-saionariy is rejeced in only one of he en 5-year ime periods esed for i $. For he KPSS ess, he null hypohesis ha he series is saionary is rejeced for all four series in all of he en 5-year ime periods a he 1 percen confidence level. Table 2B presens he resuls of he uni roo ess for he naural logarihm of $/JY he Japanese yen fuures exchange rae, f, he naural logarihm of he Japanese $/JY yen spo exchange rae, s, he 3-monh U.S. dollar LIBOR, i $, and he 3- JY monh Japanese yen LIBOR, i. For he ADF and PP ess, he null hypohesis ha he series is non-saionary is no rejeced in any of he en 5-year ime $/JY $/JY periods esed for f and s and is rejeced in only one of he en 5-year ime periods esed for i $. The null hypohesis of non-saionariy is rejeced by boh JY he ADF and PP ess in hree of he en 5-year ime periods esed for i. For he KPSS ess, he null hypohesis ha he series is saionary is rejeced for all four series in all of he en 5-year ime periods a he 1 percen confidence level. Table 2C presens he resuls of he uni roo ess for he naural logarihm of $/SF he Swiss franc fuures exchange rae, f, he naural logarihm of he Swiss $/SF franc spo exchange rae, s, he 3-monh U.S. dollar LIBOR, i $, and he 3- SF monh Swiss Franc LIBOR, i. For he ADF and PP ess, he null hypohesis ha he series is non-saionary is no rejeced in any of he en 5-year ime $/JY $/JY periods esed for f and s and is rejeced in only one of he en 5-year ime periods esed for i $ SF and i. For he KPSS ess, he null hypohesis ha he series is saionary is rejeced for all four series in all of he en 5-year ime periods a he 1 percen confidence level. The resuls presened in Tables 2A, 2B, and 2C srongly sugges ha all he individual daa series used in our model are non-saionary and possess a uni roo. The KPSS null hypohesis of saionariy is rejeced for all series in all en 5-year ime periods esed a he 1 percen confidence level. -INSERT TABLES 2A, 2B, AND 2C HERE- 3 The AR specral densiy esimaor a frequency zero for he PP and KPSS ess is deermined using he Barle kernel sum of covariances. The bandwidh parameer for he kernel-based esimaors is deermined using he Newey-Wes (1994) procedure. 8
Coinegraion Tess Coinegraion resuls when a linear combinaion of a se of non-saionary series is saionary. Johansen s (1991, 1995) mehod for deermining wheher nonsaionary series are coinegraed ess he resricions imposed by coinegraion on he unresriced vecor auoregression (VAR) involving he series. Equaion (7) represens a VAR of order ρ. ρ X = Α X + ΒY + Ε (7) i= 1 i i X is a vecor of non-saionary, I(1) variables, Y is a vecor of deerminisic variables, and is a vecor of innovaions. Equaion (8) represens anoher way E o wrie he VAR where Π = A i I and =. ρ i= 1 ρ Γ i A j j= i+ 1 ρ 1 X = ΠX + Γ X + ΒY + Ε (8) 1 i i= 1 i Assuming r represens he number of coinegraion relaions or rank, if Π has reduced rank r < k, hen here exis k r marices κ and ω each wih rank r such ha Π = κω and ω X is saionary. Each column of ω is he coinegraing vecor and he elemens of κ are known as he adjusmen parameers in a vecor error correcion model. Johansen s mehod is o esimae he Π marix in an unresriced form, hen use a race es o es wheher he resricions implied by he reduced rank of Π can be rejeced. The race es is a likelihood raio es saisic ha is based on eigenvalues. To deermine he number of coinegraing relaions, r, he es proceeds sequenially from r = 0 o r = k -1 unil i fails o rejec. In words, he null hypohesis of no coinegraion is esed agains he alernaive hypohesis of full rank. If ha is rejeced, he null hypohesis of one coinegraing relaion is esed agains he alernaive hypohesis of full rank. The process would be repeaed unil he null hypohesis of some number of coinegraing relaions, r k -1, canno be rejeced. Criical values for he race saisic can be found in Oserwald-Lenum (1992). The exac form of he Johansen coinegraion es depends on he assumpion one makes concerning he possible deerminisic componens of he sysem. From Equaion (5), he coinegraing relaion should conain a consan o accoun for he marked-o-marke adjusmen. Also, o allow for shor run rends in he level of he variables (paricularly he ineres raes), we specify he error-correcion componen o have an inercep ( α 0 ) in Equaion (9). 9
Tables 3A, 3B, and 3C presen he Johansen coinegraion es resuls for he BP. JY, and SF fuures exchange rae, respecively, based on he relaionship suggesed in equaion (3). The Johansen coinegraion es resuls indicae ha a coinegraing relaionship exiss in each of he en 5-year esimaion periods for all hree currencies. We rely on he race es and is 5 percen criical value o idenify he number of coinegraing relaions. 4 The number of saisically significan coinegraing equaions ranges from one o hree depending on he ime period esed. These es resuls srongly suppor he predicions of our model ha coinegraion should exis beween a foreign currency fuures exchange rae, he corresponding spo exchange rae, and shor-erm ineres raes in he wo counries of exchange. -INSERT TABLES 3A, 3B, AND 3C HERE- 4 The race es and he maximum eigenvalue es idenify he same number of coinegraing vecors in seven of he en sub-periods. Lukepohl, e al (2001) compare he power of he race and maximum eigenvalue ess. They find ha, in general, he wo ess show similar power, alhough in some cases he race es has greaer power. Overall, hey recommend he race es be used. Because of his, we repor boh he race es and he maximum eigenvalue es and when he wo ess disagree we use he number of coinegraing vecors indicaed by he race es. 10
Error Correcion Models A (vecor) error correcion model is a resriced VAR designed for use wih nonsaionary series ha are known o be coinegraed. An error correcion model has coinegraion relaions buil ino he specificaion so ha i resrics he long-run behavior of he endogenous variables o converge o heir coinegraing relaionships while allowing for shor-run adjusmens. The error correcion model used in his paper is presened in equaion (9) where n is he number of coinegraing equaions. The noaion refers o he change in he level of he variable. 5 f n 2 2 2 2 d f, T α 0 + β j CE j + χ i f -i, T + δ i s i + φi i i,t + γi i i,t + η j= 1 i= 1 i= 1 i= 1 i= 1 = (9) The form of he coinegraing equaion, CE j, is presened in equaion (10) where he subscrip j corresponds o he number of he coinegraing vecor and j varies from 1 o 3 across he sub-ime periods. j 0j 1j f -1, T 2j s -1, T d 3j i -1, T f 4ji -1,T CE = Θ + Θ + Θ + Θ + Θ (10) The firs coinegraing vecor is normalized wih Θ 11 = 1. The second coinegraing vecor (when presen) is normalized wih Θ 22 = 1. Finally, he hird coinegraing vecor (when presen) is normalized wih Θ 33 = 1. The acual number of coinegraing vecors, n = 1, 2, or 3, corresponds o he Johannsen coinegraion es resuls presened in Tables 3A. 3B, and 3C. The error correcion model in equaion (9) is esimaed using five years of daily daa and hen used o forecas f,t in he following one-year ou-of-sample esing period. Equaion (11) hen defines a pricing error, x, as he difference beween he acual change, f,t, and he expeced change prediced by he model, E( f,t ), presened in equaion (9). x,t ( f ) = f E (11),T A posiive x suggess he foreign currency fuures exchange rae is higher han wha is warraned given he predicions of he model. Conversely, a negaive x suggess he foreign currency fuures exchange rae is lower han wha is warraned given he predicions of he model. To conrol for volailiy differences over he differen esimaion periods, we sandardize x based on Brown and Warner s (1980) mean-adjused reurns model as shown in equaion (12) where AERC is he abnormal exchange rae change. 5 While we only forecas he change in he foreign currency fuures exchange rae, o avoid exogeneiy issues we esimae he coinegraing error-correcion sysem using he Johansen full-informaion maximumlikelihood esimaion. In addiion, we resric ourselves o one-day-ahead forecass and, herefore, he forecass only rely on predeermined variables. 11
AERC x = (12) σ f ( f d / ) To idenify he exreme one-day price changes, we sor from highes o lowes he enire disribuion (combining all en esimaion periods) of he abnormal exchange rae changes for each currency. We hen choose he highes 5 percen as winners and he lowes 5 percen as losers. The resul is 244 observaions of exreme one-day price changes for each currency; 122 of hese observaions are defined as winners and 122 of hese observaions are defined as losers. In oal for all hree currencies, here are 732 observaions wih 366 winners and 366 losers. The saisical significance of he pricing error is deermined using he saisic in equaion (13) where n is he sample size, AERC i is he sandardized mispricing for even i on day, σ AERC is he sandard deviaion of he abnormalexchange rae changes for he 5 year pre-even period. Z = (13) 1 AERC i n σaerc The saisical significance of he average pricing error will be calculaed for day -3, day -2, day -1, day 0, day 1, day 2, and day 3. The saisical significance of he cumulaive average pricing error will also be calculaed for days -1 o -3, days -1 o -2, days 1 o 2, and days 1 o 3. The saisical significance will be repored for he full sample of winners followed by resuls for sub-samples peraining o economic announcemens, poliical announcemens and uninformed announcemens. This will be repeaed for losers. Two cross-secional regression equaions are used o assess wheher he marke s degree of mispricing is relaed o he ype of informaion (economic, poliical, and uninformed) while conrolling for oher facors. The oher facors are he iniial price change (sandardized mispricing on day 0) and he degree of leakage (sandardized mispricing on day -1). The regression model in equaion (14) is run separaely for winners and losers where AERC i,1 3 is he cumulaive sandardized mispricing for days 1 o 3 for even i, IPC i is he iniial mispricing (AERC i,0 ) on day zero for even i, LEAK i is he mispricing (AERC i,-1 ) on day -1 for even i, PN i is a dummy variable equal o 1 (0 oherwise) if even i corresponds o a news announcemen ha is poliical in naure, UN i = a dummy variable equal o 1 (0 oherwise) if even i does no correspond o a news announcemen, and ε i is he error erm. AERC = β + β + β + β + β i,1 3 0 1IPCi 2LEAKi 3PNi 4UNi i (14) + ε Resuls -INSERT TABLE 4 HERE- 12
Resuls for he degree of mispricing are disclosed in Table 4. The firs column breaks down he sample of losers and he sample of winners ino sub-samples based on he ype of informaion associaed wih he exreme fuures price changes. The second column shows he sample size. The nex eleven columns disclose he resuls of assessing he saisical significance of he mispricing during he hree day period surrounding day zero, which is he day of he exreme, one-day price change. Bold-ype indicaes saisical significance. For winners (op half of Table 4), here is subsanial evidence of pre-even mispricing especially on day -2. All of he signs on he saisically significan degrees of mispricing are posiive excep for he sign peraining o he uninformed sample on day -1. This suggess winners are associaed wih a subsanial degree of leakage before he even day (day 0). We can assess he degree of over- or under-reacion by examining he signs on he degree of mispricing for he pos even period (day 1, day 2, day 3, days 1 and 2, and days 1-3). For he full sample of winners he resuls do no sugges marke paricipans under-reaced (or over-reaced) o new favorable informaion peraining o currency fuure prices. When examining he full sample of winners here is no evidence supporing hypohesis one, which suggess he marke will under-reacion he new informaion. When examining he sample of evens corresponding o economic announcemens only, here is some evidence of hypohesis one; for day wo he sign on he degree of mispricing is posiive and saisically significan suggesing he marke under-reaced on he even day. When examining he degree of mispricing during he pos even period for poliical announcemens all of he signs on he degree of mispricing are negaive and saisically significan. This suggess he marke overreaced on he day favorable poliical informaion was released. When examining he degree of mispricing during he pos even period for uninformed winners all of he signs are negaive and saisically significan excep for day 3. This provides evidence ha marke paricipans over-reaced o informaion ha was no publicly released; his finding suppors hypohesis wo. For losers (boom half of Table 4), here is subsanial evidence of pre-even mispricing especially on day -2. All of he signs on he saisically significan degrees of mispricing are posiive. This suggess ha unfavorable informaion releases, like favorable informaion releases, were preceded by subsanial increases in foreign currency fuures prices. For he sample of all losers and he sample of losers peraining o economic announcemens he resuls do no sugges marke paricipans under-reaced (or overreaced) o new unfavorable informaion peraining o currency fuure prices. None of he pos-even degrees of mispricing are saisically significan. For he sample of losers peraining o poliical announcemens all of he pos-even signs are negaive and saisically significan excep for day 1. This suggess he marke under-reaced when unfavorable news peraining o currency fuures prices was released. This finding suppors hypohesis one ha exreme price changes will be followed by price changes in he same direcion (under-reacion). 13
For he sample of uninformed losers here is some evidence he marke overreaced a he ime unfavorable news was released. For day 3, he sign on he sandardized mispricing is posiive and saisically significan. This suppors hypohesis wo. For winners and losers our daa suppors hypohesis wo ha exreme changes in foreign currency prices will be associaed wih reversals when no news corresponds o he price change (uninformed evens). The evidence for winners is sronger han he evidence for losers. This finding is in agreemen wih Larson and Madura s (2001) sudy examining currency spo raes. For winners and losers peraining o poliical announcemens he evidence suggess he marke is overly opimisic when pricing new poliical informaion ha is eiher favorable or unfavorable. Tha is, for each sample, he degree of mispricing during he pos-even period is negaive and saisically significan excep for on day one for losers. This suggess ha during our sample period marke paricipans were opimisic abou foreign currency, or pessimisic abou he U.S. dollar. Overall, i appears ha conrolling for he ype of announcemen a he ime of exreme price changes is advanageous when rying o pinpoin when he marke for foreign currency fuures conracs over- and under-reacs o new informaion. -INSERT TABLE 5 HERE- Resuls for he leas squares esimaes are disclosed in Table 5. Resuls for winners appear in he op half of he able and resuls for losers appear in he boom half of he able. The main purpose of his analysis is o deermine wheher poliical and uninformed evens are associaed differen degrees of mispricing while conrolling for oher facors. The oher facors are he iniial price change and degree of leakage. The coefficiens on IPC and Leak are no saisically significan for boh winners and losers. Therefore, his daa does no suppor hypoheses hree and four ha higher iniial price changes and higher degrees of leakage will each be associaed wih higher degrees of overreacion. For winners he coefficiens on PN (poliical news) and UN (uninformed) are each negaive and saisically significan. This confirms he resuls in able four ha poliical evens and uninformed evens are associaed wih overreacion while economic evens are no even when conrolling for oher facors. For losers he coefficien on PN is negaive and he -saisic is -1.62. Therefore, his resul does no confirm he resuls found in Table 5. I seems imporan o noe ha he F-saisic is no significan. Conclusion This paper addresses marke overreacion and under-reacion for foreign currency fuures conracs. Using a 5-year moving window mehod and he foreign currency fuures pricing model of Amin and Jarrow (1991), we find repeaed evidence of coinegraion among he fuures price, he spo exchange rae, and ineres raes over en differen esimaion periods. An error-correcion model is hen used o develop a series of prediced fuures price changes. We assess wheher he marke overreaced or under- 14
reaced o new informaion by comparing he acual fuures price change o he change prediced by he error correcion model. Our resuls sugges he ype of underlying announcemen is relevan in pinpoining when he marke over- and under-reacs o new informaion peraining o foreign currency fuures prices. Specifically, i appears he marke overreacs o non-publicized news and is oo opimisic when poliical news is released abou foreign currency. 15
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TABLE 1 IN-SAMPLE ESTIMATION PERIODS AND OUT-OF-SAMPLE TESTING PERIODS (DAILY DATA) In-Sample Esimaion Period Ou-of-Sample Tesing Period 1991-1995 1996 1992-1996 1997 1993-1997 1998 1994-1998 1999 1995-1999 2000 1996-2000 2001 1997-2001 2002 1998-2002 2003 1999-2003 2004 2000-2004 2005 Noes: Our analysis makes use of a 5-year moving window mehodology which resuls in en in-sample 5-year esimaion periods from January 2, 1990 o December 31, 2004. The error correcion model for each of hese en 5-year esimaion periods is hen used o predic changes in he foreign currency fuures exchange rae in he following 1-year ouof-sample ime period. 19
TABLE 2A UNIT ROOT TESTS Briish pound (BP): Naural logarihm of fuures exchange rae (f $/BP ), naural logarihm of spo exchange rae (s $/BP ), and 3-monh LIBOR (i BP ). Time Time period Tes f $/BP s $/BP i $ i BP period Tes f $/BP s $/BP i $ i BP 1991 o 1995 1992 o 1996 1993 o 1997 ADF saisic -2.27-2.15-2.37-3.07 ** 1996 o ADF saisic -1.80-1.69-0.70-0.76 PP saisic -2.26-2.20-2.30-3.09 ** 2000 PP saisic -1.82-1.76-0.82-0.90 KPSS saisic 2.02 *** -2.12 *** 1.05 *** 3.19 *** KPSS saisic 1.04 *** 1.06 *** 1.58 *** 0.83 *** ADF saisic -2.34-2.28-0.45-2.81 * 1997 o ADF saisic -1.58-1.55 2.86 1.01 PP saisic -2.39-2.37-0.45-2.79 * 2001 PP saisic -1.55-1.55 2.61 0.69 KPSS saisic 0.90 *** 0.99 *** 3.11 *** 1.72 *** KPSS saisic 3.65 *** 3.66 *** 1.03 *** 2.67 *** ADF saisic -2.36-2.43-1.40-0.60 1998 o ADF saisic -1.60-1.52 1.09-0.99 PP saisic -2.39-2.35-1.38-0.71 2002 PP saisic -1.56-1.51 0.88-1.01 KPSS saisic 2.62 *** 2.57 *** 2.77 *** 1.60 *** KPSS saisic 3.13 *** 3.09 *** 2.91 *** 3.47 *** 1994 o ADF saisic -2.79 * -2.54-4.02 *** -1.54 1999 o ADF saisic -0.75-0.65 0.32-1.08 1998 PP saisic -2.72 * -2.51-3.95 *** -1.55 2003 PP saisic -0.78-0.71 0.20-1.10 KPSS saisic 2.74 *** 2.78 *** 0.82 *** 2.43 *** KPSS saisic 1.04 *** 1.05 *** 3.90 *** 3.76 *** 1995 o ADF saisic -3.05 ** -2.84 * -2.26-0.90 2000 o ADF saisic 0.02 0.07-1.95-1.63 1999 PP saisic -2.89 ** -2.80 * -2.27-1.04 2004 PP saisic 0.02 0.04-1.80-1.54 KPSS saisic 2.08 *** 2.07 *** 1.33 *** 0.57 ** KPSS saisic 3.33 *** 3.38 *** 3.64 *** 2.83 *** For he ADF and PP ess, he null hypohesis is ha he series is non-saionary and has a uni roo. Rejecing he null hypohesis suggess he series may be saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in MacKinnon (1991, 1996). The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% MacKinnon criical values, respecively. For he KPSS es, he null hypohesis is ha he series is saionary. Rejecing he null hypohesis suggess he series is non-saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in KPSS. The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% KPSS criical values, respecively. 20
TABLE 2B UNIT ROOT TESTS Japanese yen (JY): Naural logarihm of fuures exchange rae (f $/JY ), naural logarihm of spo exchange rae (s $/JY ), and 3-monh LIBOR (i JY ). Time Time period Tes f $/JY s $/JY i $ i JY period Tes f $/JY s $/JY i $ i JY 1991 o 1995 1992 o 1996 1993 o 1997 ADF saisic -1.25-1.25-2.37-1.99 1996 o ADF saisic -1.97-1.90-0.70-1.68 PP saisic -1.20-1.23-2.30-1.98 2000 PP saisic -2.00-1.96-0.82-1.76 KPSS saisic 3.90 *** 3.88 *** 1.05 *** 3.83 *** KPSS saisic 0.95 *** 0.94 *** 1.58 *** 2.26 *** ADF saisic -1.44-1.36-0.45-2.46 1997 o ADF saisic -1.65-1.57 2.86-1.19 PP saisic -1.41-1.36-0.45-2.47 2001 PP saisic -1.68-1.67 2.61-1.29 KPSS saisic 2.26 *** 2.16 *** 3.11 *** 3.98 *** KPSS saisic 1.11 *** 0.94 *** 1.03 *** 2.52 *** ADF saisic -1.11-0.90-1.40-2.65 * 1998 o ADF saisic -1.87-1.83 1.09-2.11 PP saisic -1.11-0.92-1.38-2.59 * 2002 PP saisic -1.89-1.89 0.88-2.12 KPSS saisic 1.21 *** 1.26 *** 2.77 *** 3.78 *** KPSS saisic 0.80 *** 0.79 *** 2.91 *** 2.29 *** 1994 o ADF saisic -1.16-1.08-4.02 *** -1.25 1999 o ADF saisic -1.65-1.53 0.32-2.65 * 1998 PP saisic -1.18-1.13-3.95 *** -1.25 2003 PP saisic -1.59-1.54 0.20-2.71 * KPSS saisic 3.12 *** 3.15 *** 0.82 *** 2.95 *** KPSS saisic 1.32 *** 1.22 *** 3.90 *** 1.41 *** 1995 o 1999 ADF saisic -1.29-1.24-2.26-4.65 *** 2000 o ADF saisic -1.74-1.66-1.95-2.50 PP saisic -1.31-1.29-2.27-4.66 *** 2004 PP saisic -1.75-1.68-1.80-2.77 * KPSS saisic 2.55 *** 2.56 *** 1.33 *** 2.00 *** KPSS saisic 1.00 *** 1.03 *** 3.64 *** 1.71 *** For he ADF and PP ess, he null hypohesis is ha he series is non-saionary and has a uni roo. Rejecing he null hypohesis suggess he series may be saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in MacKinnon (1991, 1996). The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% MacKinnon criical values, respecively. For he KPSS es, he null hypohesis is ha he series is saionary. Rejecing he null hypohesis suggess he series is non-saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in KPSS. The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% KPSS criical values, respecively. 21
TABLE 2C UNIT ROOT TESTS Swiss franc (SF): Naural logarihm of fuures exchange rae (f $/SF ), naural logarihm of spo exchange rae (s $/SF ), and 3-monh LIBOR (i SF ). Time Time period Tes f $/SF s $/SF i $ i SF period Tes f $/SF s $/SF i $ i SF 1991 o 1995 1992 o 1996 1993 o 1997 ADF saisic -1.14-1.13-2.37-0.04 1996 o ADF saisic -1.97-1.93-0.70-0.92 PP saisic -1.15-1.17-2.30-0.11 2000 PP saisic -1.97-1.94-0.82-0.74 KPSS saisic 2.24 *** 2.15 *** 1.05 *** 3.93 *** KPSS saisic 3.45 *** 3.45 *** 1.58 *** 1.23 *** ADF saisic -1.50-1.48-0.45-1.37 1997 o ADF saisic -1.97-1.95 2.86-1.23 PP saisic -1.48-1.50-0.45-1.32 2001 PP saisic -1.96-1.96 2.61-1.13 KPSS saisic 2.92 *** 2.83 *** 3.11 *** 3.68 *** KPSS saisic 3.50 *** 3.46 *** 1.03 *** 2.64 *** ADF saisic -1.42-1.32-1.40-1.76 1998 o ADF saisic -1.36-1.28 1.09-0.55 PP saisic -1.41-1.32-1.38-1.78 2002 PP saisic -1.34-1.29 0.88-0.50 KPSS saisic 1.19 *** 1.15 *** 2.77 *** 4.00 *** KPSS saisic 1.86 *** 1.78 *** 2.91 *** 0.95 *** 1994 o ADF saisic -1.66-1.57-4.02 *** -1.30 1999 o ADF saisic -0.57-0.41 0.32-0.13 1998 PP saisic -1.66-1.57-3.95 *** -1.23 2003 PP saisic -0.56-0.45 0.20-0.14 KPSS saisic 1.61 *** 1.67 *** 0.82 *** 3.54 *** KPSS saisic 1.86 *** 1.92 *** 3.90 *** 2.09 *** 1995 o 1999 ADF saisic -0.83-0.74-2.26-3.18 ** 2000 o ADF saisic 0.30 0.16-1.95-0.31 PP saisic -0.79-0.73-2.27-3.20 ** 2004 PP saisic 0.21 0.24-1.80-0.35 KPSS saisic 3.64 *** 3.64 *** 1.33 *** 2.79 *** KPSS saisic 4.17 *** 4.19 *** 3.64 *** 3.96 *** For he ADF and PP ess, he null hypohesis is ha he series is non-saionary and has a uni roo. Rejecing he null hypohesis suggess he series may be saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in MacKinnon (1991, 1996). The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% MacKinnon criical values, respecively. For he KPSS es, he null hypohesis is ha he series is saionary. Rejecing he null hypohesis suggess he series is non-saionary. The es saisics presened above are compared o he criical values deermined by he mehodology provided in KPSS. The ***, **, and * noaion indicaes ha he compued es saisic exceeds he 1%, 5%, and 10% KPSS criical values, respecivel 22