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Foreign exchange market efficiency andcointegrationMontserrat Ferré & Stephen G. HallPublished online: 07 Oct 2010.

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Foreign exchange market e� ciency and

cointegration

MONTSERRAT FERREÂ * AND STEPHEN G. HALL{

London Business School, Sussex Place, Regent’s Park, London NW1 4SA, UK and

{ Imperial College Management School, 53 Princes Gate, London SW7 2PGE-mail: walsh.ferre@skynet.be and { s.g.hall@ic.ac.uk.

The analysis of market e� ciency in the foreign exchange market adopted a newapproach after Granger (Oxford Bulletin of Economics and Statistics, 48(3), 1986)stated that assets in an e� cient market could not be cointegrated. If they were, therewould be a market ine� ciency since there would be Granger causality running atleast in one direction and thus one price could be used to forecast the other. Theinterpretation that the literature has given to the relationship between cointegrationand market e� ciency has been that noncointegration is a necessary and su� cientcondition for market e� ciency. In the authors’ opinion, the fact that two spotexchange rates are cointegrated does not necessarily imply that ine� ciency exists.In this article, it is argued that when the economy is composed of N exchange ratesand the closed system is analysed without dynamics, as it is the case when consider-ing the no-arbitrage condition, the Granger Representation Theorem (GRT) doesnot tell one anything about e� ciency. Further, when a subset J of the N exchangerates is considered, then the GRT becomes irrelevant for e� ciency. To illustratethese hypotheses will be the objective of this article along with that of developinga framework to test for e� ciency when cointegration is present.

I . INTRODUCTION

The analysis of market e� ciency in the foreign exchange

market adopted a new approach after an article by C.Granger was published in 1986. In particular, Granger

(1986) stated that assets in an e� cient market could not

be cointegrated. If they were, there would be a marketine� ciency since there would be Granger causality running

at least in one direction and thus one price could be used toforecast the other. Given the advances in the study of time

series with unit roots and the new ways to test for cointe-

gration that appeared in the late 1980s, the literature onmarket e� ciency adopted Granger’s statement and cointe-

gration techniques as a new way to test for e� ciency.Some of the literature that followed focused on whether

foreign exchange markets were e� cient by applying coin-

tegration techniques. Tests of the market e� ciency hypoth-esis through cointegration techniques were applied to the

usual analysis of spot and forward rates of the same curr-

ency, but they were also applied in a new fashion: to spot(or forward) rates of diVerent currencies. In the former

case, spot and forward prices should be cointegrated forconsistency with the e� ciency hypothesis and furthermore,

tests of restrictions on the cointegrating parameters shouldbe carried out to test that the lagged forward rate is an

unbiased predictor of the current spot rate. In the lattercase, when analysing the cointegration relationship

between two spot rates, one is testing whether the foreignexchange market is cross-sectionally e� cient. Under this

approach, the view has been adopted that two spot pricesfrom e� cient markets cannot be cointegrated.

There have been, however, studies that criticize the useof cointegration techniques to test for market e� ciency.

For instance, Sephton and Larsen (1991) point out that

the results of the Johansen test depend critically on themodel speci®ed or the period analysed. Others, like

Applied Financial Economics ISSN 0960±3107 print/ISSN 1466±4305 online # 2002 Taylor & Francis Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/0960310011009005 5

Applied Financial Economics, 2002, 12, 131±139

131

* Corresponding author.

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Hakkio and Rush (1989) indicate that the market e� ciencyhypothesis is in fact a joint hypothesis that agents are riskneutral and that they use all available information ration-

ally, so that the expected returns to speculators are zero.Thus, the violation of either hypothesis can lead to rejec-tion of the joint hypothesis, but the rejection of the jointhypothesis does not necessarily imply market ine� ciency.

A study by Crowder (1994) analyses whether the stationarylinear combination of spot rates that comprises the coin-tegrating relationship is proxying for a stationary and time-

varying forward risk premium in some way. However,Crowder obtains evidence that the error correction term,which is stationary by de®nition, could not be serving as a

proxy for the risk premium due to their diVering orders ofintegration. Thus, Crowder concludes that the predicta-bility implied by cointegration in the spot exchange ratesis consistent with a violation of the conditions for market

e� ciency.Another issue of concern in the analysis of e� ciency has

to do with ®xed exchange rate systems. Obviously, if

exchange rates are ®xed, one would not be surprised of®nding cointegration among them. Some authors, likeHakkio and Rush (1989), Copeland (1991) and BaVes(1994) argue that in the case of ®xed exchange rates,

there could be cointegration with e� ciency, given thatthese currencies are not diVerent assets.

In the authors’ opinion, the most convincing questioning

of the use of cointegration techniques to test for markete� ciency comes from the study of Dwyer and Wallace(1992). These authors criticize the de®nition of e� cientmarkets as markets in which changes in asset prices are

unpredictable because this de®nition does not have mucheconomic content and has little connection to the existenceof arbitrage pro®ts. Fama1 (1991) already pointed out thatchanges in stock prices can be predictable in an e� cient

stock market. Dwyer and Wallace consider a more usefulde®nition of an e� cient market as `one in which there areno risk-free returns above opportunity cost available to

agents given transaction costs and agents’ information’.The authors point out that this de®nition is in accordancewith Levich (1985) and Ross (1987), who argue that markete� ciency can be de®ned more usefully as lack of arbitrage

opportunities . Dwyer and Wallace argue that with markete� ciency de®ned as the lack of arbitrage opportunities,there is no general equivalence between market ine� ciencyand cointegration. In particular, they argue that the no

arbitrage condition on cross rates assuming that transac-tion costs are zero implies that the exchange rates betweenthree countries are cointegrated:

s12 ¡ s13 ¡ s32 ˆ 0 …1†

with sij being the logarithm of the spot price of country’s j’s

currency in terms of country’s i’s currency. They use the

interesting theoretical case where s12 and s13 are I(1) and s32

is stationary. This implies that s12 and s13 are cointegrated

with a coe� cient of minus one.

Following the line of reasoning set up by Dwyer and

Wallace, it is interesting to comment on the study carried

out by Karfakis and Parikh (1994). Karfakis and Parikh

use monthly data on the exchange rate of the Australian

dollar against the US dollar, Japanese yen, UK pound,German mark and French franc, from 1975 to 1990.

They test for cointegration among the ®ve diVerent

Australian rates using the Johansen procedure and they

obtain evidence of four cointegrating vectors. The authors

conclude that this is evidence against the market e� ciencyhypothesis. However, Karfakis and Parikh do not test for

the coe� cient of the cointegrating vector, which wouldprovide evidence of what kind of relationship exists

between the diVerent exchange rates. For instance, suppose

they obtained that the four cointegrating vectors responded

to the following structure of relations (note that this is

precisely the example posed by Dwyer and Wallace wherethe stationary term s32 is represented by eit):

US$

A$ˆ Yen

A$‡ e1t

US$

A$ˆ

UK

A$‡ e2t

US$

A$ˆ DM

A$‡ e3t

US$

A$ˆ FF

A$‡ e4t …2†

All these relationships would be giving evidence of a long-

run no-arbitrage condition between the US$ and the other

exchange rates, and therefore one would not think of the

presence of market ine� ciency in this environment.

Another study, by BaVes (1994) , argues that market e� -

ciency does not rule out predictable exchange rate move-

ments but only rules out arbitrage opportunities frompredictable exchange rate movements. BaVes indicates

that in turn, this is a result of the fact that a common set

of fundamentals determines exchange rates which is part of

the information set of all agents. He shows how purchasing

power parity2 (PPP) is a su� cient condition for the no-

arbitrage condition to hold. If sij represents the spot price

of country’s j’s currency in terms of country’s i’s currency,pi

t represents prices in country i, and PPP holds in three

countries:

132 M. Ferre and S. G. Hall

1 Fama (1991), page 1583 and after.2 The theory of PPP states that the exchange rate between two countries’ currencies equals the ratio of the two countries’ price levels.

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…ln s12t ¡ ln p1

t ‡ ln p2t † ¹ I…0†

…ln s23t ¡ ln p2

t ‡ ln p3t † ¹ I…0†

…ln s31t ¡ ln p3

t ‡ ln p1t † ¹ I…0† …3†

Adding the relations in Equation 3, and given that linearcombinations of I(0) variables are I(0), he obtains that theno-arbitrage condition is I(0):

…ln s12t ‡ ln s23

t ‡ ln s31t † ¹ I…0† …4†

The ®nding of cointegration according to the literaturewould mean ine� ciency, but this would at the same timecontradict the no-arbitrage condition.

Caporale and Pittis (1996) re-examine the relationshipbetween cointegration and market e� ciency in the case ofn-dimensional systems. They argue against the commonlyheld view that if cointegration is found in a vector of pricesfrom n markets, that implies ine� ciency of all n markets.They point out that in a multivariate system with cointe-gration, there may still be some assets for which there ise� ciency. They argue that in a cointegrated system with rcointegrating vectors, only r of the n markets will beine� cient.

In the authors’ opinion, the fact that two spot exchangerates are cointegrated does not necessarily imply that inef-®ciency exists. To illustrate this hypothesis will be theobjective of this article along with that of developing aframework to test for e� ciency when cointegration is pres-ent. The next section develops our analysis of e� ciency andcointegration and Section III provides an application ofthis analysis. Finally, Section IV concludes.

II . EFFICIENCY AND COINTEGRATIONREVISITED

The interpretation that the literature has given to the rela-tionship between cointegration and market e� ciency hasbeen that noncointegration is a necessary and su� cientcondition for market e� ciency. The argument has beenthat the presence of cointegration implies that an errorcorrection form exists, and this has been interpreted inthe literature as evidence that current prices are predictableusing last’s period deviation from the long-run cointegra-tion relationship, indicating the presence of unexploitedpro®t opportunities. It will be argued in this section thatthis interpretation has not been accurate.

The crux of the problem is the equalization by the litera-ture of the concepts of forecastability and ine� ciency. Theconcept of e� ciency has to do with lack of extraordinaryreturns ± or no arbitrage opportunities. Why is forecast-ability considered to be inconsistent with an e� cientmarket? If knowledge of the last period’s value of the vari-able X helps to predict this period’s value of the variable Y,

should then one conclude that there is ine� ciency? Notethat there may not be extraordinary returns to be madewhen considering a `wider’ model: other asset marketsmay react in such a way as to negate any extraordinaryreturns. Assume for example that last period’s US$/German mark exchange rate predicted that the US$/UK£rate was going to rise. If (uncovered) interest rate parityholds, then buying UK pounds last period ± and investingthem in securities ± to take advantage of the anticipatedappreciation does not yield extraordinary gains because theinterest rate on pound securities has fallen relative to dollarsecurities.

In general terms, the theory of e� cient markets is con-cerned with whether prices at any point in time `fullyre¯ect’ available information. The theory only has empiri-cal content, however, within the context of a more speci®cmodel of market equilibrium ± that is, a model that speci-®es the nature of market equilibrium when prices `fullyre¯ect’ available information. The empirical literature isbased on the assumption that the conditions of marketequilibrium can be stated in terms of expected returns.This assumption is the basis of the `fair game’ e� cientmarkets models. In our particular market, the foreignexchange market, this fair game should be related to thelack of arbitrage opportunities. If the German mark wereselling for US$0.30 in New York and US$0.35 in London,pro®ts could be made through arbitrage and thus onewould conclude that this market would be ine� cient.

It is argued therefore that ®nding cointegration is not ameasure of e� ciency. In the ®rst place, a theoretical ex-ample will be developed where the lack of cointegrationamong spot exchange rates under the no-arbitrage con-dition would imply ine� ciency. The relationships betweene� ciency and cointegration under two perspectives willalso be reviewed in the next lines. In the ®rst place, a closedsystem with no dynamics will be analysed and it will beargued that the existence of cointegration and of an errorcorrection mechanism (ECM) does not imply ine� ciency.This analysis can be applied to the analysis of all types ofassets ± although it is particularly useful for the case of spotexchange rates. How to test for e� ciency in this set-up willalso be illustrated.

In the second place, a partial system will be analysed,which is the normal case in empirical studies. Here, it willbe claimed that the forecastability of variables and e� -ciency cannot be analysed through a cointegration analysis.How omitting variables can lead to rejection of e� ciency inan e� cient system will be illustrated.

Lack of cointegration and ine� ciency in spot markets: anexample.

In the following lines an example will be developed thatdraws on the original argument made by Dwyer andWallace (1992). In this particular example, lack of cointe-

Foreign exchange market e� ciency 133

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gration among the exchange rates would imply ine� ciencygiven that the no-arbitrage condition would be violated.

Consider a three-country economy that is e� cient. Thus,the no arbitrage condition holds between the two exchangerates and their cross-rate, expressed in logarithms:

s12t ¡ s13

t ˆ s32t …5†

Further, assume that both s12t and s13

t are martingales:

Et¡1…s12t ¡ s12

t¡1† ˆ 0

Et¡1…s13t ¡ s13

t¡1† ˆ 0 …6†

which could be modelled as:

s12t ˆ s12

t¡1 ‡ "1t …7†

s13t ˆ s13

t¡1 ‡ "2t …8†

with "1t ¹ N…0; ¼21† and "2t ¹ N…0; ¼2

2†.Consider two possible scenarios: (i) when s32

t is station-ary, and (ii) when s32

t is non-stationary.

(i) If s32t is stationary, the no-arbitrage condition

implies that s12t and s13

t are cointegrated CI(1, 1)with cointegrating vector (1, -1).

Subtracting s12t¡1 and adding s13

t¡1 to both sides ofthe no-arbitrage condition Equation 5, one obtains:

s12t ¡ s12

t¡1 ¡ …s13t ¡ s13

t¡1† ˆ s32t ¡ s12

t¡1 ‡ s13t¡1 …9†

and thus,

"1t ¡ "2t ˆ s32t ¡ s12

t¡1 ‡ s13t¡1 …10†

If s12t and s13

t are cointegrated with cointegratingvector (1; ¡1), this implies that an error correctionform exists such that:

¢s12t ˆ ¬0 ¡ ¬1…s12

t¡1 ¡ s13t¡1† ‡ ¸t …11†

and from Equation 10:

¢s12t ˆ ¬0 ¡ ¬1…s32

t ¡ "1t ‡ "2t† ‡ ¸t …12†

Expression (12) is an ECM derived from an e� cientmarket with cointegration. Further, the ECM inEquation 12 does not indicate any type of forecast-ability for the variable s12

t .

(ii) If s32t is nonstationary, the no-arbitrage condition

implies that there is a cointegration relationshipbetween the 3 variables that satis®es:

s12t ¡ s13

t ¡ s32t ˆ ¹t …13†

with ¹t stationary and zero mean error term, and thecointegrating vector is (1, ¡1, ¡1). It is assumedthat s32

t also follows a martingale diVerence and inparticular:

s32t ˆ s32

t¡1 ‡ "3t …14†

Subtracting s12t¡1 and adding s13

t¡1 and s32t¡1 to both

sides of the no-arbitrage condition (13):

s12t ¡ s12

t¡1 ¡ …s13t ¡ s13

t¡1† ¡ …s32t ¡ s32

t¡1† ˆ ¹t ¡ s12t¡1

‡ s13t¡1 ‡ s32

t¡1 …15†

and thus:

"1t ¡ "2t ¡ "3t ˆ ¹t ¡ s12t¡1 ‡ s13

t¡1 ‡ s32t¡1 …16†

And given that s12t , s13

t and s32t are cointegrated, an

ECM exists:

¢s12t ˆ ¬0 ¡ ¬1…s12

t¡1 ¡ s13t¡1 ¡ s32

t¡1† ‡ ¸t …17†

and from Equation 16:

¢s12t ˆ ¬0 ¡ ¬1…¹t ¡ "1t ‡ "2t ‡ "3t† ‡ ¸t …18†

which, again, shows that the ECM in a closedsystem with no dynamics does not necessarilyimply forecastability.

In this example in particular, if there was no cointegrationamong the exchange rates in cases (i) or (ii), it would beconcluded that there is ine� ciency.

A closed system

First it will be illustrated that an ECM is an expressioncreated mathematically. This expression can be obtainedfrom any cointegrated system with or without e� ciency,and therefore, the simple fact that an ECM exists will nottell us anything about e� ciency. However, the precise formof the ECM will tell us something about e� ciency.

Consider an economy with only two assets, Xt and Yt,that have the following cointegration relationship:

Yt ˆ ¿Xt ‡ "t …19a†

Xt ˆ Xt¡1 ‡ ¸t …19b†

with "t, ¸t stationary error terms. Observe that this is aclosed model with no dynamics linking Xt and Yt. Thisexpression is compatible with an e� cient market: there isnothing in expression (19) that indicates ine� ciency. Theimportant point to notice is that from an expression withno dynamics such as Equation 19, it is always possible toobtain an ECM. Thus, cointegration and e� ciency can co-exist.

It will now be shown that one can rewrite expression (19)as an ECM. Subtracting Yt¡1 in both sides of Equation19a:

¢Yt ˆ ¡Yt¡1 ‡ ¿Xt ‡ "t …20†

and adding and subtracting ¿Xt¡1 to the right hand-side ofEquation 20:

134 M. Ferre and S. G. Hall

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¢Y ˆ ¡Yt¡1 ‡ ¿¢Xt ‡ ¿Xt¡1 ‡ "t

ˆ ¡…Yt¡1 ¡ ¿Xt¡1† ‡ ¿¢Xt ‡ "t …21†

which is an ECM representation. The fact that an e� cientexpression such as Equation 19 has been expressed into anECM such as Equation 21 should not mean any ine� ciencyif the true model is given by Equation 19.

More formally, expressing Equation 19 through matrixform:

1 ¡¿0 1

³ ´Yt

Xt

³ ´ˆ 0 0

0 1

³ ´Yt¡1

Xt¡1

³ ´‡ 1 0

0 1

³ ´"t

¸t

³ ´

…22†

and knowing that a system like:

AZt ˆ BZt¡1 ‡ ¹t …23†

can be written as:

A¢Zt ˆ …B ¡ A†Zt¡1 ‡ ¹t

ˆ CZt¡1 ‡ ¹t …24†

then Equation 22 becomes:

1 ¡¿

0 1

Á !¢Yt

¢Xt

Á !ˆ

¡1 ¿

0 0

Á !Yt¡1

Xt¡1

Á !

‡1 0

0 1

Á !"t

¸t

Á !…25†

which is the structural form ECM. The presence of the sameparameter ¿ in the ®rst row of the matrices A and C is whatis indicating e� ciency in the system in structural form.

To obtain the reduced form the following operations willbe performed in Equation 24:

¢Zt ˆ A¡ICZt¡1 ‡ A¡I¹t

ˆ ¦Zt¡1 ‡ D¹t …26†

with ¦ ˆ A¡IC and D ˆ A¡I :Thus the reduced form ECM of Equation 25 is:

¢Yt

¢Xt

Á !ˆ

¡1 ¿

0 0

Á !Yt¡1

Xt¡1

Á !‡

1 ¿

0 1

Á !"t

¸t

Á !

ˆ ¦Yt¡1

Xt¡1

Á !‡ D

"t

¸t

Á !…27†

The presence of the same parameter ¿ in the matrices ¦and D is, again, indicating e� ciency in the reduced formsystem. However, the interpretation in this case is not asstraightforward given that ¿ is also present in the errorterms.

Therefore, the presence of the same parameter ¿ is whatis indicating the presence of e� ciency in the system. Toclarify this result, consider the expression obtained in

Equation 21, but now assume that the parameters thataVect Xt¡1 and ¢Xt are diVerent:

¢Yt ˆ ¡…Yt¡1 ¡ ¿Xt¡1† ‡ ®¢Xt ‡ "t …28†

Note that if ¿ 6ˆ ®, by operating in Equation 28 it can bewritten that:

Yt ˆ ¿Xt¡1 ‡ ®¢Xt ‡ "t

ˆ ®Xt ‡ …¿ ¡ ®†Xt¡1 ‡ "t …29†

Now it is observed that there is ine� ciency in the expres-sion that links Yt and Xt given that past values of Xt canhelp to predict Yt. Therefore, it can be argued that when¿ 6ˆ ® in the ECM, there is ine� ciency. Note that Yt and Xt

cointegrate, given that an ECM exists.Thus, with this simple case it has been illustrated that the

existence of an ECM is not what determines ine� ciency,but rather what form this ECM takes. Therefore, in aclosed system with no dynamics, the GrangerRepresentation Theorem alone cannot be used as a `test’for the presence of e� ciency.

Note that one can test whether the same parameter ¿ isaVecting Xt¡1 and ¢Xt in the ECM expression (21) or (25),and thus a testable expression has been developed for e� -ciency in foreign exchange markets. It should be pointedout that an econometric problem could arise when imple-menting this test through a single equation estimation. If Yt

is not exogenous, the presence of ¢Xt on the right-handside of the equation results in simultaneous equation bias.For example, consider the following speci®cation ofEquation 19 where ¿ is set equal to 1:

Yt ˆ Xt ‡ "t

Xt ˆ Xt¡1 ‡ ¸t …30†

Xt is a random walk and determines endogenous Yt.Renormalizing as an ECM, one obtains:

¢Yt ˆ ¡…Yt¡1 ¡ Xt¡1† ‡ ¢Xt ‡ "t …31†

Noting that …Yt¡1 ¡ Xt¡1† ˆ "t¡1 and ¢Xt ˆ ¸t, Equation31 can be expressed as:

¢Yt ˆ ¡"t¡1 ‡ ¸t ‡ "t …32†

But if the researcher runs the regression:

¢Xt ˆ …Yt¡1 ¡ Xt¡1† ‡ ®¢Yt ¡ "t …33†

where the true coe� cients on the terms are equal to 1, thecorrelation between ¢Yt and "t biases the estimated coe� -cients away from 1. Note that cov…¢Yt; "t† ˆcov…¡"t¡1 ‡ ¸t ‡ "t† ˆ var…"t† which is obviously nonzeroand hence OLS is biased. This increases the likelihood thatthe equality will be erroneously rejected. Note, however,that this problem only appears if the researcher runs asingle equation estimation. If the researcher estimates atwo-equation full information maximum likelihood(FIML) system, then the problem disappears.

Foreign exchange market e� ciency 135

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Therefore, we will proceed in the following way to testfor e� ciency. First an `unrestricted’ expression ofEquations 21 or 25 will be estimated by full informationmaximum likelihood (FIML). By unrestricted it is meantthat we will not be imposing the parameter that aVects Xt¡1

to be equal to the parameter that aVects ¢Xt. Secondly, a`restricted’ expression of Equations 21 or 25 will be esti-mated by FIML where the parameter that aVects Xt¡1 isthe same as the parameter that aVects ¢Xt. If a likelihoodratio test cannot reject that the unrestricted and restrictedmodels are the same, the e� ciency between the assets Xt

and Yt will not be able to be rejected. This framework willbe applied to exchange rates of the European MonetarySystem in Section IV. Note, however, that for spotexchange rates it is required that the parameter ¿ beequal to 1 in order to satisfy the no-arbitrage condition.

A partial system.

In this section it will be argued that the analysis of e� -ciency should not be carried out through cointegration.This is due to the fact that when dealing with the intract-able dimension of the real economy, one has to deal withonly a subset of the variables. The eVects of the omittedvariables in the results of the analysis are not negligible. Infact, as will be shown below, they can lead to the wrongconclusions.

The eVects of dealing with a partial system will be illu-strated by considering an economy composed of only threeassets: Xt, Yt and Zt.

Assume that the three assets have the following expres-sions:

Xt ˆ ¬Zt ‡ "1t

Yt ˆ ­ Zt ‡ "2t

Zt ˆ Zt¡1 ‡ "3t …34†

with "it ¹ N…0; ¼2i †, for i ˆ 1; 2; 3. If the true relationship is

like Equation 34, one would conclude that there is e� -ciency between Xt and Yt. Note that Xt and Zt, Yt andZt, and Xt and Yt are pairwise cointegrated.

Nonetheless, if only the assets Xt and Yt are analysed,ignoring the presence of the third asset Zt in the economy,and a regression analysis is carried out, any of these tworesults would be obtained (depending on what variableproxied the omitted variable Zt):

(1) if Xt proxies for ¬Zt:

Xt ˆ Xt¡1 ‡ ¸1t

Yt ˆ ­

¬Xt¡1 ‡ ¸2t …35†

(2) or, if Yt proxies for ­ Zt:

Xt ˆ ¬

­Yt¡1 ‡ ¾1t

Yt ˆ Yt¡1 ‡ ¾2t …36†

Observe that Xt and Yt would still be cointegrated, but itwould be concluded that there is ine� ciency given that onevariable can help to forecast the other. For instance, in the®rst case, if Xt proxies for ¬Zt, one would conclude thatpast values of Xt help to forecast Yt. However, from thetrue model Equation 34, it is known that this is not true.

The main point in this argument is that when a partial ormarginalized analysis is carried out, what the true relation-ships are among the variables is not known and thereforeone will not be able to say anything about the forecastabil-ity and e� ciency of these variables. Therefore, when a sub-set J of the N exchange rates is considered, the GRTbecomes irrelevant for e� ciency.

Unfortunately, it must be acknowledged that empiricalmodels always have to deal with partial systems given theintractable dimension of the economy. Therefore, research-ers should acknowledge the eVects of omitted variables onthe results of their analysis.

III . AN APPLICATION

The above exposition has illustrated why in the analysis ofa partial system one might reach the wrong conclusionsabout e� ciency. However, in the real world, when carryingout empirical analysis, the possible eVects of not being ableto include all the variables of interest must be acknowl-edged. Therefore, in empirical applications we will alwaysassume that we are dealing with a closed system hopingthat this is close to the true situation.

In this section the previous analysis will be applied toexchange rates of the European Monetary System, and twopairs of exchange rates that are cointegrated will be ex-amined. It is realized that three exchange rates from theEMS constitute a very little subset of the real economy.However, this analysis is carried out because the maininterest in this section is to apply the testing procedurethat is outlined in the preceding section. In particular itcan be demonstrated that the presence of cointegrationdoes not say anything about e� ciency. It will be shownthat in two diVerent markets where there is cointegration,one will turn out to be e� cient and the other one ine� cientaccording to the proposed testing procedure.

Two cases will be analysed: (i) the monthly exchangerates for the Belgian franc (BF) and the German mark(DM) in terms of the US dollar for the period fromFebruary 1984 to April 1997, and (ii) the monthly exchangerates for the Austrian schilling (AS) and the German mark(DM) in terms of the US dollar for the period fromJanuary 1980 to April 1997. As shown in Table 1, boththe BF and the AS are cointegrated with the DM.

136 M. Ferre and S. G. Hall

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To test whether these exchange rates come from an e� -cient market now follows. First, the testing procedure forthe Belgian franc and the German mark will be developed.According to the reasoning developed above, this is doneby testing whether the same parameter ¿ is aVecting ¢DMand DM-1 in the following expression:

¢BF ˆ C ‡ ¿¢DM ¡ …BF¡1 ¡ ¿DM¡1† …37†

Note that in the case of spot exchange rates we will require

¿ ¡ 1 for e� ciency understood as lack of arbitrage oppor-tunities.The ®rst step in the analysis is to estimate the unrestrictedand the restricted models by full information maximumlikelihood (FIML). Second, a likelihood ratio test of theunrestricted and restricted models will be carried out to seewhether the restricted model can be rejected, which is theone that imposes e� ciency.

The unrestricted model will be:

¢BF ˆ C ‡ ®¢DM ¡ …’BF¡1 ¡ ¿DM¡1† …38†

The FIML estimates of the parameters of Equation 38 areshown in Table 2.

The restricted model will be:

¢BF ˆ C ‡ ¿¢DM ¡ …’BF¡1 ¡ ¿DM¡1† …39†

The FIML estimates of the restricted model Equation 39are presented in Table 3. The likelihood ratio (LR) test isgiven by:

LR ˆ 2 £ …510:95 ¡ 473:48† ˆ 74:94

The LR test statistic will follow a À2…m†, where m is thenumber of constraints (one in this example). In this case,the LR test is bigger than 3.84, the 5% signi®cance levelvalue of the À2…1† distribution, which implies a rejection ofthe restricted model with ® ˆ ¿, and thus a rejection ofe� ciency between the Belgian franc and the Germanmark. Furthermore, whether the parameter ¿ is equal to

one is tested at the 5% signi®cance level and, we are able toreject it:

¿ ¡ 1

Sd…¿† ˆ0:83 ¡ 1

0:0244ˆ ¡7 < ¡2:9;

where ¡2:9 is the t0:05…n ¡ 2† value. Equally, if we test atthe 5% signi®cance level whether the parameter l is equalto 1, we are able to reject it:

l ¡ 1

Sd…l†ˆ 0:84 ¡ 1

0:0243ˆ ¡6:55 < ¡2:9

Therefore, the no-arbitrage market condition is violated.The same analysis has also been carried out for the

Austrian schilling (AS) and the German mark. In particu-lar, FIML estimation of the unrestricted model,

¢AS ˆ C ‡ ®¢DM ¡ …’AS¡1 ¡ ¿DM¡1† …40†

gives the estimates shown in Table 4. It can be observedthat the parameter ’ is close to 1, and both ® and ¿ arequite close in value, which indicates that there might bee� ciency between these two exchange rates. In order totest for this, the restricted model is estimated by FIML:

¢AS ˆ C ‡ ¿¢DM ¡ …’AS¡1 ¡ ¿DM¡1† …41†

which gives the estimates reported in Table 5. The likeli-hood ratio test is:

LR ˆ 2 £ …849:675 ¡ 849:32† ˆ 0:7 < À2…1† ˆ 3:84

which implies a failure to reject the restricted model with

® ˆ ¿. Therefore, e� ciency between the Austrian schillingand the German mark cannot be rejected. Furthermore,

’ ˆ 1 at the 5% signi®cance, we fail to reject the hypoth-esis that ¿ ˆ 1 at the 5% signi®cance level,

¿ ¡ 1

Sd…¿†ˆ 0:993 ¡ 1

0:0075ˆ ¡0:94 > ¡2:9

Foreign exchange market e� ciency 137

Table 1. Bilateral cointegration trace tests results

Variables: Trace test 95% critical value

DM-AS 36.8 17.9DM-BF 35.7 17.9

Table 2. FIML estimates of the unrestricted model

Parameter Estimate t-statistic

C 1 6.64® 0.897 41.4’ 0.331 6.64¿ 0.327 6.62

Note: Log of likelihood function ˆ 510.95.

Table 3. FIML estimates of the restricted model

Parameter Estimate t-statistic

C 2.55 34.4’ 0.84 34.6¿ 0.83 34.1

Note: Log of likelihood function ˆ 473.48.

Table 4. FIML estimates of the unrestricted model for the Austrianschilling and the German mark

Parameter Estimate t-statistic

C 1.82 8.6® 0.993 132.7’ 0.932 8.62¿ 0.93 8.6

Note: Log of the likelihood function = 849.675

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and that

’ ¡ 1

Sd…’† ˆ0:995 ¡ 1

0:00765ˆ ¡0:65 > ¡2:9;

and so the arbitrage condition is ful®lled.The model has led to rejection of e� ciency between the

Belgian franc and the German mark, and to acceptance ofe� ciency between the Austrian schilling and the Germanmark. Although the Belgian currency has been a memberof the European Monetary System since its inception inMarch 1979, and the Austrian currency only joined it in1995, the results obtained above do not appear to be, in theauthors’ opinion, counterintuitive. In fact, the ®nding ofine� ciency between the Belgian currency and the Germanone might be explained by the dual exchange rate systemmaintained by Belgium throughout the 1980s.

On the other hand, Austria had followed a policy ofpegging its exchange rate to the German mark. DeGrauwe3 (1996), reports that Austria had followed a policyof an implicit commitment: `the authorities were com-mitted to a particular range of the exchange rate with theGerman mark’ since the start of ¯oating exchange rates.Isard4 (1995) states that `Austria had, de facto, pegged itscurrency to the Deutsche mark since 1981’.

IV. CONCLUSION

This paper has analysed the relationship between cointe-gration and e� ciency in the foreign exchange market in anew light: cointegrated exchange rates do not necessarilyresult from an ine� cient market. The interpretation thatthe literature has given to the relationship between cointe-gration and market e� ciency has been that no cointegra-tion is a necessary and su� cient condition for markete� ciency. The crux of the problem has been the identi®ca-tion by the literature of the concepts of forecastability andine� ciency. In general terms, the theory of e� cient mar-kets is concerned with whether prices at any point in time`fully re¯ect’ available information. The empirical litera-ture is based on the assumption that the conditions ofmarket equilibrium can be stated in terms of expectedreturns. This assumption is the basis of the `fair game’

e� cient markets model to which Fama (1970) referred to.In the particular market, the foreign exchange market, thefair game should be related to the lack of arbitrage oppor-tunities. The concept of e� ciency has to do with lack ofextraordinary returns ± or no arbitrage opportunities.

In the spot foreign exchange market, we have arguedthat e� ciency should be understood as lack of arbitrageopportunities. Therefore, in that market we have shownthat the no arbitrage condition for I(1) variable impliescointegration.

A framework has also been developed that allows one totest for e� ciency between cointegrated exchange rates. Thisframework has been applied to two exchange rates of theEuropean Monetary System cointegrated with the Germanmark, and it has been shown that ine� ciency was presentin one of the relationships but not in the other one.

This analysis leads to the main conclusions. In the ®rstplace, when the economy is composed of N exchange ratesand the closed system without dynamics is analysed, as it isthe case when considering the no-arbitrage condition, theGranger Representation Theorem (GRT) does not tell any-thing about e� ciency. In fact, it is, in the authors’ opinion,what parameters appear in the error correction mechanismthat will tell something about the presence of e� ciency.

Secondly, when a subset J of the N exchange rates isconsidered, then the GRT becomes irrelevant for e� ciency.We have shown how omitting variables can lead us to rejecte� ciency in an e� cient system.

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138 M. Ferre and S. G. Hall

Table 5. FIML of the restricted model

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Foreign exchange market e� ciency 139

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