the cointegration relationships among g-7 foreign exchange rates

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International Review of Financial Analysis 17 (2008) 446–460

The cointegration relationships among G-7foreign exchange rates

Heejoon Kang ⁎

Kelley School of Business Indiana University Bloomington, IN 47405 USA

Received 25 October 2006; received in revised form 18 December 2006Available online 2 February 2007

Abstract

A searchmethod is applied to foreign exchange rates of G-7 countries, in terms of theUS dollar, to estimatecointegration relationships. The method searches numerically, by strictly following the definition of thecointegration, a particular linear combination of nonstationary series in order to make it a stationary series.The list of those exchange rates which are cointegrated from the new method is very different from thosederived from the conventional maximum likelihood estimation or ordinary least squares methods. The newmethod also provides confidence intervals for cointegration coefficients. From the confidence intervals, itis determined that certain G-7 currencies expressed in terms of the mark or the pound become stationary.© 2007 Elsevier Inc. All rights reserved.

JEL classification: C32; F31Keywords: Cointegration vector; Maximum likelihood estimation; Ordinary least squares; Search method

1. Introduction

When a linear combination of two or more nonstationary, I(1), time series becomesstationary, I(0), then they are cointegrated. Ever since Engle and Granger (1987) introduced theconcept, cointegration has been widely investigated, because many time series in economicsand business are I(1), and many I(1) series are indeed cointegrated. Cointegration has importantimplications towards long-run relationships such as predictability, causality, and marketefficiency among the time series in question. Berkowitz and Giorgianni (2001), Cushman, Lee,and Thorgeirsson (1996), Diebold, Gardeazabal, and Yilmaz (1994), Lajaunie and Naka (1997),Sephton and Larsen (1991), and others have investigated whether various foreign exchange

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447H. Kang / International Review of Financial Analysis 17 (2008) 446–460

rates are cointegrated, with varying conclusions. Both the ordinary least squares (OLS) methodand the maximum likelihood estimation (MLE) technique are commonly used in the literature totest and estimate cointegration relationships. Engle and Granger (1987) advocate the OLSmethod; Stock and Watson (1993), Wright (2000), and others use dynamic OLS methods; andJohansen (1988, 1991) and Johansen and Juselius (1990) have developed the MLE technique.

The objective of this paper is to develop and then apply a new search method to find thecointegration relationships among foreign exchange rates of the G-7 countries, in terms of theUS dollar. It will be shown that the new method finds some cointegration relationships thatthe conventional methods fail to disclose. A linear combination, y1t−λy2t, becomes I(0)between two I(1) series of y1t and y2t in the cointegration analysis. The first step of cointegrationanalysis is to find such λ values. The “Search” method is based on this definition. A value λ isnumerically searched by a trial-and-error method to make y1t−λy2t an I(0) series. If no λ valuesmake y1t−λy2t an I(0) series, then the two I(1) series are not cointegrated. A good methodshould find a cointegration relationship that other methods fail to find, because the concept ofthe cointegration relies upon the fact that any linear combination of the nonstationary seriesmay happen to become a stationary series. The Search method will be shown to be better thanthe traditional MLE and OLS methods in this regard.

As will be demonstrated later, the OLS method is inadequate to find a cointegrationrelationship. In addition to the fact that it often fails to identify the cointegration relationships,there is a difficulty of the direction of normalization. Between y1t and y2t, it is possible to regresseither y1t on y2t or y2t on y1t. Since the definition of the cointegration does not offer or require anyparticular direction of the normalization, two alternatives are equally valid. Unless the coefficientof determination, R2, is equal to one, the regression slope coefficients from the two regressionequations are not the reciprocals of each other. Therefore, there are potentially two distinctcointegration relationships. Moreover, as will be demonstrated later, one normalization oftenindicates that two nonstationary time series are cointegrated, whereas the other normalizationindicates otherwise. Even if both normalization schemes indicate a cointegration relationship,their numerical values are not reciprocals of each other because the value of R2 in the cointegrationregression is smaller than one.With more than two I(1) series, the situation gets more complicated.Among three I(1) series, for example, there will be three alternative normalizations. The nature ofthe cointegration relationships critically depends on the direction of the normalization.

The MLE method makes use of a vector autoregressive error-correction model (ECM) in theestimation and the test of cointegration. There is no longer a normalization difficulty. However,the estimation and the test of cointegration are performed indirectly through the ECM and byusing the characteristic roots of the canonical correlation matrix. The estimation of thecointegration coefficient assumes that the smallest characteristic root is zero. In practice, thesmallest characteristic root is not exactly zero. Hence, the MLE method only approximatelyestimates the cointegration coefficient and the extent of the approximation depends on themagnitude of the smallest characteristic root. The method, as will be shown later, often fails todetect a cointegration relationship. Equally importantly, the cointegration relationships indicatedin the model are erroneous, showing that the particular cointegration coefficients estimated do notsatisfy the cointegration condition. That is, the cointegrated coefficients estimated by the MLEmethod often do not produce a cointegration relationship, especially when the smallest char-acteristics roots are far from zero.

The new Search method does not suffer from any of those difficulties. First, there is nonormalization problem. It does not matter if a cointegration relationship is searched between y1tand y2t or between y2t and y1t. In fact, the cointegration coefficients from alternative normalizations

448 H. Kang / International Review of Financial Analysis 17 (2008) 446–460

are reciprocals of each other so that the unique cointegration relationship is independent of thenormalization direction. Second, by design, the Search method directly implements the definitionof the cointegration. It guarantees to find a cointegration relationship if it is indeed there. In fact, theSearch finds additional cointegration relationships that other MLE and OLS methods fail to find.Third, unlike theMLE andOLSmethods, the Searchmethod also finds confidence intervals for thecointegration relationships. In particular, confidence intervals are similarly searched to make surethat any cointegration coefficients between the intervals will make the linear combination of thegiven I(1) series still cointegrated.

Irreducible cointegration relationships exist, according to the definition given by Wickens(1996), in a minimum set of cointegrated I(1) time series. A removal of any I(1) series from the setof cointegrated time series would make the set no longer cointegrated. If an I(1) series is added toa set of cointegrated time series, the new set will also be cointegrated possibly with multiplecointegration relationships. Irreducible cointegration is important, because while cointegrationrelationships in the multiple cointegration are not well defined and thus are unstable, as shown byKang (2002), those in the irreducible cointegration are well defined and stable.

Monthly foreign exchange rates fromMarch 1973 to December 2001, the last month before theadoption of the euro, reveal far more bivariate cointegration relationships than what the OLS andMLE methods find. The cointegration list of the foreign exchanges of the Japanese yen, Germanmark, British pound, French franc, Italian lira, and the Canadian dollar from the Search method isconsiderably more extensive than those from the OLS and MLE methods. The Search method isthen applied to three foreign exchange rates to identify all trivariate irreducible cointegrationrelationships. Moreover, it turns out that some foreign exchange rates, in terms of dollars, arecointegrated with the cointegration coefficients of [1, −1], from the direct inspection of thecointegration confidence intervals from the Search method. This suggests that certain foreignexchange rates in terms of the currencies other than the US dollar are stationary, I(0), instead ofbeing nonstationary, I(1). In fact, either the lira in terms of the mark or the Canadian dollar interms of the pound is an I(0) time series. Whether a foreign exchange rate is I(1) or I(0) thereforedepends on the choice of a numeraire currency.

The structure of the paper is as follows. Section 2 describes in detail how the new Searchmethod directly implements the concept of the cointegration. Section 3 applies the new method,along with the MLE and OLS methods, to the foreign exchange rates of the Japanese yen, Germanmark, British pound, French franc, Italian lira, and the Canadian dollar, all in terms of the UnitedStates dollar. The Search method is shown to find more cointegration relationships than thetraditional OLS and MLE methods. The fact that certain foreign exchange rates in terms of theGerman mark or the British pound becomes I(0) is elaborated in Section 4. Brief concludingremarks are given in Section 5.

2. Search method

The concept or the definition of cointegration does not imply any causal directions between y1tand y2t. In fact, no causal directions are necessary among cointegrated I(1) series. Consider twocointegrated series of y1t and y2t, following Engle and Granger (1987), which contain a common I(1) zt as in

y1t ¼ kzt þ e1t;y2t ¼ zt þ e2t;zt ¼ zt−1 þ e3t;

ð1Þ


where λ≠0 and ε1t, ε2t, and ε3t are I(0). In (1), zt determines (or causes) both y1t and y2t, but y1tand y2t are not necessarily causally related. Both y1t and y2t are I(1) due to the dominance of ztover ε1t and ε2t. In general, a linear combination of y1t−γy2t, which is (λ−γ)zt+ε1t−γε2t, is alsoI(1) because of the dominant presence of zt. A particular combination of y1t−λy2t, where γ isequal to λ, becomes ε1t−λε2t. This linear combination in turn is I(0) by construction. Hence, y1tand y2t in Eq. (1) are cointegrated with the cointegration coefficient, [1, −λ].

If zt in Eq. (1) were known or observed, then the fact that y1t and y2t are cointegrated would betrivial, because we know how exactly y1t is determined and how y2t is determined. A linearcombination between the two series that becomes I(0) would be a by-product in the sense thatboth series happen to share the common I(1) determinant of zt. It is important to note that thecointegration relationship of y1t−λy2t is still I(0) even if zt is not observed.

Before we discuss a search method to estimate a cointegration vector, [1, −λ], testing ahypothesis to determine if [1, −λ0] is indeed a cointegration vector is also important. This test isequivalent to a test if xt (=y1t−λ0y2t) is I(0). Standard unit root tests can be applied. Among manyavailable test procedures, the Augmented Dickey-Fuller (ADF) test procedure will be adoptedhere. If the ADF test statistic shows that xt is I(0), say, at the 5% significance level, then y1t and y2tare cointegrated with the cointegrating vector, [1, −λ0]. The ADF tests depend on the number ofaugmented lagged terms in the equation. First, construct xt as y1t−λ0y2t and Δxt as xt−xt−1. Forthis Δxt, estimate and investigate the following regression equation:

Dxt ¼ aþ bxt−1 þ d1ðDxt−1Þ þ d2ðDxt−2Þ þ d3ðDxt−3Þ þ d4ðDxt−4Þ þ et; ð2Þ

where εt is the regression disturbance term. In the above, four lag terms are used on the right-handside. The order of lag length should be determined such that the regression residualsbecome white noise. In the literature, different lag terms are often used in the ADF testprocedure. The t-statistic of the coefficient, β, of xt−1 plays a key role in the cointegration test.The ADF statistic is expressed as b / se(b), where b is the OLS estimate of β and se(b) is itsstandard error. It appears to be the same as the usual t statistic to test the hypothesis that β=0.0,but its distribution is not Student's t. Instead, we have to look at the ADF table for the criticalvalues. If the value is smaller than the critical ADF test statistic, then the series, xt, is stationarywithout a unit root. For instance, when the number of observations is 346, the critical value at the5% significance level is −2.88.

Since the definition of the cointegration involves whether a linear combination is I(1) or I(0), adirect application of the definition calls for the test of xt (=y1t−λy2t) to evaluate for the presence ofa unit root. There are many different procedures for the unit root test: Phillips–Perron test, simpleDickey–Fuller test, variance ratio test, and numerous other approaches. In this paper, wewill adoptthe Augmented Dickey–Fuller test, which is the most commonly used method in the literature.

In the estimation of the cointegration vector, the search method uses a trial-and-error techniqueto find a value of b, with which a linear combination, xt=y1t−by2t, is investigated whether it is I(1)or I(0). In particular, the value of b is searched such that y1t−by2t becomes “most” stationary. Themost stationary series here, by definition, implies that it has the lowest ADF statistic. When theintercept term is included, the 5% critical ADF statistic for 346 observations in our applications is−2.88. If the ADF statistic of the most stationary series searched happens to be smaller than −2.88,then xt is I(0) without a unit root and consequently y1t and y2t are cointegrated. An ADF statisticgreater than −2.88 would indicate that xt(=y1t−by2t) is I(1), implying that y1t and y2t are notcointegrated (with the cointegration vector of b). Since b is searched to make the ADF statistic the


smallest, if that particular b does not make y1t and y2t cointegrated, then no other linearcombinations between y1t and y2t will be I(0).

In the test and estimation of cointegration, a grid search method is adopted. First, all the values ofb between −10 and +10 at the interval of 1.0 are tried to find the smallest ADF statistic. That is,generate y1t− (−10)y2t, y1t− (−9)y2t, y1t− (−8)y2t,…, y1t− (10)y2t. Each of these generated series isinvestigated by using the regression equation in Eq. (2) to calculate the corresponding ADF statistic.Arrange those ADF statistics to find the smallest ADF value. If the ADF statistic occurs at theboundary, say, at −10, then the boundary is further broadened, from −10 to, say, −100. When thesmallest ADF statistic is found at, say, b=1.0, within the interval, then a finer gridwith the interval of0.1 is tried between the interval from 0.0 to 2.0, surrounding the optimum b in the previous iteration.

In the next iteration, a series of new variables are generated: y1t− (0.0)y2t, y1t− (0.1) y2t, y1t− (0.2)y2t,…, y1t− (2.0)y2t. As before, each of these series is investigated by using a regression equation inEq. (2) to calculate the correspondingADF statistic. Arrange thoseADF statistics to find the smallestADF value. Suppose the new value of b is 1.3 that corresponds to the smallest ADF statistic. Then,use of a finer interval around 1.3, from 1.2 to 1.4, will find a more accurate value of b. This searchcontinues by reducing the grid size until a desirable accuracy is accomplished. In this paper, thesearch is continued until the interval of 10−6 is reached. Once the boundary values of, say, −10 and10, and the degree of desired accuracy are set, an analyst does not have to intervene or look into thesearch process at all. The optimal λ value and its corresponding, smallest ADF statistic will beproduced at the end of the search.

The definition or the concept of the cointegration does not require any specific direction of thenormalization. If [1, −b] is a cointegration vector between y1t and y2t, then the cointegration vectorof [−1 /b, 1] should also make the two series cointegrated. In other words, the cointegrationrelationships should not depend on the direction of normalization. Moreover, the ADF statisticfrom the cointegration vector of [1, −b] and the other ADF statistic from [−1 /b, 1] should beexactly the same. Both cointegration vectors are searched in this paper to make sure that they areindeed reciprocals of each other and, additionally, that the solution found is globally, not merelylocally, optimum. It should be noted that the Search method tries to find [1, −b] to make xt(=y1t−by2t) most stationary. If the optimum [1, −b] produces an ADF statistic, say, −2.50, then xt is I(1).In turn, y1t and y2t are not cointegrated.

In case y1t and y2t are found to be cointegrated because the smallest ADF statistic is indeedbelow −2.88, then a range of cointegration vector has also been searched to find the confidenceinterval of the cointegration vector. For instance, all the values of b within the range of0.6bbb1.5 may produce ADF statistics below −2.88. That is, the boundary values of 0.6 and 1.5,in this example, would produce the ADF statistics of −2.88 exactly. In this hypothetical situation,the 95% confidence interval for the cointegration vector will be between 0.6 and 1.5. Theconfidence interval constructed in this way can be used to test a hypothesis on the cointegrationvector. For instance, a hypothesis if [1, −0.8] is a cointegration vector will not be rejected at the5% significance level, because 0.8 lies within the 95% confidence interval.

For three I(1) series in the cointegration investigation, the Search becomes more computerintensive, but the extension is straightforward. Try two values of b1 and b2 in the generation ofy1t−b1y2t−b2y3t in a specified range of, say, from −10 to 10, at the interval of, say, 1.0, for bothparameter values. The grid sizes are made finer and finer around those values with the smallestADF statistic in previous iterations. When the global minimum is found, then the direction of thenormalization does not matter. That is, the same smallest ADF will be found regardless of thenormalization direction and the cointegration coefficients will be such that all threenormalizations will provide exactly the same cointegration relationships.


As before, if [1, −b1, −b2] produces the smallest ADF statistic, then either [−1 /b1, 1, b2 /b1] or[−1 /b2, b1 /b2, 1] should also produce exactly the same, smallest ADF statistic.

Once the optimal values of b1 and b2 are found and the time series in question are cointegrated,confidence intervals can be found in a similar way as in the bivariate case. For the given value ofthe optimum b1, the range of b2 is searched to make the series cointegrated. Then, similarly, forthe given value of the optimum b2, the range of b1 is searched to make the series cointegrated.Instead of the grid search method used in this paper, one can potentially develop moresophisticated and efficient Gauss or Newton search algorithms. Such algorithm may not bestraightforward, however, because we try to find λ with the smallest corresponding ADF. HereADF is not a simple function of the parameter in the model as ADF depends on the estimate andits standard error. Tasks of algorithm development will be left for other numerical analysts andcomputer programmers. It is reassuring to note that the grid search method is simple toimplement, computationally easy to understand, and a foolproof search method.

Before we apply the new method to find cointegration relationships among G-7 foreignexchange rates, the traditional OLS and MLE methods will be re-evaluated in their performanceof estimating cointegration relationships. The following model is used with known parametervalues to generate and analyze the data.

y1t ¼ zt þ xt;y2t ¼ zt − xt;;zt ¼ zt−1 þ e1t;xt ¼ gxt−1 þ e2t;

ð3Þ

where ε1t, and ε2t are mutually independent and separately iid (independently and identicallydistributed) standard normal variables. By construction, zt is an I(1) series, in particular, a randomwalk series, and xt is an I(0) series. The starting values of zt−1 and xt−1 are set to zero. The numberof observations is set to 200. To minimize the impact of the initial starting values, both zt and xt,are generated for 400 observations and then the first 200 of them are discarded in the analysis.

Once zt and xt are generated, y1t and y2t are subsequently generated by using Eq. (3). Both y1tand y2t are tested at the 5% significance level to make sure they are individually I(1)series through the ADF test with two lag terms. Moreover, by using the ADF test with two lagterms, y1t−y2t term is also tested, again at the 5% significance level, to verify that the two seriesare indeed cointegrated with the true cointegration vector of [1, −1]. A total of 10,000 replicationsare conducted in the sampling experiment. Three different g values of 0.9, 0.95, and 0.98 are tried.The data generation and the subsequent data analysis are conducted by using Fortran.

For y1t and y2t generated by Eq. (3), OLS, MLE, and Search methods are used to estimate thecointegration relationships and the cointegration relationships estimated are further evaluated fortheir accuracies. The results are given in Table 1. The autocorrelation parameter, g, is 0.90 inModel 1. When y1t is regressed on y2t and OLS is used, the mean slope coefficient over 10,000replications is 0.7342 with the standard deviation of 0.2012. The intercept term has beenestimated in every method, but its results are not reported for brevity, because its value does notchange the existence or the nature of the cointegration relationships. Throughout, the ADF testprocedures are adopted by using two lags in determining whether a particular series has a unitroot or not. In 1373 out of 10,000 replications, OLS estimated values do not produce anycointegration. That is, although time series are generated to be cointegrated, over 13% of time,OLS fails to find the cointegration relationship.

The mean squared error (MSE) is presented along with R2. For Model 1, the mean value of R2

is 0.5519 and the MSE of b1 is 0.1112. Next, y2t is regressed on y1t and OLS is used again in

Table 1Evaluation of cointegration relationships from OLS, MLE, and Search

A. Data generationy1t= zt+xt, y2t=zt−xt, zt= zt−1+ε1t, and xt=gxt−1+ε2t, where ε1t and ε2t are separately, independently and identicallystandard normal deviates.

Model 1 Model 2 Model 3Value of g 0.90 0.95 0.98

B. OLS estimationy1t=a1+b1y2t+e1t and y2t=a2+b2y1t+e2t.b1 0.7342 (0.2012) 0.6816 (0.2361) 0.6440 (0.2712)No cointegration 1373 2656 3390(b1−1.0)2 0.1112 0.1571 0.2003R2 0.5519 (0.2114) 0.4846 (0.2294) 0.4424 (0.2393)b2 0.7358 (0.2015) 0.6827 (0.2360) 0.6399 (0.2688)No cointegration 1341 2661 3348(b2−1.0)2 0.1104 0.1563 0.2019

C. MLE estimationCointegration coefficient, r, in y1t−ry2t.r 1.0297 (0.2611) 1.0461 (0.3193) 1.0657 (0.5721)No cointegration 20 64 115(r−1.0)2 0.0690 0.1041 0.3316

D. Search estimationCointegration coefficient, λ, in y1t−λy2t.λ 1.0289 (0.2582) 1.0424 (0.3170) 1.0551 (0.3406)No cointegration 0 0 0(λ−1.0)2 0.0675 0.0984 0.1190

E. Comparison: the number of replications in which Search λ is more accurate than MLE r.Search better than MLE 5221 5297 5264

Note: Standard deviations are given in parentheses. No cointegration shows the number of cases out of 10,000 replicationsthat are not cointegrated by the augmented Dickey–Fuller test with two lags.


estimating cointegration relationships. Because of the symmetry between y1t and y2t, the behaviorof b2 is similar to that of b1 with the mean value of 0.7358. In 1341 cases, OLS fails to find anycointegration relationships.

In Panel C, MLE is used in the estimation. The mean cointegration coefficient is found to be1.0297 with the standard deviation of 0.2611. The mean value fromMLE is much closer to the trueparameter value of 1.0 than that from OLS, and the MSE of 0.0690 fromMLE is almost half of thatfrom OLS. In 20 out of 10,000 replications, MLE does not identify any cointegration relationships.

The results from the Search method are given in Panel D. The mean cointegration coefficient is1.0289 with the standard deviation of 0.2582. The MSE is 0.0675, which is the smallest of allestimates. Without exception, cointegration is found in each and every of 10,000 replications.This is expected because the Search method is designed to find a cointegration relationship aslong as one is present. Finally, in Panel E, the number of replications in which Search is betterthan MLE in the sense that Search produces the cointegration coefficient closer to the true value isrecorded. For Model 1, Search λ is better than MLE r in 5221 replications.

As we move from Model 1 to Model 2, and then to Model 3; the autocorrelation coefficient, g,increases from 0.90 to 0.95, then to 0.98. Consequently, R2 in the OLS estimation goes down.Both b1 and b2 are less accurate with larger MSE. The number of replications in which two timeseries are not cointegrated with the value of cointegration coefficients estimated also increases.For instance, no cointegration is found in either 3390 or 3348 replications for Model 3 dependingon the OLS normalization direction.

Table 2Bivariate cointegration relationships among G-7 exchange rates

A. Mark and poundSearch method (with 2 lags)⁎⁎⁎Mark — 1.356721 [0.963, 2.065] Pound, ADF=−2.99

B. Mark and francSearch method (with 2 lags)⁎⁎⁎Mark — 0.979510 [0.764, 1.204] Franc, ADF=−3.15MLE method (with 2 lags)Mark — 0.41662 Franc, but ADF=−2.28MLE method (with 4 lags)Mark — 0.42566 Franc, but ADF=−1.60

C. Mark and liraSearch method (with 2 lags)⁎⁎⁎Mark — 1.024880 [0.617, 1.639] Lira, ADF=−3.42Search method (with 4 lags)⁎⁎⁎Mark — 1.022209 [0.890, 1.182] Lira, ADF=−2.96MLE method (with 2 lags)Mark — 0.58429 Lira, but ADF=−2.80MLE method (with 4 lags)Mark — 0.71880 Lira, but ADF=−2.53

D. Pound and francSearch method (with 2 lags)⁎⁎⁎Pound — 0.545810 [0.364, 0.704] Franc, ADF=−3.01OLS method (with 2 lags)⁎⁎⁎Pound — 0.667157 Franc, R2=0.671, ADF=−2.93(But Franc — 1.006165 Pound, R2=0.671, ADF=−2.25)

E. Pound and liraSearch method (with 2 lags)⁎⁎⁎Pound — 0.320660 [0.253, 0.377] Lira, ADF=−2.94Search method (with 4 lags)⁎⁎⁎Pound — 0.324881 [0.195, 0.419] Lira, ADF=−3.08MLE method (with 4 lags)⁎⁎⁎Pound — 0.32731 Lira, ADF=−3.08OLS method (with 4 lags)⁎⁎⁎Pound — 0.410011 Lira, R2=0.763, ADF=−2.91(But Lira — 1.860443 Pound, R2=0.763, ADF=−2.20)

F1. Pound and C$, up to December 2001Search method (with 4 lags)⁎⁎⁎Pound — 0.828870 [0.391, 1.227] C$, ADF=−3.13OLS method (with 4 lags)⁎⁎⁎Pound — 0.929476 C$, R2=0.495, ADF=−3.11(But C$ — 0.532958 Pound, R2=0.495, ADF=−2.05)

F2. Pound and C$, up to December 2004Search method (with 4 lags)⁎⁎⁎Pound — 0.814547 [0.223, 1.337] C$, ADF=−3.21MLE method (with 4 lags)⁎⁎⁎Pound — 0.95766 C$, ADF=−3.13

F2. Pound and C$, up to December 2004OLS method (with 4 lags)⁎⁎⁎Pound — 0.873240 C$, R2=0.487, ADF=−3.20(But C$ — 0.557803 Pound, R2=0.487, ADF=−2.46)

G. Lira and C$Search method (with 2 lags)

(continued on next page)


⁎⁎⁎Lira — 2.521447 [2.083, 2.945] C$, ADF=−3.05G. Lira and C$

Search method (with 4 lags)⁎⁎⁎ Lira — 2.539764 [1.830, 3.226] C$, ADF=−3.31MLE method (with 4 lags)⁎⁎⁎Lira — 2.65979 C$, ADF=−3.30OLS method (with 2 lags)⁎⁎⁎Lira — 2.553724 C$, R2=0.824, ADF=−3.05(But C$ — 0.322706 Lira, R2=0.824, ADF=−2.75)OLS method (with 4 lags)⁎⁎⁎Lira — 2.553724 C$, R2=0.824, ADF=−3.31⁎⁎⁎C$ — 0.322706 Lira, R2=0.824, ADF=−3.01

Notes: Monthly data start from March 1973. Unless otherwise specified, the data period extends to December 2001. Thesymbol, ⁎⁎⁎, indicates a cointegration relationship at the 5% significance level. The values in square brackets show the95% confidence intervals.

Table 2 (continued )


In Model 3, MLE does not find the cointegration relationships in 115 replications. The MSE is0.3316. Though inaccurate in the mean cointegration coefficients, OLS cointegration coefficientestimation is relatively better than MLE with a smaller MSE. Yet, the Search method is better thanOLS and MLE in producing more accurate mean cointegration coefficients with smaller MSE. Inall three models tried, Search finds more accurate cointegration relationships than MLE in over5200 replications. As expected, the Search method truly finds the cointegration relationships andit does with a greater accuracy, while OLS and MLE even fail to find such relationships. In theMonte Carlo experiments, Fortran has been used with IMSL (International Mathematical andStatistical Libraries) subroutines.

One important outcome of the sampling experiments in Table 1 is that both OLS and MLE arenot as good as Search in estimating the cointegration coefficients. The model in Eq. (3) issymmetric so that the regression of y1t on y2t and that of y2t on y1t produces similar results in thetest and the estimation of cointegration relationships. If the true cointegration coefficient is 0.8,for instance, rather than 1.0 as in Eq. (3), the normalization scheme as to which of the two y'sshould be regressed on the other will become an additional OLS difficulty.

The Search method is better than OLS and MLE by providing cointegration relationships moreaccurately, on average, with smaller mean squared errors, and in majority of replications. At thesame time, the Search method does find a cointegration relationship in many situations in whichOLS or MLE fails to find one. Now the new method will be applied to find cointegrationrelationships in foreign exchange rates.

3. G-7 foreign exchange rates

Monthly exchange rates are retrieved from St. Louis Federal Reserve Bank database, FRED,for the flexible exchange rate period fromMarch 1973 to December 2001, prior to the adoption ofthe euro; for the Japanese yen, German mark, British pound, French franc, Italian lira, and theCanadian dollar, all in terms of the US dollar. For the Japanese yen, British pound, and theCanadian dollar, the end period is extended to December 2004 to allow for additionalinvestigations for both common period and the extended period. All the exchange rates aretransformed by taking logarithms. As typically done in the literature, the presence of unit roots isinitially investigated by means of the Augmented Dickey–Fuller test procedure. Each of the rates


has one unit root at the 5% significant level, either at the lag length of two or of four, consistentwith the findings in the literature. Berkowitz and Giorgianni (2001), Cushman et al. (1996),Diebold et al. (1994), Ferre and Hall (2002), Kellard, Newbold, and Rayner (2001), Kugler andLenz (1993), Lajaunie and Naka (1997), MacDonald and Taylor (1991), Sephton and Larsen(1991), and Zivot (2000), among numerous others have investigated and reported that variousforeign exchange rates indeed have a unit root.

Many researchers (e.g., Auyong, Gan, & Treepongkaruna, 2004; Copeland, 1991; Ferre &Hall,2002; Hakkio & Rush, 1989; Karfakis & Parikh, 1994; Rapp & Sharma, 1999) have examinedforeign exchange rates to find cointegrated relationships by using either MLE or OLS method.

The Search method, by using Fortran, is used to estimate the bivariate cointegration rela-tionships as shown by Table 2. For both OLS and MLE methods, SAS version 9.1 is used. PROCREG is used for the OLS method and PROC VARMAX is used for the MLE method. In the lattermethod, ECM is utilized and both maximum eigenvalue and trace statistics are investigated to findif series in question are cointegrated. The 5% significance level and the 95% confidence intervalsare used, throughout. Although all the regression equations and the cointegration relationshipscontain a constant term, its estimation results are not reported for brevity.

As shown in Panel A in Table 2, the Search method finds a cointegration relationship betweenthe mark and the pound, because mark −1.356721 pound is I(0) with the ADF statistic of −2.99when two lagged terms are used in Eq. (2). The cointegration vector is therefore [1, −1.356721]and the 95% confidence interval is from −2.065 to −0.963. It should be noted that the confidenceinterval includes [1, −1] indicating that the difference between mark and pound, namely, mark−pound, is I(0). More will be discussed later about this special cointegration relationship. NeitherMLE nor OLS method finds any cointegration relationships, however.

In Panel B, the mark and the franc are shown to be cointegrated from the Search method.From the trace, but not from the maximum eigenvalue, statistic; the MLE method reveals thatthe two exchange rates are cointegrated. With two lags, the cointegration coefficient is given by[1, −0.41662]. With four lags, it is given by [1, −0.42566]. When these particular linearcombinations are used in Eq. (2), however, the ADF statistics are, respectively, −2.28and −1.60. That is, the ADF statistics show that those cointegration coefficients do not lead toany I(0) series. The cointegration vector from the MLE method is therefore incorrect. It shouldbe noted that those values suggested by the MLE method are not within the 95% confidenceinterval from the Search method. From the Search method, with two lags, the confidenceinterval is given by [0.764, 1.204]. Since −0.41662 is not within the interval from −1.204to −0.764, the cointegration relationships suggested by the MLE method are expected to beinadequate. The results for the mark and the lira in Panel C are similar to those in Panel B. TheMLE method again provides incorrect cointegration coefficients.

For the previous three cases, the OLS method finds no cointegration relationships. Betweenthe pound and the franc, as shown in Panel D, however, the OLS method indicates the presence ofcointegration. When the pound is regressed on the franc, R2 is 0.671, and the ADF statistic of−2.93 indicates the presence of a cointegration relationship. In fact, the cointegration is found inthe Search method and the cointegration coefficient, [1, −0.667157] from the OLS method iswithin the 95% confidence interval from the Search method. Yet, the OLS regression of the francon the pound produces the ADF statistic of −2.25, indicating that there is no cointegration. Theproduct of the two slope coefficients, 0.667157 and 1.006165, is the same as the value of R2.Since R2 is not equal to one, the two slope coefficients are not reciprocals of each other. This isa serious shortcoming of the OLS method. The estimation results are critically sensitive tothe choice of the left-hand side variable in the regression. Cointegration is found in one


normalization, but not in the other. On the other hand, the MLE method does not detect anycointegration relationships.

For the pound and the lira, all three methods find the presence of a cointegration relationship,although the Search method again finds cointegration relationships with both two and four lags.As before, the OLS methods present the contrasting results by indicating that there is acointegration in one normalization, but not in the other normalization. The MLE method finds acointegration with only four lags.

There are two, albeit overlapping, time periods for the pound and the Canadian dollar inPanel F. In the first period from March 1973 to December 2001, the Search method with four lagsand one of the two OLS methods again with four lags find a cointegration relationship. But theMLE method does not find any. In the extended time period up to December 2004, all threemethods find a cointegration relationship, although the OLS regression of the Canadian dollar onthe British pound does not find any. None of the methods detects any cointegration relationshipswhen only two lags are used in Eq. (2), however. The Search method shows that the cointegrationcoefficient confidence interval includes [1, −1] in both time periods.

Finally, the case of the lira and the Canadian dollar is given in Panel G. With two lags, theSearch method and only one of the two OLS methods show a cointegration relationship. Withfour lags, however, all three methods reveal a cointegration relationship. In this case, both OLSnormalizations indicate the presence of the cointegration. Of course, the point estimates of thecointegration coefficients implied by the two OLS method are different, because they are notreciprocals of each other. As expected, those cointegration relationships found in the MLE andOLS methods are contained within the 95% confidence intervals found from the Search method.

Out of all possible pairs of exchange rates, those not shown in Table 2 are not cointegrated. Forinstance, there is no cointegration, with either two or four lags, between the yen and any otherexchange rates. Those currencies replaced by the euro are more frequently cointegrated. Bothfranc and lira are cointegrated with the mark, although there is no cointegration between the francand the mark. The pound is also cointegrated with the franc and the lira, in addition to theCanadian dollar. The cointegration relationships between lira and C$ are more commonlydetected by all three methods.

Next, trivariate irreducible cointegration relationships are investigated. Since the lira and the C$are cointegrated, an addition of any foreign exchange to this pair will also be cointegrated. Forinstance, the addition of the mark will result to a multiple cointegration relationship with thecointegration rank of two, because the lira and the mark are cointegrated. As Kang (2002) hasshown, cointegration relationships are not well defined with unstable cointegration vectors whenthere is a multiple cointegration relationship. Even if a third exchange rate is not cointegrated witheither lira or C$, those three exchange rates will still be cointegrated because the lira and the C$ arealready integrated. Such cases are not considered in the investigation of the irreduciblecointegration relationships. In Table 3, all those trivariate irreducible cointegration relationshipsfound by the Search method are listed. By definition of the irreducible cointegration, a removal ofany one exchange rate will leave the other two exchange rates no longer cointegrated.

The Search method finds, with both two and four lags, a cointegration relationship among yen,mark and C$, as shown in Panel A. The MLE method does not find any cointegrationrelationships. In two out of the three normalizations, the OLS method indicates a cointegrationrelationship. When the Canadian dollar is chosen for the left-hand side variable, the OLS methoddoes not produce any cointegration relationships. It should be noted that those R2 values are forthe regression equations among the three exchange rates, but not for the subsequent ADFregression equations in Eq. (3).

Table 3Trivariate irreducible cointegration relationships among G-7 exchange rates

A. Yen, mark, and C$Search method (with 2 lags)⁎⁎⁎Yen — 1.455414 [0.941, 2.695] Mark+1.728899 [−2.933, −0.945] C$, ADF=−3.64Search method (with 4 lags)⁎⁎⁎Yen — 1.329971 [1.010, 1.758] Mark+1.665178 [−2.247, −1.181] C$, ADF=−3.18OLS method (with 2 lags)⁎⁎⁎Mark — 0.626180 Yen — 0.810032 C$, R2=0.751, ADF=−3.41⁎⁎⁎Yen — 1.153849 Mark+1.523157 C$, R2=0.864, ADF=−3.25(But C$+0.416892 Yen — 0.408537 Mark, R2=0.673, ADF=−2.87)OLS method (with 4 lags)⁎⁎⁎Yen — 1.153849 Mark+1.523157 C$, R2=0.864, ADF=−3.03⁎⁎⁎Mark — 0.626180 Yen — 0.810032 C$, R2=0.751, ADF=−2.95(But C$+0.416892 Yen — 0.408537 Mark, R2=0.673, ADF=−2.60)

B. Yen, mark, and poundSearch method (with 4 lags)⁎⁎⁎Yen — 1.497908 [0.918, 2.078] Mark+2.012629 [−4.253, −1.333] Pound, ADF=−3.15OLS method (with 4 lags)⁎⁎⁎Pound+0.528414 Yen — 0.770185 Mark, R2=0.525, ADF=−3.14(But Yen — 1.456979 Mark+0.993118 Pound, R2=0.823, ADF=−2.37)(But Mark — 0.535683 Yen — 0.532171 Pound, R2=0.780, ADF=−2.33)

C. Yen, pound, and francSearch method (with 4 lags)⁎⁎⁎Yen+6.835818 [−13.376, −5.056] Pound — 3.361910 [1.142, 4.842] Franc, ADF=−3.23OLS method (with 4 lags)⁎⁎⁎Pound+0.140372 Yen — 0.625064 Franc, R2=0.768, ADF=−3.09(But Yen+2.699118 Pound — 1.100570 Franc, R2=0.314, ADF=−1.96)(But Franc — 0.134133 Yen — 1.139194 Pound, R2=0.720, ADF=−2.37)

D. Mark, pound, and francSearch method (with 4 lags)⁎⁎⁎Mark+3.642961 [−6.203,−2.783] Pound — 2.337256 [1.397, 3.017] Franc, ADF=−3.12MLE method (with 4 lags)Mark+1.77372 Pound — 1.41111 Franc, but ADF=−2.68OLS method (with 4 lags)⁎⁎⁎Pound+0.271764 Mark — 0.763323 Franc, R2=0.768, ADF=−3.01(But Mark+1.080246 Pound — 1.074551 Franc, R2=0.383, ADF=−2.03)(But Franc — 0.356217 Mark — 1.005834 Pound, R2=0.797, ADF=−2.46)

Note: see notes to Table 2.


Among the yen, mark, and pound variables as shown in Panel B, one of the three OLSequations finds a cointegration relationship in addition to the Search method, with four lags. It isinteresting to note that those two normalizations which do not find any cointegration relationshipshave higher R2 values than the one normalization which produces the cointegration relationships.This shows that the numerical value of R2 does not tell us which one may produce thecointegration relationships. In fact, the one normalization that produces the cointegration happensto show the lowest R2 value. In contrast, larger R2 values are associated with the cointegrationrelationship in Panel A.

The results for the yen, pound, and franc in Panel C are very similar to those for the yen, mark,and pound in Panel B. The MLE method does not identify any cointegration results. Only one outof the three OLS normalizations reveals a cointegration relationship. Finally, although the MLEmethod indicates a cointegration relationship among the mark, pound, and franc, the particular


cointegration coefficients from the method do not lead to a cointegration relationship. Again, onlyone of three OLS normalizations indicates a cointegration relationship.

Overall, the Search method does find more cointegration trivariate cointegration relationships.The OLS method is very sensitive to the direction of the normalization. The MLE method doesnot find any trivariate cointegration relationships. Needless to say, those cointegration coefficientsobtained through the Search method do show that the particular linear combination of the givenforeign exchange rates indeed becomes a stationary series. This result is expected because theSearch method is designed to find the smallest ADF statistics among all possible linearcombinations and the cointegration relationships are identified when the corresponding ADFstatistics are below the critical value given by the ADF test procedure.

4. Stationary exchange rates

All the exchange rates thus far been analyzed are expressed in terms of the US dollars.Moreover, as commonly done in the literature, logarithmic transformations are taken for eachexchange rate. One distinct feature of the Search method is that it can also provide confidenceintervals for the cointegration relationships.

In Table 2, a few bivariate cointegration relationships contain [1, −1] in their 95% confidenceintervals. For instance, Panel A in Table 2 shows that log (mark/US$)−1.356721 log(pound/US$)is stationary. The confidence interval for the coefficient is from −2.065 to −0.963. Hence log(mark/US$)− log(pound/US$) is stationary at the 5% significance level by using two lags inEq. (2). Here, log(mark/US$)− log(pound/US$) is the same as log(mark/pound) or − log(pound/mark). The exchange rate of the German mark expressed in terms of the British pound or thatof the pound expressed in terms of the mark is therefore stationary. The ADF statistic from twolags is −2.90 as shown by Table 4.

Panel B in Table 4 shows that the (log of) the franc in terms of the mark or its reciprocal isalso stationary. Similarly, the lira in terms of the mark or its reciprocal is stationary. Finally,the Canadian dollar in terms of the pound or its reciprocal is stationary in both sample periodswhen four lags are used in the ADF test procedure. In the literature and at the beginning of thepresent paper, foreign exchange rates are shown to be I(1) nonstationary series. Correctlyspeaking, those G-7 currencies in terms of the US dollar are nonstationary. When some of thosecurrencies are expressed not in the US dollar but in other currencies, they become stationarywithout a unit root. Stationary foreign exchange rates are found between the Canadian

Table 4Stationary exchange rates

A. Pound in terms of Mark (or Mark in terms of pound)Log (Pound/Mark) with 2 lags, ADF=−2.90

B. Franc in terms of Mark (or Mark in terms of Franc)Log (Franc/Mark) with 2 lags, ADF=−3.14

C. Lira in terms of Mark (or Mark in terms of Lira)Log (Lira/Mark) with 2 lags, ADF=−3.42Log (Lira/Mark) with 4 lags, ADF=−2.96

D1. C$ in terms of Pound (or Pound in terms of C$), up to December 2001Log (C$/Pound) with 4 lags, ADFS=−3.08

D2. C$ in terms of Pound (or Pound in terms of C$), up to December 2004Log (C$/Pound) with 4 lags, ADF=−3.16

Note: see notes to Table 2.


dollar and the British pound in addition to some European currencies. The importance ofnumeraire currency in the cointegration analysis and the fact that European currencies behavedifferently from other currencies are also noted in Kugler and Lenz (1993) and MacDonald andTaylor (1991).

5. Conclusion

The Search method illustrated here directly applies the concept and the definition ofcointegration. It tries to find if there indeed exists a linear combination of I(1) series whichbecomes I(0). The method is shown to be superior to the conventional OLS and MLE methods inthe estimation of the cointegration vector. The Search method finds certain cointegrationrelationships that the OLS and MLE methods fail to find.

It is well known that unit root test procedures, including the Augmented Dickey–Fullerprocedure used in this paper, are sensitive to the lag lengths, the presence of the intercept term, orthe inclusion of an additional time trend in Eq. (2). Only lag lengths of two and four are adopted inthis paper. Such sensitivity is not unique for the Search method, however. The conventional OLSand MLE methods are also sensitive to all such factors. No matter what the lag length used inEq. (2), the Search method will find, by design, certain cointegration relationships that the MLEand OLS methods may fail to find. Likewise, the fact that the Search method will find morecointegration relationships than the other conventional methods will remain true whether theADF procedure includes other factors such as a time trend or other seasonal dummy variables.

The G-7 foreign exchange rates in terms of the United States dollar demonstrate the impor-tance of the use of the Search method. Foreign exchange rates are shown to be cointegrated in farmore cases than previously obtained from the use of the conventional methods. The new methodhas identified a few bivariate cointegrations and irreducible trivariate cointegration relationships.The Search method also produces the confidence intervals for the cointegration coefficients. Itturns out that certain cases of the confidence intervals in bivariate cointegration relationshipsinclude [1, −1], which indicates that the foreign exchanges expressed in terms of certaincurrencies, other than the US dollar, may be I(0) stationary series. In fact, the foreign exchangerates of pound/mark, franc/mark, lira/mark, and C$/pound are all I(0) stationary time series.

Since cointegration implies an important long-run relationship among the cointegrated timeseries, further study is needed among those particular foreign exchange rates that are found to becointegrated. The Search method is shown to be an effective method to find such cointegrationrelationships.

References

Auyong, H. H., Gan, C., & Treepongkaruna, S. (2004). Cointegration and causality in the Asian and Emerging ForeignExchangeMarkets: Evidence from the 1990s financial crisis. International Review of Financial Analysis, 13, 479−505.

Berkowitz, J., & Giorgianni, L. (2001). Long-horizon exchange rate predictability? The Review of Economics andStatistics, 83, 81−91.

Copeland, L. S. (1991). Cointegration tests with daily exchange rate data. Oxford Bulletin of Economics and Statistics, 53,185−198.

Cushman, D. O., Lee, S. S., & Thorgeirsson, T. (1996). Maximum likelihood estimation of cointegration in exchange ratemodels of seven inflationary OECD countries. Journal of International Money and Finance, 15, 337−368.

Diebold, F. X., Gardeazabal, J., & Yilmaz, K. (1994). On cointegration and exchange rate dynamics. The Journal ofFinance, 49, 727−735.

Engle, R. F., & Granger, C. W. J. (1987). Cointegration and error correction: Representation, estimation and testing.Econometrica, 55, 251−276.


Ferre, M., & Hall, S. G. (2002). Foreign exchange market efficiency and cointegration. Applied Financial Economics, 12,131−139.

Hakkio, C. S., & Rush, M. (1989). Market efficiency and cointegration: An application to the Sterling and DeutschmarkExchange Markets. Journal of International Money and Finance, 8, 75−88.

Johansen, S. (1988). Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control, 12,231−254.

Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models.Econometrica, 59, 1551−1580.

Johansen, S., & Juselius, K. K. (1990). Maximum likelihood estimation and inference on cointegration—With applicationsto the demand for money. Oxford Bulletin of Economics and Statistics, 52, 169−210.

Kang, H. (2002). Unstable multiple cointegration relations in the term structure of interest rates. Seoul Journal ofEconomics, 15, 31−54.

Karfakis, C. I., & Parikh, A. (1994). Exchange rate convergence and market efficiency. Applied Financial Economics, 4,93−98.

Kellard, N., Newbold, P., & Rayner, T. (2001). Evaluating currency market efficiency: Are cointegration tests appropriate?Applied Financial Economics, 11, 681−691.

Kugler, P., & Lenz, C. (1993). Multivariate cointegration analysis and the long-run validity of PPP. The Review ofEconomics and Statistics, 75, 180−184.

Lajaunie, J. P., & Naka, A. (1997). Re-examining cointegration, unit roots and efficiency in foreign exchange rates.Journal of Business Finance and Accounting, 24, 363−374.

MacDonald, R., & Taylor, M. P. (1991). Exchange rates, policy convergence, and the European Monetary System. TheReview of Economics and Statistics, 73, 553−558.

Rapp, T. A., & Sharma, S. C. (1999). Exchange rate market efficiency: Across and within countries. Journal of Economicsand Business, 51, 423−439.

Sephton, P. S., & Larsen, H. K. (1991). Tests of exchange rate efficiency: Fragile evidence from cointegration tests.Journal of International Money and Finance, 10, 561−570.

Stock, J. H., & Watson, M. W. (1993). A simple estimator of cointegrating vectors in higher order integrated systems.Econometrica, 61, 783−820.

Wickens, M. R. (1996). Interpreting cointegrating vectors and common stochastic trends. Journal of Econometics, 74,255−271.

Wright, J. H. (2000). Confidence sets for cointegrating coefficients based on stationary tests. Journal of Business andEconomic Statistics, 18, 211−222.

Zivot, E. (2000). Cointegration and forward and spot exchange rate regressions. Journal of International Money andFinance, 19, 785−812.

the cointegration relationships among g-7 foreign exchange rates

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