Cointegration between exchange rates: a generalized linear cointegration model
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Journal of Multinational Financial Management8 (1998) 333352
Cointegration between exchange rates: a generalizedlinear cointegration model
Yan-Xia Lin a,*, Michael McCrae b, Chandra M. Gulati aa School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522,
Australiab Department of Accounting and Finance, The University of Wollongong, Northfields Avenue, Wollongong,
NSW 2522, Australia
Accepted 28 February 1998
Traditional cointegration techniques assume a simple linear model of the long run equilib-rium dynamics between eligible series. This linear limitation is convenient, but restricts theidentification of more complex or non-linear forms of cointegration. This paper expands thesimple linear cointegration model (SLCM) to a more generalised linear cointegration model(GLCM) which can identify more complex long-run relationships that may not be detectableunder simple linear cointegration models. An application of the GLCM to a vector of dailyspot exchange rates for five major currencies over the 19941995 period confirms that complexlinear forms of cointegration can exist even where current simple linear cointegration modelsreveal no evidence of cointegration. 1998 Elsevier Science B.V. All rights reserved.
JEL classification: C40; C62; F31
Keywords: Exchange rate equilibrium; Generalized cointegration; Nonlinear cointegration
Since the 1973 Australian exchange rate float, a large literature has developed onthe stochastic modelling of exchange rate movements and the detection of potentiallong run equilibrium relationships between individual rates. Such equilibrium maybe implied by economic theories of rate determination such as the purchasing powerparity and real interest rate models or by the existence of spheres of influence suchas trade blocs. Cointegration analysis oVers one method of detecting such long runequilibrium tendencies between exchange rates in the spot or forward markets oracross markets.
* Corresponding author. E-mail: firstname.lastname@example.org
1042-444X/98/$ see front matter 1998 Elsevier Science B.V. All rights reserved.PII S1042-444X ( 98 ) 00035-8
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Under cointegration, time series are in long run equilibrium if forces within amodel keep them in close proximity so that some linear combination of them is astationary series (Layton and Tan, 1992; Granger, 1981; Granger and Weiss, 1983;Engle and Granger, 1987; Phillips and Ouliaris, 1986, 1987; Johansen, 1988, 1991;see Johansen and Juselius, 1990 for a review of associated topics). This techniqueprovides a precise statistical definition to the concept of a stable long-run equilibriumassociation between high frequency asset prices such as exchange rates.
In spot or forward currency markets, cointegration between exchane rate seriesimplies that long-term trends adjust according to an equilibrium constraint, whilethe transitory or cyclical components of the series fit into an error correction model(ECM) that takes into account past levels, first diVerences and any errors resultingfrom over- or under-adjustment to the long run equilibrium situation. This propertyimplies that the ECM contains predictive information about the future paths ofcomponent series.
However, advances in the modelling of exchange rate time series imply that thesimple linear cointegration model (SLCM) used in traditional cointegration analysismay fail to identify more complex, long run equilibrium relationships under suchsystems as chaotic or nonlinear time series models. In this paper, we extend thedefinition of SLCM to include higher order expressions such as the cubic andquadratic form in a generalised linear cointegration model (GLCM). The GLCMprovides a means of identifying more complex forms of long run equilibrium relation-ships between rates that cannot be detected under the SLCM. As part of thisanalysis, the necessary conditions for cointegration under the GLCM are developed,along with a procedure for identifying GLCM relationships. The procedure is appliedto a time series vector of daily exchange rates for five major currencies between1994 and 1995. The results show that complex linear cointegration between ratesmay occur even when no cointegration is detected by the SLCM.
2. Modelling exchange rate processes
Since Meese and Singleton (1982), a consensus developed that exchange rate timeseries are generally well characterised as I(1) processes exhibiting high volatility andnon-stationarity; for example, random walk processes which could wander withoutbound in response to random shocks (Corbae and Ouliaris, 1988; Mark, 1990;McNown and Wallace, 1989; see Bleaney and Mizen (1995) for a survey). Trendingand periodicity could be handled by model modification. For instance, most highfrequency asset prices exhibit seasonal volatility patterns. This periodicity requiresmodels in which the conditional heteroskedacity is explicitly designed to capture therepetitive seasonal time variation in the second-order moments (Bollerslev andGhysels, 1996).
However, recent work suggests that evidence for exchange rates as I(1) randomwalk processes is equivocal (Hsieh, 1989). Non-stationarity has been frequentlyrejected for very long spans of annual data in several studies (Bleaney and Mizen,1996; Cheung and Lai, 1993; Abuaf and Jorion, 1990; Edison and Klovland, 1987).
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While such findings may indicate the low power of the relevant tests on relativelyshort spans of data, it also raises the possibility that non-linear models may captureaspects of the dynamics not picked up by linear models (Meese and Rose, 1991;Diebold and Nason, 1990, Hsieh, 1989; Davidson, 1985; see Hsieh (1989) for anearly survey).
As early as 1989, Hsieh (1989) proposed that exchange rates act as purely deter-ministic processes that look random. Under these chaotic models, exchange ratechanges are seen as nonlinear stochastic functions of their own past, chaotic deter-ministic systems which show the same properties as stochastic systems. Using theBrock et al. (1987) test that data are IID, Hsieh found strong evidence for nonlineardependencies in the daily exchange rates, for the period of 19741983, among fivemajor currenciesthe British pound, the Canadian dollar, the Deutsche mark, theJapanese yen and the Swiss franc. The nonlinearity in the daily exchange rates waslargely due to conditional heteroskedasticity. Hsieh concluded that nonlinear modelsof short-term exchange-rate determination are needed to include these dynamicswithin time series models rather than making adjustments to statistical tests toeliminate the condition prior to diagnostic testing.
3. Implications for cointegration modelling
The use of cointegration by Baillie and Bollerslev (1989a) to identify a long runequilibrium cointegration-type relationship between seven major spot currenciessparked a large number of studies on the issue (Baillie and Bollerslev, 1989b; Hakkioand Rush, 1991; Sephton and Larsen, 1991; Diebold et al., 1994). However, theempirical evidence for cointegration between exchange rates remains incoiicltisive.For instance, Diebold et al. (1994) replicated the Baillie and Bollerslev (1989a) data,but found little evidence of cointegration between rates. A simple martingale modelstill dominated the error correction model in out of sample forecasts.
These cointegration models continue to be based on a SLCM representation oflong run equilibrium relationships between eligible rate series. This SLCM persists,despite advances in cointegration theory such as allowances for structural breaks,fractional cointegration and the inclusion of diVerent order series within the integ-ration equation. Structural breaks occur when changes in the underlying regime orpolicy environment cause permanent changes in the slope of the exchange rate seriesor shift the series up or down. Periodicity in the deviations from the estimatingcointegrating relationship is dealt with through the concept of fractional cointegra-tion, under which the exchange rates may be tied together through a long memoryI(d)-type process, rather than an I(0) process. In these cases, the inclusion of anerror correction term would only reduce the MSE of predictions from a simplemartingale model in the very long run, so that little, if any, predictive improvementwill be noticeable.
Traditionally, cointegration applies only to I(1) processes (Banerjee et al., 1993;Stock and Watson, 1993). Lutkepohl (1991) and Flores and Szafarz (1996) proposea broader definition that allows the cointegration expression to capture long run
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equilibrium relationships which may occur between economic variables of diVerentintegration orders. Their extension allows the standard ideas and methods of identi-fying cointegrated series to apply if the cointegration process is stationary. If theorders of elements are diVerent, then diVerent cointegration vectors may be generatedfrom a given one.
However, the SLCM may not be suYciently general to capture more complexnonlinear equilibrium dynamics within the cointegration model itself. Under theSLCM, these dynamics are often treated as undesirable complications requiringadjustments to residual data prior to diagnostics for white noise processes (e.g.autocorrelation and conditional heteroskedasticity). The challenge is to treat theexistence of such factors as evidence of mis-specification and to then identify thefunctional form which best characterises the cointegrating process. For example, ina GLCM process, movements towards equilibrium over time may tend to follow apath best characterised as a complex linear or nonlinear process.
4. A generalized linear cointegration model
The extension of SLCM-type cointegration functions to encompass the generalcase includes two possibilities: generalized linearity in the long run, cointegrationrelationship, and non-linearity in the ECM (see Granger and Hallman, 1991; Grangerand Swanson, 1996). It is the first type of generalized linearity, nonlinearity in thevariables within the cointegration equation, which concerns us here. To date, thisproblem has been solved by adjusting test procedures to allow for residual autocorre-lation, non-constant variance and lack of independence in the error term rather thanre-examining the structure of the underlying model for mis-specification.
We take the alternative approach by developing a generalized linear cointegrationmodel (GLCM ) to expand the form of the cointegration expression to a moregeneralized set of long run equilibrium functions. This removes the linear restrictionby allowing various classes of non-linear functions to be used as the structural formof the cointegration function. The following definitions summarise cointegration asdefined in Engle and Granger (1987) and Granger and Weiss (1983).
4.1. Definition 1
A series Xt with no deterministic component, which has a stationary, invertible,ARMA representation after diVerencing d times, is integrated of order d and denotedby Xt~I(d).
4.2. Definition 2
Let Xt=(X1,t,,, Xn,t) be a (transposed) time series vector. If there exists a vectorb such that bX
t~I(0), we call X1,t,,, Xn,t are cointegrated and b is called cointe-
grated vector.Under the traditional linear cointegration model there are usually four major
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steps in the application of unit root and cointegration techniques. First, unit roottests are used to establish whether elements of a time series vector are stationary ornon-stationary. Second, linear cointegration expressions are used to identify anylong run equilibrium relationships between eligible series. Third, the estimates ofthe long-run parameters are used in error correction models to estimate the short-run or dynamic disequilibrium relationships. Fourth, the robustness of the estimateddisequilibrium relationships, as reflected in the error correction models, are deter-mined through standard diagnostic tests. In the context of extending traditionalcointegration model we will focus on the first two steps.
Since cointegration is restricted to I(1) time series, unit root tests are traditionallyused to establish the rank of time series. Two popular tests are the Dicky Fuller(DF ) test for the first-order process and the Augmented Dicky Fuller (ADF ) testfor higher-order autoregressive processes (Dickey and Fuller, 1981).
Tests for a simple linear cointegration between eligible I(I ) series in a time seriesvector, say (X, Y ), involve a cointegration regression of the form:
tto test if the residual e
tis I(0) or not.
The Johansen test for cointegration is used in this paper in preference to theGranger and Engle method, since it is considered as econometrically superior to theearlier tests. The Johansen method treats the hypothesis of cointegration about avector time series X as a hypothesis of reduced rank of the long run impact matrixP=ab (Johansen, 1988; Johansen and Juselius, 1990). The procedure is brieflydescribed below.
Consider the model (not including seasonal term)
1Xt1+,PkXtk+m+et , t=1,,,T (1)
are i.i.d with distribution Np(O, L). Johansens method uses the maximum
likelihood technique to provide estimates of the number of cointegrated vectors andlikelihood ratio tests on ab with a x2 distribution as an approximation to thedistribution of the likelihood ratio test. Thus, if
has full rank, then the vector process X is stationary. If the rank of P is zero, thenthere is no cointegrated vector b such that bX is I(0). If the rank of P is greaterthan 0, but not full rank, then there exists a cointegrated vector b such bX is I(0).The technique avoids the information loss due to diVerencing under the Grangerand Engle method, but places linear restrictions on both the cointegration vectorsand the weights a (Hall, 1989).
5. Generalized cointegration model
The cointegration relation concept proposed under traditional analysis assumes asimple linear expression for the long term constraining force or tendency operating
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among given time series, say, X1,t, X2,t,,, Xn,t. This linear restraint will bindX1,t, X2,t,,, Xn,t together and constrain long term changes in these time series sothat they behave like an I(0) process. This simple linear cointegration paradigmprovides a suYciently large and cornplex range of models to suit the needs of manyanalysts.
However, simple linear cointegration may not be appropriate to all relationshipsamong economic data series or asset price time series. Economic theory suggestsseveral nonlinear models of such time series behaviour together with several practicalexamples (Granger and Terasvirta, 1993). For...