Download - Cointegration between exchange rates: a generalized linear cointegration model

Journal of Multinational Financial Management8 (1998) 333–352

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

Abstract

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 1994–1995 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

1. Introduction

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 offers one method of detecting such long runequilibrium tendencies between exchange rates in the spot or forward markets oracross markets.

* Corresponding author. E-mail: [email protected]

1042-444X/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved.PII S1042-444X ( 98 ) 00035-8

334 Y.-X. Lin et al. / Journal of Multinational Financial Management 8 (1998) 333–352

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 differences 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).

335Y.-X. Lin et al. / Journal of Multinational Financial Management 8 (1998) 333–352

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 1974–1983, among fivemajor currencies—the 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 different 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


equilibrium relationships which may occur between economic variables of differentintegration 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 different, then different cointegration vectors may be generatedfrom a given one.

However, the SLCM may not be sufficiently 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 differencing 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 vector

b such that b∞Xt~I(0), we call X1,t,,, X

n,t are cointegrated and b is called cointe-grated vector.

Under the traditional linear cointegration model there are usually four major


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:

Xt=bY

t+e

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)

Xt=P

1Xt−1+,P

kXt−k+m+e

t, t=1,,,T (1)

where et

are i.i.d with distribution Np(O, L). Johansen’s 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

P=−(I−P1−,−P

k)

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 b∞X is I(0). If the rank of P is greaterthan 0, but not full rank, then there exists a cointegrated vector b such b∞X is I(0).The technique avoids the information loss due to differencing 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


among given time series, say, X1,t, X2,t,,, Xn,t. This linear restraint will bind

X1,t, X2,t,,, Xn,t together and constrain long term changes in these time series so

that they behave like an I(0) process. This simple linear cointegration paradigmprovides a sufficiently 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 instance, nonlinear relationships existbetween US real GNP and the US Department of commence quarterly index ofleading indicators. We suspect that similar complex situations exist for exchangerate series as well.

If the long term relationship between eligible exchange rate series is complex linearrather than simple linear, then fitting an SLCM-type expression will not necessarilydetect any long run relationship due to mis-specification of the relationship. Theonly way to represent these more complex constraining relationships among say,X1,t, X2,t,,, X

n,t using cointegration is to broaden the simple, linear cointegratingexpression used in traditional analysis to include the more general case. We nowexpand the definition of simple linear cointegration given in definitions one and twoto include generalized linear models.

5.1. Definition 3

Let X1,t, X2,t,,, Xn,t be time series. If there exist a vector b∞= (b1, b2, ,, b

k) and

a function vector

f (x1,x2,,,x

n)=[ f

1(x1,x2,,,x

n), f

2(x1,x2,,,x

n),,, f

k(x1,x2,,,x

n)]∞

such that b∞f is I(0), where fi(X1,t, X2,t, ,, X

n,t) is I(1), for i=1, 2, ,, k, thenX1,t, X2,t, ,, X

n,t are cointegrated under the relationship f.The validity of this extension can be shown by setting k=n and f

i(x1,,, x

n)=x

iin Definition 3 which then reduces it to Definition 1. However, the meaning of‘constraint’ in Definition 3 is more general than in Definition 1. In Definition 3, weallow f

ito be any function of given time series, subject only to f

ibeing I(1) time series.

In defining generalized linear cointegration, we retain the concept of ‘linearity’ inthe sense of a long term, linear, constraining force acting between the function timeseries f1, f2,,, f

k. The fundamental difference between generalized linear cointegra-

tion and simple linear cointegration is that f1, f2,,, fk

are functions of X1,t, X2,t,,, X

n,t. The ‘linear internal force’ is only executed on f1, f2, ,, fk

instead of onlyon X1,t, X2,t, ,, X

n,t. The nonlinear components are then assigned to f1, f2,,, fk.

This extension has the important property that the technique developed for thesimple linear cointegration can still be used in generalized linear cointegration, butover a wider range of economic time series applications.


6. Identifying cointegration relationships with generalized linearity components

6.1. Identifying the rank order of elements

Generalization of the linear model requires modification of the tests used toestablish whether the necessary conditions for cointegration are met. Simple linearcointegration only requires a test of the I(1) of each time series in the vector, whilecointegration under the GLCM requires a test capable of identifying the order ofall elements in the cointegration expression, not just the component time seriesthemselves; for instance, cross-products and squares of series. One method of doingthis is to inspect the ACF plots for each element in a given time series vector toidentify their rank order, and then to confirm the I(0) or I(1) status of each complexelement by running Dicky Fuller test for each element (e.g. see Box et al., 1994 fordetails of ACF plotting and inspection).

Unfortunately, where particular complex elements of an I(1) time series vectorare found to be I(1), this result cannot be generalized to all time series. For example,simple random walk time series that are I(1) may not have squares and cross-products that are also I(1). Theoretically, the squares and cross-products of a simplerandom walk time series are not I(1). This is shown in Appendix 1. However, incertain situations, the squares and cross-products of a simple random walk timeseries can be practically accepted as I(1). However, the above analysis must berepeated for each variable before proceeding to cointegration tests, since the I(1)nature of cross-products or powers of the component variables of a proposed GLCMcannot be assumed.

6.2. Identifying a proposed GLCM expression

The identification of an eligible time series vector now leads to a second issue.The techniques for identifying SLCM type cointegration within a given vector maynot work for GLCM relationships. Cointegration between time series underDefinition 1 requires a strong linear relationship between series. Such a relationshipimplies similar trends and moving patterns between the series. Attributes that canbe identified through inspection of graphic plot of all the time series. However, thistechnique is unsuitable for identifying non-linear relationships typical of the GLCM,since non-linear relationships between series may not show any identifiable patternin the time series. An alternative is to inspect the ‘scatter plots’ of eligible elementswhich will reflect general relationships between series. This technique can beillustrated by simulation of time series from a model of the form:

Xt=X

t−1+dt

Yt=X

t+0.2X2

t+e

twhere t=1, 2,,, 500, and d

tare i.i.d. normal distributed with mean 0 and variance

4, et

are i.i.d. normal distributed with mean 0 and variance 1, and dt

and et

aremutually independent.


Table 1Johansen’s cointegration test on X

t, Y

t, X2

t

Ho:rank=p −Tlog(1−mu) using T−nm 95% −T sum lg(1−mu) using T−nm 95%

p==0 390.4** 388** 21.0 395.4** 393** 29.7p<=1 4.698 4.67 14.1 5.021 4.99 15.4p<=2 0.3223 0.3204 3.8 0.3223 0.3204 3.8

Standardized beta∞ eigenvectorsy x x21.000 −0.9995 −0.19990.07206 1.000 0.005189−2.103 0.9899 1.000

Potential nonlinear relationships that remain unidentified by inspection of a timeseries plot, will show up in the scatter plot Y

tvs X

t. The construction of Y

t,

Yt=X

t+0.2X2

t+e

t, indicates that Y

tand X

tcannot be cointegrated under Definition

1, but they, may be cointegrated under Definition 3. Indeed, inspection of ACFplots indicate that all components (X

t, X2

tand Y

t) are I(1) for all practical purposes.

A fact which is confirmed by unit root tests.If we define f1=Y, f2=X and f3=X2, then Y

tand X

tare cointegrated under the

relationship f=( f1, f2, f3) by Definition 3 since

Yt−X

t−0.2X2

t=e

tis I(0). The cointegrated vector is (1, −1, −0.2). This result can be confirmed byrunning a cointegration test on the data using Johansen’s method (Table 1). Theestimated cointegrated vector is (1, −0.9995, −0.1999) which is very close to thetrue cointegrated vector of (1, −1, −0.2).

6.3. Are strict normality requirements essential?

As mentioned before, Johansen’s method requires the residual vectors in Eq. (1)to be independent with identical multi-normal distribution. In this example, theresiduals for each equation satisfy the independence requirement, but not strictnormality conditions. The graphics testing for the normality condition test showthat only one of the three equations have anything like normally distributed residuals(Fig. 5).

This example raises the question of whether strict normality conditions are essen-tial to the Johansen approach. For large sample sizes, strict normality conditionscan be relaxed when estimating the cointegrating vector, since asymptotic normalityconditions on the residuals will still provide accurate hypothesis testing results forcointegrated vector. However, for inference purposes, the assumptions about theindependence and lack of correlation of residual vectors are more important thanthe normality condition.


7. Identifying GLCM-type relationships in exchange rate data

We now illustrate the techniques for identifying GLCM-type cointegration byreference to a vector of time series data for the log of daily exchange rates of theUS dollar against the Japanese yen, the German mark, the Singapore dollar and

Fig. 1. ACF plots for X1 ($US/J Yen) autocorrelations.

Fig. 2. ACF plots for X1 ($US/J Yen) (period(s) of differencing=1) autocorrelations.


Fig. 3. ACF plots for X21

($US/J Yen) autocorrelations.

Fig. 4. ACF plots for DX21

($US/J Yen) (period(s) of differencing=1) autocorrelations.

the Swiss franc, respectively, for the period 03/01/94 to 18/10/95. The series arereferred to as X1, X2, X3 and X4, respectively.

The first task is to identify the order for each exchange rate time series in thevector. All four series are found to be strictly I(1) and eligible for consideration.The second task is to establish whether all complex elements in a proposed GLCM-


type model are also I(1). As an illustration, assume a quadratic cointegrationexpression made up of the the above series as vector elements, their squares andtheir cross-products. For the expression to be valid all the elements must be strictlyI(1). Although a lack of absolute theoretical compliance in this regard may not benecessary for practical applications; a close approximation may be sufficient. Weillustrate for the square (X2

1of the US/yen series (X1).

Consider the ACF plots of X1,t

,DX1,t=X

1,t−X

1,t−1,X21,t

and D(X21,t

)=X21,t−X2

1,t−1. The ACF plots are presented in Figs. 1–4. The plots in Figs. 1 and 3represent the autocorrelations of ‘X1’ and ‘X2

1’ time series before differencing. The

plots of these series are slow to approach zero. However, after differencing, the plots(Figs. 2 and 4) quickly approach and stabilize around zero. From these results wecan accept that, for practical purposes, both X1 and X2

1are I(1). This proposition

is supported by the output from a Dickey-Fuller unit root test as shown in Table 2.A comparison of the t-adf results with the relevant critical values also show thatboth X1 and X2

1are I(1). The application of these tests to the squares and cross-

products of the remaining series (X2, X3 and X4) established all complex elementsas I(1).

7.1. Identifying an appropriate structural model

The second task involves identification of a GLCM-type relationship. This requiresscatter plot inspection of the elements associated with the time series vector ofexchange rates. This procedure illustrates how generalized linear cointegration mayexist despite the absence of simple linear cointegration. Cointegration tests areapplied using Johansen’s technique with asymptotic normality and independenceconditions rather than strict conditions Fig. 5.

The scatter plots for X1 against X2, X3 and X4, respectively indicate, only weaklinear relationships between X1, X2, X3 and X4. For convenience, only the scatterplot of X1 vs X2 is given (Fig. 6). The scatter plots give no indication of an SLCMtype relationship between component time series. The output from Johansen test(Table 3) confirms this impression). The null hypothesis: p=0, cannot be rejectedat the 0.05 level of significance (Table 3). There is no evidence of a linear cointegra-tion relationship between X1, X2, X3 and X4 under Definition 1. However, referenceto scatter plots indicates a potential relationship containing non-linear componentsbetween X1 and other exchange rates.

The initial problem is to identify an appropriate model to describe the relationship

Table 2DF unit root test for X1, X2

1DX1, DX2

1. Unit root tests 2–447; critical values: 5%=−1.94 1%=2.57

t−adf beta Y_1 sigma lag t−DY−lag t−prob F~prob

X 0.57046 1.0009 0.0070037 0X2 −0.12732 0.99963 0.0034591 0dif(X ) −22.144** −0.056583 0.0069952 0dif(X2) −22.318** −0.064787 0.0034520 0


Fig. 5. Residuals of Johansen’s cointegration equation for Y, X and X*Y: (a) Y; (b) X; (c) X*Y.


Fig. 6. Scatter plot of X1 with X2 ($US/J Yen, $US/DM). Legend: A=1 obs, B=2 obs, etc.

between X1, X2, X3 and X4. We now consider a model incorporating quadratic andcross-product terms as follows:

X1,t=d+a

2X2,t+a

4X4,t+a

2,2X22,t+a

3,3X23,t

+a4,4

X24,t+a

2,3X2,t

X3,t+a

2,4X2,t

X4,t+e

t, (2)

The criteria for choosing a fitted model is mainly based on the R2 value (thecoefficient determination) and the ACF plot of {e

t}. Thus, in Eq. (2), both R2 and

the ACF plot of {et} suggest that X1,t, X2,t, X4,t, X2

2,t,X2

3,t,X2

4,t, X2,tX3,t and X2,tX4,t

are possibly cointegrated. However, the cointegrated vectors cannot be estimated bydirect application of the ordinary least-squares method to the suggested model. Thedifficulty arises because {e

t} in model Eq. (2) may not be independent and of equal

variance for each t. The usual OLS procedure can now only be used as an initialapproximation of the generalized form of f in relation to Definition 3.

Eq. (2) suggests that X1, X2, X3 and X4 may be a GLCM-type relationship of the

Table 3Johansen’s cointegration test on X1, X2, X3, X4

Ho:rank=p −T log(1−mu) using T−nm 95% -T sum lg (1-mu) using T-nm 95%

p==0 22.55 22.35 27.1 36.49 36.17 47.2p<=1 10.35 10.26 21.0 13.94 13.82 29.7p<=2 3.235 3.207 14.1 3.594 3.562 1 15.4p<=3 0.3588 0.3556 3.8 0.3588 0.3556 3.8


form:

f=( f1, f2, f3, f4, f5, f6, f7, f8),

where

f1(X1,X2,X3,X

4)=X

1, f2(X1,X2,X3,X4)=X

2,

f3(X1,X2,X3,X

4)=X

4, f4(X1,X2,X3,X4)=X2

2,

f5(X1,X2,X3,X

4)=X2

3, f6(X1,X2,X3,X4)=X2

4,

f7(X1,X2,X3,X

4)=X

2X3, f8(X1,X2,X3,X4)=X

2X4.

As previously stated, all elements must be I(1). Unit root tests (Dickey–Fuller tests)and ACF plots confirm that all components of X are I(1), where

Xt=(X

1,t,X2,t

,X4,t

,X22,t

,X23,t

,X24,t

X2,t

X3,t

,X2,t

X4,t

).

Johansen’s method provides a formal test for a cointegrated vector b such thatb∞X

tis I(0) and shows the following results (Table 4).

The results suggest that there is precisely one cointegrated vector in the estimatedmodel. The estimate for the cointegrated vector is

b∞=(0.00727,1.000,−1.477,−2.077,1.341,−8.138,−3.000,11.100).

Table 4Johansen’s cointegration test for selected series and complex elements

Ho:rank= −T log using 95% −T sum lg using 95%p (1−mu) T−nm (1−mu) T−nm

p==0 70.04** 68.81** 51.4 181.8** 178.6** 156.0p<=1 33.88 33.29 45.3 111.7 109.8 124.2p<=2 28.19 27.69 39.4 77.86 76.5 94.2p<=3 20.08 19.73 33.5 49.68 48.81 68.5p<=4 13.78 13.53 27.1 29.6 29.08 47.2p<=5 10.67 10.48 21.0 15.82 15.55 29.7p<=6 5.099 5.01 14.1 5.153 5.063 15.4p<=7 0.05438 0.05343 3.8 0.05438 0.05343 3.8

Standardized beta∞ eigenvectorsX2 X1 X4 X1^2 X3^2 X4^2 X2X3 X2X41.000 0.007271 −1.477 −2.077 1.341 −8.138 −3.000 11.1099.47 1.000 −211.8 −1027.0 −720.7 −772.0 1593.0 694.5−0.5170 0.03604 1.000 −8.928 −8.198 4.131 18.13 −4.146−0.2972 −0.03685 0.3966 1.000 0.6795 2.048 −1.125 −2.538−0.02218 −0.0002195 0.05238 1.485 1.000 0.5181 −2.190 −0.71650.1656 −0.9439 4.693 −72.97 −43.05 1.000 105.7 13.840.01760 −0.004349 0.08356 −0.5578 −0.4916 0.1320 1.000 0.15660.1439 −0.01263 −0.05049 15.18 14.15 −0.6394 −29.49 1.000


The ACF plot for

b∞X=0.00727X1,t+X

2,t−1.477X

4,t−2.077X2

2,t+1.341X2

3,t−8.138X2

4,t−3.00X

2,tX3,t+11.10OX

2,tX4,t

indicates that b∞X is indeed I(0). Thus, although there was no evidence of an SLGIMtype relationship, there is a GLCNI type generalized linear cointegration betweenthe series. This relationship would remain undetected under assumptions of linearity.

8. Discussion

Application of the extended model to a series of daily exchange rate data supportedthe contention that a generalised linear form of cointegration may exist, even wherethere is no evidence of simple linear cointegration. The results imply that the processof exchange rate adjustment to an identified long term equilibrium may be character-ised by nonlinear models as well as simple linear expressions. Furthermore, theseGLCM-type processes may not be identified under a simple linear model.

The identification of these nonlinear processes increases our insight about theprocess or path of movement towards equilibrium. The importance of this sort ofextension is equivalent to the extension of linear regression to generalized linear.regression. The challenge is to interpret the meaning of extended forms of cointegra-tion in the light of: (i) expectations about future movements in the equilibriumposition; and (ii) economic theories about long run equilibrium relationships betweenexchange rate series.

On procedural matters, we have suggested GLCM-compatible methods for identi-fying the rank order of elements in a generalized linear cointegration expression,formulating the cointegration expression and then testing its validity and robustness.The procedures include the following steps: (i) testing the rank order of time serieswithin a given times series vector using accepted techniques; (ii) formation of aproposed cointegration function; (iii) test for the I(1) rank order of all proposedGLCM elements through ACF plots and unit root tests; (iv) test for potentialvalidity of the cointegration expression through inspection of scatter plots; (v)calculation of coefficient estimates using Johansen’s method; (vi) final validation ofefficiency of model relative to alternative models.

A necessary condition underlying the GLCM is an I(1) rank for all elementsentering the cointegration expression. We have shown that this condition is satisfiedfor one particular GLCM model that contains squares and cross-products of thecomponent series. At present, the elements of each new model must be individuallytested for rank order. A theoretical analysis of the conditions under which higherpowers and cross-products of I(1) series are also I(1) may help to make thisrepetitive procedure redundant.

The statistical aspect of the analysis also suggests several extensions. We defineda GLCM that extends the application of cointegration to a much wider class ofequilibrium dynamics than under the siinple linear model. There is a need to identify


the general classes of functions permissible under the GLCIM and to formallyinvestigate whether the I(1) requirement is satisfied for all elements in each model )in order to overcome the need for repetitive analysis as part of each investigation.

Our sequential procedures for establishing cointegration need further refinementto ensure the rigour and generality of each step. This includes formalizing theidentification of potentially non-linear series through ACF plots and testing thenecessity for strict normality and i.i.d. distribution of residuals in the cointegrationexpression. Two other issues suggest themselves. The form and derivation of errorcorrection models under the GLCM needs investigation. The forecasting efficiencyof these models relative to linear cointegration counterparts and naive martingaleforecasts also requires analysis. We are currently pursuing these issues

9. Conclusion

(1) The enlarged definition proposed by Flores and Szafarz (1996) is: ‘‘Definition.A multivariate process x

ithe p components of which have different orders of

integration, the highest being d1, is said to be cointegrated if there is a non-trivial linear combination of its components, with at least a non-zero scalarmultiplying one d1 component, which is integrated of order d*<d1. … thedefinition implies that at least two non-zero scalars must be associated withtwo I(d1) components.’’ (break inserted) (pp. 193–194).

(2) The main computer statistics package used in this paper is PcGive.

Acknowledgements

We are grateful for the helpful comments received from Dr A. Moreau, Dr E.Wilson and several participants of both the 10th Annual Banking and FinanceConference (UNSW ) and the Economics Department Seminar Series, University ofWollongong, and two anonymous referees. The assistance of Dr Wolf Friesling ofthe Commonwealth Bank is also appreciated.

Appendix A

Consider the simplest I(1) time series

Xt=X

t−1+et

and

Yt=Y

t−1+dt,

where X0=0, Y0=0 and where t=1, 2,,, and et

and dt

are i.i.d with distributionN(0, s1) and N(0, s2), respectively, e

tand d

tare mutually independent.


Property 1Let Z

t=X2

t−X2

t−1. Then

E(Zt)=Var(e

t)s2

COV(Zt,Zs)=0,t≠s,

and

var(Zt)=2(2t−1)s4

1.

ProofSince Z

t=X2

t−X2

t−1, we have

Zt=e

t(et+2 ∑

i=1t−1

et−i).

Because et

are i.i.d, it turns out that

EZt=E [e

t(et+2 ∑

i=1t−1

)]et−i)]=Ee2

t=s2 ,

Var(Zt)=EZ2

t−(EZ

t)2

=E [e4t+4e3

t∑i=1t−1

et−i+4( ∑

i=1t−1

et−i)2]−s4

1

2s41+4s4

1(t−1)

=2(2t−1)s41

and, if t≠s,

Cov(Zt,Zs)=E(Z

t,Zs)−E(Z

t)E(Z

s)=0,

as required.Property 2

Let Zt=X

tYt−X

t-1Yt−1. Then:

EZt=0,

Var(Zt)=2(2t−1)s2

1s22

COV(Zt,Zs)=0, if t≠s.

The proof of Property 2 is simple and omitted. Properties 1 and 2 point out thatRandom Walk square and cross-product is not I(1). However, following practicaltesting procedure, in certain situation and within certain limit, we still can approxi-mately accept that square and cross-product are I(1). Following are examples. Twosets of data are simulated from the following models:

Xt=X

t−1+et


(a)

(b)

Fig. 7. (a) ACF plots for X2 (period(s) of differencing=1) autocorrelations; (b) ACF plots for X*Y(period(s) of differencing=1) autocorrelations.

and

Yt=Y

t−1+dt,

where t=1, 2,,, 500, et~N(0, 0.5), d

t~N(0, 0.3), and e

tand d

tare independent.


Table 5DF unit root test for X2, X*Y, DX2, D(X*Y ). Unit root tests 2–499; critical values: 5%=−1.94, 1%=−2.57

t−adf beta Y1 sigma lag t−DY_lag t−prob F-prob

X2 −1.2974 0.99428 6.2522 0X*Y −0.80618 0.99734 4.1523 0diff(X2) −22.018** 0.017099 6.2618 0diff(X*Y ) −22.308** −0.00066276 4.1550 0

The ACF plots for X2t

and XtYt. ACF plots in Fig. 7 and the output in Table 5 all

suggest that we can accept that X2t

and XtYt

are I(1) in practice.

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