Asymmetric Cointegration

Yann Schorderet

No 2003.01

Cahiers du département d’économétrie

Faculté des sciences économiques et sociales Université de Genève

Février 2003

Département d’économétrie Université de Genève, 40 Boulevard du Pont-d’Arve, CH -1211 Genève 4

http://www.unige.ch/ses/metri/

asymmetric cointegration∗

By Yann Schorderet†

February 2003

Abstract

By decomposing a time series into its positive and negative partialsums, a nonlinear version of cointegration is introduced. This generaliza-tion of cointegration is appealing to model asymmetric behavior. Exami-ning the properties of this peculiar form of cointegration, we show thatthe stochastic structure of the relation calls for specific estimation andtesting procedures. The theory is applied to historical exchange ratesbetween the US dollar and some countries belonging to the EuropeanMonetary system.

Keywords : Nonlinear transformations, integrated processes,cointegration, asymmetry, exchange rates.

JEL classification : C22, F31.

1. Introduction

Since Granger (1981, 1983), Engle and Granger (1987), two or more stochas-tically trending series moving together over time are said to be cointegrated.From then on, the concept of cointegration and its applications in a variety ofareas in economics have been the subject of an enormous amount of literature.However, most of it has been devoted to the study of linear relationships.

The natural extension to the classical cointegration theory deals with non-linearities in the cointegration relation. In that way, the contributions in Gran-ger and Lee (1989), Granger and Hallman (1991), Granger (1995, 1997), Gran-ger and Swanson (1996), Gregory and Hansen (1996), Balke and Fomby (1997),

∗An earlier version of this paper was entitled ”A Nonlinear Generalization of Cointegra-tion : A Note on Hidden Cointegration”.

†Department of Econometrics, University of Geneva, Uni Mail, Bd du Pont-d’Arve 40,CH-1211 Geneva 4, Switzerland, E-mail : yann.schorderet@metri.unige.ch. This article waswritten in part while the author was a Visiting Scholar at the University of California,San Diego. I thank the Economics Department at UCSD for its hospitality and all theparticipants in the UCSD econometrics seminar for helpful discussions. I am in debt toClive Granger, Jaya Krishnakumar, Elvezio Ronchetti, Gawon Yoon for useful comments onearlier version of this paper. All remaining errors are my own responsibility. This researchwas supported by the Swiss National Science Foundation.

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Siklos and Granger (1997), Aparicio and Escribano (1998), Enders and Gran-ger (1998), Escribano and Pfann (1998), Enders and Siklos (2001), Park andPhillips (2001) provide new insights into the idea of cointegration and errorcorrection modelling.

In this paper, we explore a different point of view by considering possiblestationary relationships between nonstationary components of data series ra-ther than the series themselves. Explicitly, by distinguishing its positive andnegative increments, any time series can be broken down into its initial va-lue and its negative and positive cumulative sums. In that way, a randomwalk process can be decomposed into two random walk processes with drift.Examining the multivariate combinations arising from this decomposition, weinvestigate a new form of cointegration. This nonlinear version of cointegrationis intuitively appealing to model asymmetric behavior. We show that althoughbeing apparently linear in some particular variables, the model does not standfor a conventional cointegration relation.

The remainder of this paper is organized as follows. The next section pre-sents the peculiar series on which is based the cointegration relation we havebeen interested in. Focusing on the case where only one component of eachoriginal series appear in the model, we discuss in section 3 the nonlinear pro-perties of the equilibrium error and their implications on statistical inference.We show that the stochastic structure of the cointegration relation calls forspecific estimation and testing procedures. The technique suggested is illus-trated in Section 4 where the model is fit to data of real exchange rates. Someconcluding remarks are provided in Section 5.

2. Definitions and conceptual framework

Consider the following decomposition of a time series, say {xt}Tt=0

xt = x0 + x+t + x−t (1)

where x0 is the initial value of the process and1

x+t =

t−1∑i=0

1{4xt−i > 0}4xt−i (2)

x−t =t−1∑i=0

1{4xt−i < 0}4xt−i (3)

11{·} is the indicator function taking the value 1 if the event in brackets occurs and zerootherwise. 4 denotes the first-difference operator.

2

{x+t }T

t=1 and {x−t }Tt=1 are respectively the cumulative sums of the positive and

negative shocks defining the level of the original series at time t.Assume now that {xt} is a random walk process with a possible linear time

trend in mean. The proposition below states the main properties characterizingits components (2) and (3).

Proposition : Let {xt}Tt=1 be generated according to xt = xt−1 + ζt where

ζt ∼ iid (δ; σ2ζ ) of probability density function fζt(ζt) such that P (ζt > 0) > 0,

P (ζt < 0) > 0 with finite first and second moments. Let {x+t }T

t=1, {x−t }Tt=1 be

the resulting components defined as (2) and (3). Then

(a) {x+t } (respectively {x−t }) is monotonic increasing (decreasing), that is

x+t ≥ x+

t−1 ≥ 0 (x−t ≤ x−t−1 ≤ 0) t = 2, ..., T .

(b) {x+t } ({x−t }) follows a random walk process with positive (negative) drift.

(c) Cov (4x+t ;4x−t ) > 0.

Proof : See the appendix.

It follows from property (a) and (b) that {x+t } and {x−t } are nonstationary

series. In this respect, remark that {x+t } and {x−t } have a linear time trend in

mean even though {xt} does not. When {xt} is a random walk process, {x+t }

and {x−t } are integrated of order one. Property (c) emphasizes that {x+t } and

{x−t } are not independent.In this framework, suppose that two time series, say {x1,t} and {x2,t},

are not linearly cointegrated but there exists a linear combination of theircomponents, say

zt = β0 x+1,t + β1 x−1,t + β2 x+

2,t + β3 x−2,t (4)

which has a stationary distribution. This leads to the following definition.

Definition : Let {x1,t} and {x2,t} be two nonstationary time series and de-fine {x+

j,t} and {x−j,t} according to (2) and (3) j = 1, 2. {x1,t} and {x2,t} are saidto be asymmetrically cointegrated if there exists a vector β′ = (β0, β1, β2, β3),β 6= 0, such that {zt} in (4) is a stationary process.

In reference to the classical cointegration theory, zt will be called the equi-librium error.

The intuition behind the model is that the relation between some econo-mic variables might not be the same whenever they increase or decrease. Forinstance, setting β0 = β1 in (4) can be used to model asymmetric response

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of x1 to x2 in the sense that x1 reacts differently according to whether thesign of the growth rate of x2 is positive or negative. Setting further β2 = β3

yields the relation examined by Engle and Granger (1987). Hence, model (4)includes the classical cointegration relation as the particular case where thereis no asymmetry at all.

In what follows, we will focus on the situation where only one component ofeach original series appears in (4). This may be seen as a cointegration relationwhich ”operates”only in one direction. We show that although seemingly linearin the peculiar variables introduced above, the model stands for a nonlinearcointegration relation. Indeed, interpreting the model as a standard cointegra-tion relation neglects the intrinsic nonlinear properties of the series involved.Because of these, the regression function is nonlinear in the parameters andthe usual techniques of statistical inference are misspecified.

3. A nonlinear cointegration relation

Consider the asymmetric cointegration relation

x+1,t = β+ x+

2,t + z1,t t = 1, ..., T (5)

where {x+1,t} and {x+

2,t} are defined as (2) with respect to some time series{x1,t} and {x2,t} while {z1,t} denotes a stationary series. In the conceptualframework (4), the model stands as the case where β1 = β3 = 0, β+ = −β2/β0.Since x+

j,t → ∞ as t → ∞ j = 1, 2, the cointegrating parameter β+ takes apositive value.

The idea of such a model has been suggested by Granger and Yoon (2002).By considering (5) as a conventional linear cointegration relation between x+

1,t

and x+2,t, these authors do not take into account the nonstandard features of

the stochastic structure of the model. Here we aim to look into these aspectsin detail and discuss them in the context of statistical inference. We begin byemphasizing the nonlinear properties characterizing the equilibrium error z1,t.

A. Nonlinear properties of the equilibrium error

In view of the features of the left hand side variable, z1,t must comply with thefollowing inequality

x+1,t ≥ x+

1,t−1 ⇐⇒ z1,t ≥ x+1,t−1 − β+ x+

2,t t = 2, ..., T (6)

while for t = 1

x+1,1 ≥ 0 ⇐⇒ z1,1 ≥ −β+4x+

2,1 (7)

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As in the censored regression model developed by Tobin (1958) where thelimited dependent variable is viewed as the censoring process of some latentvariable with unbounded support, conditions (6) and (7) can be modeled bydefining z1,t as the outcome of some underlying disturbance, say ε1,t.

z1,t =

{max{−β+4x+

2,t; ε1,t} if t = 1max{x+

1,t−1 − β+ x+2,t; ε1,t} if t = 2, ..., T

(8)

ε1,t stands as the generating disturbance of the equilibrium error.Noting that

x+1,t−1 − β+ x+

2,t = z1,t−1 − β+4x+2,t (9)

(8) can be rewritten as

z1,t = max{z1,t−1 − β+4x+2,t; ε1,t} t = 1, ..., T (10)

with z1,0 = 0. {z1,t} follows a nonlinear autoregressive process. Substitutingrecursively yields

z1,t = max{ ε1,t ; ε1,t−1 − β+4x+2,t ; ε1,t−2 − β+4x+

2,t−1 − β+4x+2,t ; ... ;

ε1,1 − β+4x+2,2 − β+4x+

2,3 − ...− β+4x+2,t−1 − β+4x+

2,t ;z1,0 − β+4x+

2,1 − β+4x+2,2 − ...− β+4x+

2,t−1 − β+4x+2,t}

(11)where z1,0 = 0.

Basic properties of the stochastic process {z1,t} can be found in Hellandand Nilsen (1976). The latter have shown that if {ε1,t} and {4x+

2,t} are twoindependent iid sequences with E(|ε1,t|) < ∞, E(4x+

2,t) < ∞, P (4x+2,t > 0) >

0, then {z1,t} has a unique stationary distribution.Unless otherwise stated, we will consider the set of assumptions below.

Assumption : (a) {ε1,t} ∼ iid N(0 ; σ2ε). (b) {4x2,t} ∼ iid N(δ ; σ2

4x2).

(c) E(ε1,t4x2,s) = 0 for all t and s.

The normality assumption is not essential but is useful for ease of expositionof the next results. Note that in the context discussed here, it seems naturalto let E(ε1,t) = 0 since if there is no random disturbance, we expect thecointegration relation to be in equilibrium meaning that z1,t = 0 for all t. Thisis what we get by setting ε1,t = 0 for all t.

B. The reciprocal component {x−1,t}At this stage, only one ”part”of {x1,t} has been modeled. Specifying the processof the reciprocal component {x−1,t} is required to obtain a model for {x1,t}.

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To be internally consistent, the generating process of {x−1,t} must satisfythe two conditions :

1. 4x−1,t ≤ 0

2. 4x−1,t = 0 if 4x+1,t > 0

for t = 1, ..., T .Letting

4x−1,t =

{min{0; β+4x+

2,t + ε1,t} if t = 1min{0; β+ x+

2,t − x+1,t−1 + ε1,t} if t = 2, ..., T

(12)

obviously meets condition 1. To check condition 2., subtract x+1,t−1 in both

sides of (5)4x+

1,t = β+ x+2,t − x+

1,t−1 + z1,t t = 2, ..., T (13)

that is4x+

1,t = max{0; β+ x+2,t − x+

1,t−1 + ε1,t} (14)

By a similar argument, we have for t = 1

4x+1,1 = max{0; β+4x+

2,1 + ε1,1} (15)

We stress that (12) is only a convenient way to take the intrinsic propertiesof 4x−1,t into account. Provided that conditions 1. and 2. are fulfilled, a moregeneral definition of 4x−1,t could have been considered. Yet as we will see, thecase where (12) holds has interesting properties and this is what we will assumein the rest of the paper. In this respect, remark that (12) can be rewritten as

4x−1,t = ε1,t − z1,t t = 1, ..., T (16)

meaning that

x−1,t =t−1∑i=0

(ε1,t−i − z1,t−i) t = 1, ..., T (17)

C. Simulated data

It is rewarding to look at the wide variety of dynamics allowed by a cointegra-tion relation of the form (5). Towards this end, simulated data can be obtainedby the following procedure.

1. Generate a Gaussian random walk {x2,t}Tt=0 and derive the components

{x+2,t}T

t=1, {x−2,t}Tt=1.

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2. Simulate a Gaussian white noise {ε1,t}Tt=1 independent of {x2,t}T

t=0 andconstruct {z1,t}T

t=1 and {x+1,t}T

t=1 according to (10) and (5).

3. Derive {x−1,t}Tt=1 from (17) and sum the two components to get {x1,t}T

t=1.Add any constant as x1,0.

Figure 1 depicts the outcome of this exercise for β+ set to one and differentvalues of σ2

ε , T = 250. The mean and variance of {4x2,t} have been set to zeroand one respectively. The initial value x1,0 has been taken equal to x2,0. Notethat in the generating process, the first hundred values of {x2,t} and {ε1,t}have been discarded.2

It is shown that the time paths of {x1,t} and {x2,t} can be quite differentand seemingly unrelated. Interestingly, the variables can wander far from eachother and some features which could be thought to be structural changes areconsistent within the model.

Additional charts concerning the data such as the associated cointegrationrelations and the equilibrium errors can be found in the appendix.

D. Estimating the cointegrating parameter

Since {z1,t} is defined as the nonlinear process (10), the equilibrium error of (5)cannot be assumed to be normally distributed. Moreover, {z1,t} depends onsome past and current realizations of the variables appearing in the model.From then on, the cointegration relation does not stand for the regression ofx+

1,t on x+2,t and OLS on (5), although consistent, are likely to be substantially

biased in finite sample. The reasoning is similar to the one presented in theanalysis of linear cointegration relationships, yet the issue here is somewhatdifferent and does not lead to the procedures suggested in the classical case.

Explicitly, let Ft = {x1,t−1, x1,t−2, ..., x1,0, x2,t, x2,t−1, ..., x2,0}. Well-knownresults on the truncated normal distribution yield

E(z1,t | Ft) = Φ

(x+

1,t−1 − β+ x+2,t

σε

)(x+

1,t−1− β+ x+2,t) + σε φ

(x+

1,t−1 − β+ x+2,t

σε

)

(18)where Φ(·) and φ(·) denote respectively the cumulative distribution and pro-bability density functions of a standard normal variable. Accordingly, the re-

2The simulations have been carried out on a PC computer by using the statistical packageS-Plus. The code can be provided by the author upon request.

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sigma = .5

0 50 100 150 200 250

-10

010

20

sigma = .6

0 50 100 150 200 250

-10

010

2030

40

sigma = .65

0 50 100 150 200 250

-30

-25

-20

-15

-10

sigma = .8

0 50 100 150 200 250

-30

-20

-10

010

Fig. 1. Simulated pairs of asymmetrically cointegrated series. {x1,t}250t=1 (solid line) is gene-rated as described in section C where {x2,t}250t=1 (dashed line) is a Gaussian random walk.

gression model is given by

x+1,t = β+ x+

2,t + Φ

(x+

1,t−1 − β+ x+2,t

σε

)(x+

1,t−1 − β+ x+2,t)

+σε φ

(x+

1,t−1 − β+ x+2,t

σε

)+ νt

(19)

where the error νt is a martingale difference sequence with respect to Ft.Performing OLS on (5) is based on a misspecified regression since some rele-

vant variables are neglected. However, because the omitted terms are boundedwhile {x+

2,t} is monotonic increasing, the former will be asymptotically domi-nated by the latter and leaving the bounded regressors out of the specificationdoes not matter as far as the consistency of the OLS estimator is concerned.

Though the conditional mean is nonlinear in the parameters, a simple pro-cedure can be used to get an optimal estimate of β+. Combining (16) with (5)

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provides the auxiliary model

x+1,t +4x−1,t = β+ x+

2,t + ε1,t (20)

Since the regressor has a linear time trend in mean, it can be shown that underfairly general conditions, the OLS estimator of (20) is asymptotically normal(see West, 1988). Statistical inference can then proceed in the usual way.

E. Testing the null of no cointegration

In empirical work, it is of prime importance to have a statistical test of whethersome data series may be linked by a cointegration relation. By definition, thereis no cointegration if the sequence {z1,t} is not stationary.

In this respect, consider applying the Engle and Granger (1987) testingmethodology to this particular context. This leads to estimate (5) by OLSand test the unit root hypothesis in the residuals. However, it has been shownthat running OLS on the cointegration relation is expected to provide poorestimate of the parameters. Hence, any statistic built on this procedure will bebadly affected by this property. Furthermore, any linear time series model fit tothe OLS residuals of (5) will be misspecified in view of the intrinsic nonlinearproperties of {z1,t}.

By contrast, consider the auxiliary regression (20). Optimal estimate canbe straightforwardly obtained from this latter model. Assuming that {ε1,t}is a classical linear process, standard unit root testing strategy such as theaugmented Dickey-Fuller test (henceforth ADF) can be applied to the OLSresiduals of (20). Recall finally that the properties of {z1,t} depend on those of{ε1,t}. In particular, it is evident from (11) that if {ε1,t} is a unit root process,then {z1,t} is not stationary as well.

These arguments lead to base the test on the auxiliary regression ratherthan on the cointegration relation in itself. This approach can be seen as theEngle and Granger procedure applied to (20) instead of (5). Since the explana-tory variable in (20) has a deterministic trend, the critical values for the ADFtest can be found in Fuller (1976).3 Panel A of Table 1 reports the absolutevalues of the critical values for one-tailed tests of size 1%, 5% and 10%.

Due to the specificities of the series involved, we have conducted a smallMonte Carlo experiment. Simulated data of {x1,t} and {x2,t} have been gene-rated according to 4xj,t = φ4xj,t−1 + εj,t, j=1,2, where {ε1,t}, {ε2,t} are two

3When including a constant term on the right hand side of (20), the limiting distributionof the ADF t-statistic for the test of a unit root in the OLS residuals of (20) turns out tobe the usual Dickey-Fuller t-distribution for a model with constant and time. See Hansen(1992) for a detailed discussion.

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Table 1. Critical Values for the No Cointegration Test

A. Fuller’s critical valuesa

Sample size T 1% 5% 10%100 4.04 3.45 3.15250 3.99 3.43 3.13∞ 3.96 3.41 3.12

B. Monte Carlo experimentb

φ = 0 φ = 0.6Statistic 1% 5% 10% 1% 5% 10%

T = 100

DF 5.991 5.159 4.719 3.801 3.050 2.683ADF(1) 4.585 3.927 3.577 3.946 3.302 2.993ADF(2) 4.162 3.560 3.247 3.989 3.381 3.073ADF(4) 3.935 3.371 3.078 3.951 3.348 3.053

T = 250

DF 6.023 5.161 4.725 3.527 2.833 2.512ADF(1) 4.538 3.893 3.571 3.774 3.195 2.904ADF(2) 4.147 3.563 3.268 3.916 3.332 3.045ADF(4) 3.993 3.419 3.116 3.969 3.386 3.097

Notes :a Source : Fuller (1976, Table 8.5.2).b The critical values are computed by generating T observations of 4xj,t =

φ4xj,t−1 + εj,t, j = 1, 2, where ε1,t and ε2,t are mutually and serially independentstandard normal, 50,000 replications. In order to initialize the process, the first hun-dred observations have been discarded. By noting {εt} the OLS residuals of regressing(x+

1,t +4x−1,t) on x+2,t and a constant, the DF statistic is the absolute t-ratio of the OLS

estimate of γ in 4εt = γ εt−1 +errort while ADF(k) refers to the augmented regression4εt = γ εt−1 +

∑ki=1 γi4εt−i + errort, k = 1, 2, 4.

independent Gaussian white noises with mean zero and variance unity. Next,the components {x+

1,t}, {4x−1,t}, {x+2,t} have been constructed and the usual

ADF statistics have been computed on the OLS residuals of (20) including aconstant term amongst the regressors. Panel B of Table 1 reports the results

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of the simulations for 50,000 replications and different parametrizations andsample sizes. While the DF and ADF(k) k=1,2, statistics turn out to be de-pendent on the underlying data generating process, the ADF(4) statistics arereasonably insensitive to the level of serial correlation in the two variables. No-tice that the critical values associated to the ADF(4) statistics are essentiallythe same as those given in Fuller (1976).

It is interesting to compare the power of the suggested procedure withthe Engle and Granger methodology applied on (5). This has been done bysimulating 2,500 asymmetric cointegrated series following the steps describedin section C where T = 100, β+ = 1 and σε = 1. The model where {ε1,t} is nomore white noise but a first order autoregressive process has been considered aswell. In that case, two different values of the autoregressive parameter ρ havebeen investigated. Table 2 reveals the percentage of instances in which theADF(4) statistic correctly rejected the null of no cointegration for test sizesof 10%, 5% and 1%. It is shown that the statistics relying on the auxiliaryregression (20) outperform the testing strategy based on (5).

Table 2. Power Test

No cointegration test on (20) No cointegration test on (5)ρ 10% 5% 1% 10% 5% 1%0 98.7 95.9 77.4 81.4 65.8 33.30.8 49.2 32.8 10.0 32.8 18.5 4.60.9 23.6 13.3 2.6 16.0 7.6 1.3

Notes : The data have been generated according to x+1,t = x+

2,t+z1,t, z1,t = max{z1,t−1−β+4x+

2,t; ε1,t}, z1,0 = 0, ε1,t = ρ ε1,t−1 + ξ1,t, 4x2,t = ξ2,t, (ξ1,t, ξ2,t)′ ∼ iidN(0; I2),t=1,...,100. The ADF(4) t-statistics are derived from the OLS residuals of x+

1,t+4x−1,t =c + β+ x+

2,t + errort in case of the no cointegration test on (20) and x+1,t = c + β+ x+

2,t +errort in case of the no cointegration test on (5). The statistics are then comparedto the critical values given in Fuller (1976), T = 100. Each entry is the percentageof instances in which the null hypothesis of no cointegration is correctly rejected over2,500 replications.

F. Extensions

Various extensions of model (5) are possible. Adding an intercept on the righthand side yields x+

1,t = c + β+ x+2,t + z1,t. Accordingly, the process driving the

equilibrium error is given by z1,t = max{x+1,t−1 − c− β+ x+

2,t ; ε1,t}.

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On the other hand, we may think of a cointegration relation of the form

x−1,t = β− x−2,t + z2,t (21)

In that case, (10) and (17) become respectively

z2,t = min{z2,t−1 − β−4x−2,t; ε2,t} (22)

x+1,t =

t−1∑i=0

(ε2,t−i − z2,t−i) (23)

where z2,0 = 0 and ε2,t the underlying disturbance generating z2,t. Similarlyto (5), β− takes a positive value since otherwise, z2,t → −∞ as t →∞.

4. Example

As an illustration of the theory, we consider some data for which no evidenceof classical cointegration is found despite the fact that the variables shouldbe and do appear to be closely related.

Figure 2 depicts the historical paths of real exchange rates (in logarithms)between three European currencies and the U.S. Dollar, namely the DeutscheMark (DM), French Franc (FF) and Italian Lira (IL). The data have beenconstructed according to the formula

log RERj,t = log Sj,t − log Pj,t + log PUS,t , j = DM, FF, IL (24)

where RERj,t denotes the real exchange rate of currency j against the U.S.Dollar, Sj,t is the nominal bilateral exchange rate, that is the number of foreigncurrency units per U.S. Dollar, Pj,t is the foreign consumer price index andPUS,t is the U.S. consumer price index. The price series as well as the nominalexchange rates which consist of the monthly average values have been obtainedfrom International Financial Statistics (CD ROM of December 2002).4 Sincethe European Monetary System was organized in 1979, the period consideredspans January 1980 to December 1998 for a total of 228 observations.

The stylized facts about these series are the following. Although we expectthem to be stationary according to the purchasing power parity, they fail toreject the null hypothesis of standard unit root tests.5 From this perspective

4The series codes are given in the appendix.5The empirical assessment of the purchasing power parity is a longstanding issue. In

post-1973 data, it is generally assumed that real exchange rates are nonstationary. See forinstance Kim (1997).

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1980 1985 1990 1995

-0.1

0.0

0.1

0.2

0.3

Fig. 2. Natural logs of real exchange rates of the Deutsche Mark (solid line), the FrenchFranc (short dashed line), the Italian Lira (long dashed line) against the U.S. Dollar,monthly, 1980-98. The initial value for 1980 :1 has been subtracted from each observationin order to normalize each series to be zero for 1980 :1.

and in view of the rules set by the European Monetary System in order tostabilize foreign exchange among members, we may think there exist some long-run dependencies amongst them. Yet performing Johansen’s (1991) maximumlikelihood procedure provides no evidence of linear cointegration in the systemof the three variables considered.6

Examining the possibility of asymmetric behavior leads to different conclu-sions. Models (5) and (21) including an intercept on the right hand side of theequations have been successively estimated for bivariate combinations of thethree variables. As shown in panel A of Table 3, the cointegration tests in-dicate strong evidence of asymmetric cointegration in the form (21). Indeed,the test statistics lie beyond the 5% critical values reported in Table 1 with allbut one lying beyond the 1% critical values.7 These results lead into thinking

6The details of the results not reported here are available from the author upon request.7The same critical values apply when x−1,t +4x+

1,t = c + β− x−2,t + errort takes the place

13

that the series may be cointegrated only when there is real appreciation of theEuropean currencies.

The long-run economic relationships are reported in panel B of Table 3.The coefficients are the OLS estimates of the regression

log RER−i,t +4 log RER+

i,t = c + β− log RER−j,t + et (25)

where {i, j} = {DM, FF, IL} and et the regression disturbance. Due to thepresence of serial correlation in the residuals, the OLS formulas for standarderrors are incorrect. Following West (1988), consistent estimates of the stan-dard deviations can be obtained by scaling the OLS statistics by (s/σ2

e)1/2

where σe denotes the usual OLS standard error of regression and s is an esti-mator of the spectral density at frequency zero of the regression disturbance.Letting et be the OLS residual of (25), s has been computed as in Newey andWest (1987)

s = σ(0) + 2m∑

h=1

[1− (h/m + 1)] σ(h) (26)

where σ(h) = 1T

∑Tt=h+1 etet−h, m = 4 since the number of observations inclu-

ded in the regression is 227.Using these, we reject the null hypothesis that the intercept is equal to zero

in every case. The same is true of testing the cointegrating parameter to beequal to one.

Note finally that reversing the causality in the regressions does not changethe qualitative conclusions.

5. Conclusion

This paper investigates the properties of a peculiar form of cointegration emer-ging from decomposing time series into their positive and negative partial sums.Our motivation has been driven by potential asymmetries in the interactionamong main economic indicators.

Although seemingly linear in the variables resulting from the decomposi-tion introduced, the relation is shown to be nonlinear in two ways. First, itis defined in terms of nonlinear transformations of the raw data series. Se-cond, the equilibrium error follows a nonlinear autoregressive process. Thesecharacteristics imply that the model cannot be interpreted as a standard coin-tegration relation. In particular, the usual statistical procedures devoted tolinear cointegration are misspecified in this context.

of x+1,t +4x−1,t = c + β+ x+

2,t + errort in the Monte Carlo experiment.

14

Table 3. Asymmetric Cointegration Analysis of Real Exchange Rates

A. Asymmetric Cointegration Testsa

Statistic i = FF, j = DM i = IL, j = DM i = IL, j = FF

DF 8.48 7.08 7.12ADF(1) 6.42 6.09 5.92ADF(2) 4.68 4.79 4.58ADF(4) 3.61 4.18 4.00

B. Long-run Economic Relationshipsb

Estimated Equation R2 DW σe

log RER−FF,t = 0.005

(0.002)+ 0.952

(0.003)log RER−

DM,t 0.99 0.97 0.009

log RER−IL,t = 0.007

(0.002)+ 0.959

(0.003)log RER−

DM,t 0.99 0.73 0.011

log RER−IL,t = 0.008

(0.002)+ 1.011

(0.004)log RER−

FF,t 0.99 0.73 0.010

Notes :a The entries refer to the DF and ADF t-statistics computed on the OLS residuals of (25).

Remark that in every case only the coefficients of the first and second lag are significantlydifferent from zero. Using Akaike or Schwartz information criterion leads to select k = 2 inthe augmented autoregressions.

b The reported parameters are the OLS estimates of (25). Standard errors calculated asin West (1988) are in parentheses. DW , R2 and σe denote respectively the usual Durbin-Watson statistic, the sample multiple correlation coefficient and the residual standard errorresulting from OLS estimation of (25).

The theory is applied to the recent history of real exchange rates betweenthree European currencies and the U.S. Dollar. Since countries belonging tothe European Monetary System have agreed to coordinate their currenciesso that any one currency would not deviate too far from another currencybelonging to the System, we should expect the existence of some long-runrelationships amongst them. The fact that no evidence of linear cointegrationis found might be the consequence of some underlying nonlinear dynamics. In

15

that way, the framework presented provides strong support to the hypothesisof asymmetric cointegration relations.

Appendix

Proof of the Proposition

By construction, property (a) is immediate.Regarding (b), let 4x+

t be the non negative increment of the process bet-ween epochs t− 1 and t

4x+t = 1{4xt > 0}4xt (27)

Since 4xt = ζt ∼ iid of probability density function f4xt(4xt) = fζt(ζt),4x+

t ∼ iid of probability density function

f4x+t(4x+

t ) =

{P (4xt ≤ 0) if 4x+

t = 0f4xt(4xt) if 4x+

t > 0(28)

Noting that

E(4x+t ) = P (4xt > 0) E(4xt|4xt > 0) (29)

V (4x+t ) = P (4xt > 0)

(E(4x2

t |4xt > 0)−P (4xt > 0) E2(4xt|4xt > 0)

) (30)

we can write 4x+t ∼ iid

(E(4x+

t ) ; V (4x+t )

)with E(4x+

t ) > 0.Define now

v+t = 4x+

t − E(4x+t ) (31)

and rewrite (2) as

x+t = t E(4x+

t ) +t−1∑i=0

v+t−i (32)

Since v+t ∼ iid ( 0 ; V (4x+

t ) ), {x+t } is a random walk process with drift E(4x+

t ).A similar reasoning stands for {x−t }. In that case, (3) can be rewritten as

x−t = t E(4x−t ) +t−1∑i=0

v−t−i (33)

where 4x−t = 1{4xt < 0}4xt, v−t = 4x−t − E(4x−t ). Because E(4x−t ) =P (4xt < 0) E(4xt|4xt < 0) < 0, {x−t } has a downward linear time trend inmean.

Concerning property (c), notice that 4x−t = 0 if 4x+t > 0 and conversely,

4x+t = 0 if4x−t < 0. Hence, E(4x+

t 4x−t ) = 0 and we have Cov (4x+t ;4x−t ) =

−E(4x+t ) E(4x−t ) with E(4x+

t ) > 0, E(4x−t ) < 0.

16

Simulated Pairs of Asymmetrically Cointegrated Series

x1+, x2+

0 50 100 150 200 250

020

40

60

80

100

z1

0 50 100 150 200 250

-1.0

-0.5

0.0

0.5

1.0

Fig. 3. σε = .5

x1+, x2+

0 50 100 150 200 250

020

40

60

80

10

0

z1

0 50 100 150 200 250

-1.0

-0.5

0.0

0.5

1.0

1.5

Fig. 4. σε = .6

17

x1+, x2+

0 50 100 150 200 250

020

40

60

80

10

0

z1

0 50 100 150 200 250

-1.0

0.0

0.5

1.0

1.5

Fig. 5. σε = .65

x1+, x2+

0 50 100 150 200 250

02

04

060

80

100

z1

0 50 100 150 200 250

-2-1

01

2

Fig. 6. σε = .8

18

International Financial Statistics Codes

Table 4. IFS Series Codes for Real Exchange Rates

Country National Currency per U.S. Dollar Consumer Price Index

France 132..AF.ZF... 13264...ZF...Germany 134..AF.ZF... 13464...ZF...Italy 136..AF.ZF... 13664...ZF...U.S. 11164...ZF...

Source : IFS CD ROM of December 2002.

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