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Macroeconomic fundamentals and exchange rates: anon-parametric cointegration analysisEmmanuel Davradakisa Warwick Business School , University of Warwick , Coventry CV4 7AL, UK E-mail:Published online: 15 Aug 2006.

To cite this article: Emmanuel Davradakis (2005) Macroeconomic fundamentals and exchange rates: a non-parametriccointegration analysis, Applied Financial Economics, 15:7, 439-446, DOI: 10.1080/09603100500056593

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Macroeconomic fundamentals and

exchange rates: a non-parametric

cointegration analysis

Emmanuel Davradakis

Warwick Business School, University of Warwick, Coventry CV4 7AL, UKE-mail: emmanuel.davradakis@wbs.ac.uk

This paper examines in a non-parametric setup whether a long-runrelationship exists between monetary fundamentals and the dollar spotexchange rates for 19 countries. Although the Johansen’s parametricapproach failed to retrieve a long-relationship for any of the countriesconsidered, the Bierens (1997a) non-parametric approach suggests thatthere is one cointegrating relationship for the majority of the countriesconsidered. In addition, the [1,�1] cointegrating vector between thefundamentals and the log-level of the dollar exchange rate could not berejected in the non-parametric formulation.

I. Introduction

Several authors have recently examined whether

nominal exchange rate returns can be forecast by

monetary fundamentals. Chinn and Meese (1995),

Mark (1995), Chen and Mark (1996), among others,

provided empirical evidence that the long-horizon

regression approach can be used in order to predict

the spot exchange rate returns at longer horizons.

However, according to these studies the monetary

fundamentals cannot outperform in forecasting a

naıve random walk model of the spot exchange rate

for shorter horizons, a finding first reported by Meese

and Rogoff (1988).

The empirical success of the long horizon regres-

sion employed by Mark (1995) has been heavily

criticized by subsequent research on the grounds of

the methodology used and the underlining assump-

tions made. Specifically, the long-horizon regression

includes, as a regressor, the deviation of the log-

level of the spot exchange rate from its fundamental

value, which is assumed to be stationary in Mark’s

setup. However, Berkowitz and Giorgianni (2001) –

hereafter referred to as BG – argue that if this

deviation is not stationary the long-horizon regres-

sion will be a spurious regression, while the critical

values of the tests involved in the forecasting evalua-

tion generated under the assumption of cointegration

will be incorrect. Other researchers, like Kilian

(1999), emphasized that the empirical results will

depend on the assumed bootstrap data generating

process (DGP) applied in the non-parametric boot-

straps performed by Mark (1995). Subsequent

empirical results reported by Faust et al. (2003)

point out that the forecasting performance of the

monetary model crucially depends on data revisions

and on the use of real-time forecasts of future funda-

mentals compared to actual future fundamentals.

Mark and Sul (2001) formally tested for a panel

cointegration between exchange rates and their funda-

mental values in a panel of 19 countries using a

dynamic ordinary least squares approach. Their find-

ings indicated that there is cointegration. However,

their approach does not consider the possibility that

the DGP of the deviation of the log-level of spot

exchange rate from its fundamental value might be

non-linear and/or the short-run dynamics towards

the long-run relationship might be non-linear, as well,

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online # 2005 Taylor & Francis Group Ltd 439

http://www.tandf.co.uk/journalsDOI: 10.1080/09603100500056593

Applied Financial Economics, 2005, 15, 439–446

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due to transaction costs’ considerations. Otherstudies performed by Taylor and Peel (2000) andKilian and Taylor (2003) do not distinguish betweennon-linearity in the DGP compared to non-linearityin the short-run adjustment. The only attempt thatthe present author is aware of is by Sekioua (2003),who examined whether a non-linear threshold auto-regressive unit root exists in the deviation of the spotexchange from its fundamental value by means oftests, proposed by Caner and Hansen (2001).

In addition, it is very likely that the long-run equil-ibrium level implied by the monetary model will varyover time depending on the exchange rate regime inoperation. Sarno et al. (2004) provide supportiveempirical evidence of the time-varying long-runequilibrium level for six industrialized countriesafter fitting a Markov-switching vector error correc-tion model (VECM) with time varying parametersfor a data sample starting in the late 1800s.

The current study attempts to examine the cointe-gration between the spot exchange rate and its funda-mental value at a non-parametric framework in orderto account for the possibility of a non-linear DGP.The framework employed here does not assume aspecific DGP, while it enables one to test explicitlyfor the validity of the [1,�1] cointegrating vectorbetween the spot and its fundamental exchange ratevalue. Thus, the approach is general enough withoutimposing a specific DGP and without ruling out thepossibility of a non-linear DGP. To that end, thenon-parametric unit root test proposed by Breitung(2002) and the non-parametric cointegration analysisproposed by Bierens (1997a) are considered. Theformer has been employed by Maki (2003) in orderto examine whether non-parametric cointegrationexists between the nominal interest rate and theexpected inflation in Japan. Coakley and Fuertes(2001) implemented Bierens’ approach in order totest for a non-parametric cointegration between thespot exchange rate and its fundamental value basedon the purchasing power parity (PPP).

Using quarterly data on industrial productionindices, monetary aggregates and the spot exchangerate for 19 countries, the null hypothesis of a unit rootcould not be rejected for the series considered acrosscountries on the basis of Breitung’s non-parametrictest. The latter finding implies that it might be thecase that a non-linearity characterizes the underlinedDGP. Furthermore, empirical evidence is providedin favour of a single mean reversion process ofthe deviation of the spot exchange rate from itsfundamental value on the basis of the Bierens’(1997a) non-parametric cointegration analysis forthe majority of the countries considered. Most impor-tantly, the [1,�1] cointegrating vector between the

fundamental vale and the spot exchange rate couldnot be rejected for any of the countries examined,even when the non-parametric approach indicatedthat there are two instead of one cointegratingvectors.

In contrast, the Johansen approach provided lim-ited support of the mean reversion hypothesis for thecountries examined. The obvious discrepancy of thetwo tests on the possible existence of non-linearDGPs and/or non-linear short-run dynamics wasattributed due to transaction costs. As an additionalexplanation, the size distortions that the Johansenapproach experiences in the event of an erroneouschoice of optimal lag length was also considered.

The remainder of the paper proceeds as follows.In Section II, the monetary model of the exchangerate determination, along with its empirical formula-tion is briefly discussed. In Section III, the non-parametric unit root and cointegration tests used inthis study are demonstrated. Section IV provides thedataset description and presents the empirical resultsobtained after the implementation of the parametricand non-parametric approaches considered. Finally,Section V concludes the paper.

II. The Monetary Model for the ExchangeRate Determination

The starting point of the monetary model (see Neelyand Sarno, 2002) is the definition of the exchangerate as the relative price of two monies, whilemodelling the money and supply-side of these twomonies is at the heart of the monetary model.The monetary equilibria at home and abroad, respec-tively, are provided by

mt ¼ pt þ kyt � �it ð1Þ

m�t ¼ p�t þ ky�t � ��i�t ð2Þ

where mt, pt, yt and it stand for the log-levels of themoney supply, the price level, the income and thenominal interest rate, respectively at time t; k, � arepositive constants, while asterisk denotes foreignvariables and parameter values. In addition to themonetary equilibria at home and abroad, it isassumed that PPP holds continuously in time imply-ing that the following relationship holds.

st ¼ pt � p�t ð3Þ

where st is the log-level of the spot exchange ratedefined as the domestic price of a unit of foreigncurrency. The domestic money supply will determinethe domestic price level and subsequently the spotexchange rate. After subtracting Equation 2 from

440 E. Davradakis

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Equation 1, solving, for pt � p�t , substituting inEquation 3 and assuming k¼ k* and �¼ �*

st ¼ ðmt �m�t Þ � kðyt � y�t Þ þ �ðit � i�t Þ ð4Þ

is obtained. Equation 4 is the monetary model equa-tion for the determination of the exchange rateaccording to which an increase at the domestic supplywill induce a depreciation of the domestic currency,while an increase at the level of the domestic impactwill appreciate the domestic currency. The latter istrue since the price level need to fall in order tomaintain equilibrium at the domestic money market,leading to an appreciation of the domestic currencythrough Equation 3.

The monetary model in Equation 4 was consideredto be an appealing device in forecasting spot

exchange rates. However, the early attempts byMeese and Rogoff (1983a, 1983b) towards thatdirection were not fruitful pointing that the naıvenon-change forecast of the spot exchange rategenerated by a random walk could not be out-performed by Equation 4. Further research byMark (1995) indicated that the monetary modelcan be used to predict the spot exchange rate in thelong-run. Specifically, Mark (1995) considered thein and out-of-sample forecasting performance ofthe monetary model by estimating the followingformulation of Equation 4 assuming it � i�t ¼ 0

stþ‘ � st ¼ c0‘ þ c1‘ð ft � stÞ þ vtþ‘, t ð5Þ

where ft ¼ ½ðmt �m�t Þ � ðyt � y�

t� is called the funda-

mental value of the exchange rate under long-run

money neutrality (k¼ 1). Equation 5 is an error cor-rection equation where the term ( ft� st) can be con-sidered the error term. Therefore, Equation 5 impliesthat although the exchange rate and its fundamentalvalue individually are I(1), their linear combination isstationary I(0) with a cointegrating vector [1,�1]. BGpointed out that the assumption of cointegrationbetween the spot exchange rate and its fundamentalvalue, when in fact such a relationship does not exist,will produce incorrect critical values for the teststatistics considered. The latter is true because inthis event Equation 5 will be a spurious regression.Hence, it is necessary to test whether there is a long-

run relationship between st and ft and subsequentlyestimate Equation 5. In order to perform this task atthe present study, parametric and non-parametrictesting procedures are used. The latter approachdoes not assume a specific DGP but it is generalenough to account for the possibility of a non-linearDGP that will cause non-linear short-run dynamicstowards the long-run relationship.

III. Non-Parametric CointegrationFormulation

The non-parametric approach implemented in thisstudy consists of three steps. First, it is tested whethera non-parametric unit root is present at the log-levelof the spot exchange rate and its fundamentalvalue by means of Breitung (2002) test. Second, it istested whether a non-parametric long-run relation-ship exists among st and ft in the vector xt¼ [ ft, st]

0.For this purpose, Bierens’ (1997a) non-parametricapproach is employed. Finally, it is tested whetherthe [1,�1] cointegrating vector can be accepted forft and st.

Breitung’s (2002) non-parametric unit root test

In addition to the Phillips–Perron (1988) unit roottest, a non-parametric unit root test proposed byBreitung (2002) is employed. The latter is free of prob-lems concerning the specification of the short-rundynamics and the estimation of nuisance parameters.This non-parametric unit root test decomposes theunivariate process fxtg

n1 as xt ¼ �0dt þ ut where

�0dt is the deterministic part with �0 ¼ ½�1, �2� anddt ¼ ½1, t�0, while ut is the stochastic part. If the deter-ministic part was not present, xt would be consistentto the stochastic part ut. If the deterministic part wasnot present, xt would be consistent to the stochasticpart ut. Under the null xt is I(1), if as n!1,n�1=2x½�n� ) �Wð�Þ, for �>0 the constant long-runvariance, W(�) is a Brownian motion and [.] is theinteger part. The expression for ut is not specifiedgiving rise to a general DGP.

In order to avoid the specification of short-rundynamics towards stationarity and the computationof an estimate for �, Breitung proposed the followingtest statistic which is a variance ratio test

��n ¼n�4Pn

t¼1 UU2t

n�2Pn

t¼1 ""2t

ð6Þ

where ""t is the Ordinary Least Squares (OLS) resid-uals such as "" ¼ xt � ��0dt and UUt is the partial sumsuch that UUt ¼ ""1 þ � � � þ ""t. In the event that xt isI(0), the test statistic ��n converges to zero. Breitungprovided simulation evidence that non-parametricunit root test outperform the standard parametrictests.

The Bierens’ (1997a) test

The non-parametric cointegration test proposed byBierens (1997a) is based on the solutions to a general-ized eigenvalue problem like Johansen’s approach.

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However, the corresponding eigenvalue problem atBierens’ setup is formulated on the basis of tworandom matrices that are constructed withoutreference to the DGP. Specifically, Bierens assumesan observable q-variate process xt for t¼ 1, . . . , ngenerated from:

xt ¼ �0 þ �1tþ yt ð7Þ

where �0ðq� 1Þ and �1ðq� 1Þ are optional and trendterms, while yt is a zero-mean unobservable processsuch that �yt is stationary. No further specificationfor the DGP with respect to x is required. The test isbased on the following pair of random matrices thatare the weighted sums of the series considered inlevels and first differences, respectively

AAm ¼8�2

n

Xmk¼1

k21

n

Xnt¼1

cosð2k�ðt� 0:5Þ=nÞxt

!

�1

n

Xnt¼1

cos 2k�ðt� 0:5Þ=nÞxtð Þ0

!ð8Þ

BBm ¼ 2nXmk¼1

1

n

Xnt¼1

cosð2k�ðt� 0:5Þ=nÞ�xt

!

�1

n

Xnt¼1

cos 2k�ðt� 0:5Þ=nÞ�xtð Þ0

!ð9Þ

The ordered generalized eigenvalues ll1,m � � � � �

llq,m of the characteristic equation det½Pn � lQn� ¼ 0are having similar properties to the correspondingeigenvalues in the Johansen framework justifyingtheir inclusion for testing the cointegration rank r.In order for this to happen, Pn should be definedas Pn ¼ AAm, while Qn should be chosen asQn ¼ BBm þ n�2AA�1

m . Bierens (1997a) suggests thatthe lmin test llq�r0,m

to test the null hypothesis r¼ r0against the alternative r¼ r0þ 1. Bierens tabulates thecritical values for the distribution under the nullafter Monte Carlo simulations. Parameter m is alsotabulated for different significance levels and forvarious values of r0 and q in such a way that thelower end of the power of the test is maximized.

In order to consistently estimate the rank r, thegm(r0) is proposed, as well. This test takes the form:

ggmðr0Þ¼

Yqk¼1llk,m

� ��1

if r0¼0Yq�r0k¼1 llk,m

� ��1

�ðn2R0

Yqk¼q�r0þ1llk,m

� ��1

if r0¼1, ...,q�1

n2qY

qk¼1llk,m if r0¼q

8>>>>>>><>>>>>>>:

ð10Þ

When r0¼ q the optimal value of m is given by m¼ q,while when r0<q the optimal value of m is given by

tabulated values for different significance levelsand different combinations of r, q. The rank r isconsistently estimated and it is given by rr ¼argminr0�q gmðr0Þ.

Having determined the dimension of the cointegra-tion space, linear restrictions on the cointegratingvectors could be tested. For this purpose non-parametric tests could be used that employthe ordered eigenvalues solutions to the followingproblem:

det H 0AAmH � lH 0 AAm þ n�2AA�1m

� ��1

H

� �¼ 0 ð11Þ

where H spans a subspace of cointegrating vectors.The critical values for the corresponding trace andlambda-max tests used to test the restrictions aregiven in Bierens (1997a) with m¼ 2q.

IV. Data Description and Empirical Results

The dataset includes quarterly time series observa-tions from 1975:Q1 to 1997:Q4 for 19 countries,namely Austria, Australia, Belgium, Canada,Denmark, Finland, France, Germany, UnitedKingdom (UK), Greece, Italy, Japan, Korea, theNetherlands, Norway, Spain, Sweden, Switzerlandand the United States (US). Data availability deter-mined the ending and starting points of the sample.Data on nominal spot exchange rate were end ofperiod observations retrieved from the IFS CD-ROM (line code AE). The industrial productionindex (IFS line code 66) proxy the national incomefor the countries considered, while money was proxyby the money stock (line code 34) plus quasi money(line code 35) for all the countries, except Norwayand Sweden due to availability constraints. M2 wasused for Norway and M3 for Sweden from OECD’sMain Economic Indicators. The resulting series forthe monetary aggregates were not seasonally adjustedand so was done by implementing a one-sided movingaverage of the current observation plus three lags ofthe corresponding series.

In order to test for the presence of a unit root onthe nominal exchange rate and its fundamental valuest and ft, respectively, the Phillips–Perron test (PP)and the Breitung (2002) test ð��nÞ were considered.With regard to the PP test, the truncation parameter[cn]k was used in the heteroscedasticity consistentcovariance matrix of Newey–West with c¼ 5 andk¼ 0.2, following Bierens (1997b). The potentialsize distortions that the PP test might experience infinite samples was accounted for by computing thep-values following 1000 simulations of a GaussianAR(1) process for the first difference of every series

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considered. Testing results from the unit root analysis

are reported at Table 1.The PP test indicated that st and ft were both I(1)

processes for all the countries considered on the basis

of the p-values for the PP test statistic.1 However, ftfor Norway was found to be I(0) for both simulatedand non-simulated p-values. The implementation of

the Breitung test2 produced test statistics that are notstatistically significant at the 5% significance level,implying that all the series were I(1) processes con-

sistent to the empirical outcome of the PP test. Sincethe series considered are individually I(1) processesthere might be the case that a linear combination of

them is an I(0) process.In order to test whether a long-run stationary rela-

tionship exists between ft and st, it is necessary to

perform a cointegration analysis at a parametricand a non-parametric context. The parametric testconsidered was the Johansen cointegration test the

estimates the following VECM

�xt ¼ �þY

xt�1 þXLj¼1

�j�xt�j þ�dt þ "t

for t ¼ 1, . . . , n ð12Þ

where � is the error correction term; � and �dtis the coefficient vector and the deterministic vector,respectively; " is the error vector. In order to choose

the optimal lag length L, successive estimation of thecorresponding to Equation 12 vector autoregression(VAR) for different lag lengths was performed

and the L that rendered the minimum value of theAkaike information criterion was chosen.3 Estimationresults based on the Johansen test are reported in

Table 2.Empirical results in Table 2 show that the cointe-

grating relationship between the log-level of the spot

exchange rate and its fundamental value is rejectedfor 14 out of the 18 countries considered, implyingthat the use of Equation 5 is going to be a spurious

regression. Specifically, since the term ft � st is notI(0) the left-hand side of Equation 5 will be morepersistent, making both sides of the equation very

persistent. In that case, Equation 5 will become spuri-ous making any inference based on that equationto be erroneous. Because of the spurious regression

property of the aforementioned equation, the criticalvalues of the test statistics that will be used in order to

evaluate the forecasting performance of Equation 5

will be incorrect, as underlined by BG.

Furthermore, Johansen’s cointegrating analysis

indicated that there is one cointegrating relationship

between ft and st, only for Canada and Germany.

However, the [1,�1] cointegrating vector for these

countries could not be accepted as the involved

likelihood ratio test was equal to 12.53 and 3.90,

1The series considered were found to be stationary at their first differences. These results are not reported here for the sake ofbrevity, but they are available on request.2Breitung’s test was computed using the Bierens (2004) EasyReg International package.3Different information criteria did not alter the L chosen on the basis of the Akaike criterion.

Table 1. Unit root test results

PPa BRb

Australia s 0.6900 (0.6390)c 0.0750f 0.9900 (0.9740) 0.0921

Austria s 0.3700 (0.2970) 0.0564f 0.4100 (0.2420) 0.0254

Belgium s 0.2600 (0.1190) 0.0172f 0.9700 (0.9040) 0.0546

Canada s 0.5300 (0.5290) 0.0460f 0.7900 (0.7530) 0.0562

Denmark s 0.3100 (0.1730) 0.0154f 0.5800 (0.7610) 0.0862

Finland s 0.3000 (0.1240) 0.0157f 0.7600 (0.6330) 0.0174

France s 0.3100 (0.1960) 0.0166f 0.7900 (0.8020) 0.0516

Germany s 0.3500 (0.2710) 0.0533f 0.6900 (0.6120) 0.0214

Greece s 0.9100 (0.8490) 0.0957f 0.9400 (0.8770) 0.0999

Italy s 0.4600 (0.4630) 0.0536f 0.7800 (0.7470) 0.0929

Japan s 0.6800 (0.6150) 0.0877f 0.6100 (0.5980) 0.0257

Korea s 0.9300 (0.8280) 0.0561f 0.9800 (0.9650) 0.0708

Netherlands s 0.3500 (0.2350) 0.0469f 0.3400 (0.3940) 0.0137

Norway s 0.3600 (0.3050) 0.0326f 0.0100* (0.0210)* 0.0208

Spain s 0.4600 (0.4240) 0.0458f 0.8700 (0.8890) 0.0910

Sweden s 0.5300 (0.3880) 0.0481f 0.6400 (0.6980) 0.0399

Switzerland s 0.3400 (0.2840) 0.0624f 0.5100 (0.6280) 0.0157

UK s 0.1600 (0.0770) 0.0231f 0.9400 (0.8100) 0.0935

Notes:aPhillips–Perron �-test with critical value �14 at the 5%significance level.bBreitung’s ��n test with critical value 0.01 at the 5%significance level.cSimulated p-values in parenthesis.*Significant at the 5% significance level.

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respectively, with corresponding p-values equal to

0% and 4%, respectively. In addition, Table 2 reportsthat there are two cointegrating relationships between

ft and st for Austria and the Netherlands, it was tested

whether the [1,�1] cointegrating vector could beaccepted or not, by means of a likelihood ratio test.

The latter pointed out that the former cointegrating

vector could not be rejected, implying that forthese countries Equation 5 could be used in order

to evaluate the forecasting performance of themonetary model with respect to the log-level of the

spot exchange rate without the risk of estimating aspurious regression.

Bierens’ test results,4 reported in Table 3, provideempirical evidence in favour of a non-parametriccointegrating relationship between ft and st for13 out of the 18 countries considered, while the[1,�1] cointegrating vector is not rejected for these13 countries. Although the lmin test indicated thatthere are two cointegrating relationships, the gm(r0)test consistently estimated a number of cointegratingrelationships equal to 1 for these 13 countries. Thegm(r0) test is the rest that Bierens (1997a) suggests asa double-check on the findings from the lmin test.Furthermore, the [1,�1] cointegrating vector wasnot rejected for any of the countries for which onecointegrating relationship was found on the basisof the gm(r0) test.

Moreover, for Austria, UK, Japan, Norway andSwitzerland the lmin and the gm(r0) tests producedresults consistent to two cointegrating relationships.In order to test whether the [1,�1] cointegratingvector could be rejected or not for these countries, arank equal to 1 was imposed and the correspondinglinear restriction tested by computing the relatedtrace-test. The latter was not able to reject the [1,� 1]linear restriction for the variables at the x vector,implying that the [1,�1] cointegrating vector is avector that can span the cointegrating space.

The discrepancy in the inference from the Johansenand the Bierens’ tests can be attributed to the factthat it is very likely that the DGP of the variablesinvolved is non-linear resulting into non-linearshort-run dynamics towards the long-run. Severalstudies in the literature demonstrate that it is verylikely the short-run adjustment to be exhibit a non-linear process. Taylor and Peel (2000) and Kilianand Taylor (2003) demonstrate that the morethe exchange rate is misaligned compared to itsfundamental value the more market participantsand policymakers will push the exchange rate backto its fundamental value. This is justified by thepresence of transaction costs with reference to thearbitrage in international goods market.

In addition, it might be the case that the DGP itselfis a non-linear process. Towards that direction,Sekioua (2003) applied non-linear threshold auto-regressive unit root tests suggested by Caner andHansen (2001) in order to examine whether ft� st ismean reverting. Sekioua’s empirical finding indicatedthat linearity and nonstationarity are rejectedimplying a non-linear mean reversion that dependson the size of the deviation ft� st.

4Bierens non-parametric cointegration analysis was performed using the Bierens (2004) EasyReg International package.

Table 2. Johansen cointegration analysis

La

ltracer� 0r� 1

lmax

r� 0r� 1 H0: �

0¼ (1,�1)b

Australia 2 7.23 6.830.40 0.40

Austria 2 11.32 5.79 0.17 [0.67]5.54* 5.54*

Belgium 4 7.00 6.490.50 0.50

Canada 4 21.65* 20.67* 12.53 [0.00]*0.98 0.98

Denmark 4 7.34 6.281.06 1.06

Finland 4 8.15 7.760.39 0.39

France 2 8.75 7.870.88 0.88

Germany 1 15.62* 14.52* 3.90 [0.04]*1.10 1.10

Greece 1 4.44 2.631.81 1.81

Italy 2 8.43 4.923.51 3.51

Japan 3 13.33 11.641.68 1.68

Korea 4 10.85 7.253.59 3.59

Netherlands 1 10.60 6.37 1.99 [0.5]4.23* 4.23*

Norway 1 14.91 12.232.68 2.68

Spain 4 8.72 6.801.92 1.92

Sweden 4 10.14 9.560.57 0.57

Switzerland 1 6.87 6.180.69 0.69

UK 3 5.02 4.780.24 0.24

Notes:aLag order selected by the Akaike information criterion.bThe likelihood ratio test value for the linear restrictiontested with p-values in brackets.*Significant at the 5% significance level.

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Furthermore, it is necessary to be aware that thelmin test might exhibit a low power for small samplescompared to Johansen’s lmax test on the basis ofsimulation evidence reported in Bierens (1997a). Inthat event Bierens suggests that a full parametricapproach might work better than a non-parametricapproach. The same simulation evidence underlinedthe fact that once the Johansen approach is correctedfor size distortions experienced for small L, itwill have a greater power. However, without thecorrection of these size distortions generated by anincorrect choice of L for Equation 12, the Bierenstest will outperform the Johansen test in terms ofits statistical power.

V. Conclusion

An attempt was made in this study to examinewhether a long-run relationship exists between thefundamental value of the spot exchange rate andthe log-level of the latter at a parametric and a non-parametric context. The two cointegration testsconsidered were the Johansen test and the Bierens(1997a) non-parametric test. Using quarterly dataon industrial production indices, monetary aggre-gates and the spot exchange rate, empirical evidencein favour of a single mean reversion of the deviationft� st was provided on the basis of non-parametriccointegration analysis for the majority of the coun-tries considered. Most importantly, the [1,�1]cointegrating vector between ft and st could not berejected for none of the countries examined evenwhen the non-parametric approach indicated thatthere are two instead of one cointegrating vectors.

In contrast, the Johansen approach provided lim-ited support of the mean reversion hypothesis for thecountries examined. The obvious discrepancy of thetwo tests was attributed the possible existence of non-linear DGPs and/or non-linear short-run dynamicsdue to transaction costs. The size distortions thatthe Johansen’s approach experiences in the event ofan erroneous choice of optimal lag length was alsoconsidered as an additional explanation.

The analysis performed in the current study can beextended in order to exploit simultaneously the crosssection and time series information.5 To that end, thenon-parametric analysis for the panel of the 19countries considered could be replicated by modify-ing the Bierens (1997a) test in order to correspond toa panel cointegration test. This could be done byaugmenting Equation 7 by a common time effect t,constructing the corresponding random matrices AAm

5Work in progress of the author.

Table 3. Nonparametric cointegration analysis

lamin

r¼ 0/r¼ 1r¼ 1/r¼ 2

gm(r0)r0¼ 0,1,2 H0: �

0¼ (1,�1)b

Australia 0.30� 10�3* 35.03� 105 1.062.60� 10�2* 31.06� 10

0c

17.90� 101

Austria 0.06� 10�3* 19.72� 105 1.110.90� 10�2* 49.47� 100

31.80� 100

Belgium 1.84� 10�2 45.09� 101 1.9426.97� 10�2

24.14� 101

13.91� 104

Canada 0.11� 10�2* 71.75� 1012 1.042.10� 10�2* 24.80� 10�8

87.43� 10�8

Denmark 0.85� 10�3* 25.75� 104 1.132.21� 10�2* 62.44� 10

0

24.35� 101

Finland 0.30� 10�3* 66.14� 103 1.134.48� 10�2* 59.49� 100

94.85� 101

France 0.30� 10�3* 33.19� 104 1.193.93� 10�2* 15.39� 100

18.90� 101

Germany 0.79� 10�2* 11.30� 103 1.162.37� 10�2* 12.43� 102

55.49� 102

Greece 0.10� 10�3* 10.63� 105 1.056.18� 10�2

19.47� 10�1

58.98� 100

Italy 0.18� 10�2* 69.35� 106 1.031.22� 10�2* 76.03� 10�2

90.46� 10�2

Japan 0.39� 10�2* 12.97� 106 1.350.59� 10�2* 17.23� 100

48.34� 10�1

Korea 0.50� 10�3* 29.02� 105 1.131.94� 10�2* 71.87� 10�1

21.61� 100

Netherlands 0.02� 10�3* 23.37� 1011 1.131.34� 10�2* 18.74� 10

�6c

26.84� 10�6

Norway 0.05� 10�3* 89.46� 106 1.020.81� 10�2* 13.42� 10�1

70.12� 10�2

Spain 0.45� 10�3* 33.42� 104 1.249.83� 10�2

24.48� 10�1

18.77� 101

Sweden 0.03� 10�3* 41.45� 108 1.072.14� 10�2* 41.48� 10�4

15.13� 10�3

Switzerland 0.39� 10�2* 12.16� 104 1.110.80� 10�2* 10.01� 102

51.57� 101

UK 0.92� 10�2* 47.88� 104 1.190.94� 10�2* 18.38� 101

13.10� 101

Notes:aLambda-min statistic with critical values 0.017 and 0.05 for (q¼ 2,r¼ 0) and (q¼ 2, r¼ 1), respectively, at the 5 percent level. SeeBierens (1997a), Table 2.bValue of the trace test for liner restrictions with critical value form¼ 2q equal to 4.70 when r¼1 and q¼ 2 at the 5% significancelevel. See Bierens (1997a), Table 3.cThe consistent estimate of the number of cointegrating vectors isgiven by rr ¼ argminr0�2 gm(r0), m¼ 2. Values in bold denote theminimum value of the test statistic.*Significant at the 5% significance level.

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and ��m for the individual countries and averagingthem across countries. This kind of panel dataestimator is the group-mean estimator. An alternativeestimator could be the pooled estimator where therandom matrices are jointly estimated for thecountries considered. Of course, in both cases it willbe necessary to modify accordingly the trace-testthat will be used in order to test the possible linearrestrictions on the cointegrating vector.

At a parametric level, Mark and Sul (2001) focusedon a dynamic OLS estimator and they examinedthe small sample properties of weighted and non-weighted panel estimators and of the mean groupestimator. They reported findings in favour of acointegrating relationship between ft and st, whilethey re-established the ability of monetary fundamen-tals to forecast future exchange rate returns usingforecasts generated by panel regressions.

Finally, the asymptotic properties of the two testconsidered, namely the Joahnsen and the Bierenstests, should be examined in order to identify theproportion of times that the two tests individ-ually render the correct number of cointegratingrelationships in the presence of alternative non-linearshort-run dynamics. For this purpose a parametricbootstrap exercise should be performed wherethe bootstrap DGP will be determined by smoothtransition functions or functions that incorporate athreshold. These possibilities are left for a futureresearch agenda.

Acknowledgements

I would like to thank Mark P. Taylor and LucioSarno for helpful comments and suggestions.

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