Comparison of Panel Cointegration Tests†

M.Sc. Deniz Dilan KaramanHumboldt University Berlin

Institute for Statistics and EconometricsSpanduerstr. 110178 Berlin

karaman@wiwi.hu-berlin.de27 May 2004

Abstract

The main aim of this paper is the comparison of size and power properties of four residual-based andtwo maximum-likelihood-based panel cointegration tests with a DGP consisting of three variables. In addi-tion to this, size-adjusted power results are also presented. In this study Pedroni (1999)’s panel-ρ, group-ρ,panel-t, group-t statistics and Larsson, Lyhagen and Løthgren (2001)’s standardized LR-bar statistic will beconsidered. A new panel cointegration test statistic analogous to the test statistic of Larsson et al. (2001)is introduced as well, which is based on maximum-eigenvalue test statistic. The simulation results indicatethat the panel-t and maximum-likelihood-based statistics have better size and power properties among thesix panel cointegration test statistics discussed in this paper. Finally, the Fisher Hypothesis is tested withtwo different data sets for OECD countries. The results point out the existence of the Fisher relation for theresidual-based panel cointegration tests.

1 Introduction

For the last two decades in the empirical literature cointegration techniques have been awidely used method. But it was sometimes difficult to find large time series for some empiricalproblems. To solve this limited time observations problem and to make use of the advantageof the growing multiple cross-sectional dimension, it was necessary to develop unit root andcointegration tests for the pooled time series panels. One other reason for the developmentof these techniques for the panel data, was the low power of the ADF and DF unit root testsfor the univariate case against near unit root alternatives. Following this necessity, in recentyears after the extension of the univariate unit root tests to the panel data by Levin andLin (1992), Quah (1994), Breitung and Meyer (1994) and Im et al. (1997), the extensionof the cointegration tests to the panel data has grasped a wide interest in the literature.There are mainly two different approaches for the panel cointegration tests, residual-basedand maximum-likelihood-based.

McCoskey and Kao (1998), Kao (1999), Pedroni (1995, 1997, 1999) propose residual-based, while Groen and Kleibergen (1999), Larsson and Lyhagen (1999) and Larsson, Lyhagenand Løthgren (2001) propose maximum-likelihood-based panel cointegration test statistics.

†Preliminary version. Do not copy without author’s permission.

1

McCoskey and Kao (1998) derive a panel cointegration test for the null of cointegration whichis an extension of the LM test and the locally best unbiased invariant (LBUI) test for anMA root. They take Harris and Inder (1994) and Shin (1994) as a basis for their research.Kao (1999) considers the spurious regression for the panel data and introduces two types ofpanel cointegration tests, the Dickey-Fuller (DF) and augmented Dickey-Fuller (ADF) typetests. He proposes four different DF type tests, and makes use of the sequential limit theoryof Phillips and Moon (1999) for the asymptotic distributions of these tests.

Groen and Kleibergen (1999) present within maximum-likelihood framework how homoge-nous and heterogeneous cointegration vectors are estimated using the GMM estimator, andpropose a likelihood ratio test for the common cointegration rank, which is based on theseGMM estimators and the cross-dependence. Larsson et al. (2001), on the other hand, proposepanel cointegration test statistic based on cross-independence. The tests of Pedroni (1995,1999) and Larsson, Lyhagen and Løthgren(2001) will be introduced in the next section.

In this paper the properties of the residual-based panel cointegration tests of Pedroni(1999) will be compared to the properties of the maximum-likelihood-based panel cointegrationrank test of Larsson et al. (2001). In addition to this, the approach in Larsson et al. (2001)will be extended to a maximum eigenvalue based test for the panel cointegration.

In the simulation study we concentrate our attention to the changes in size, power and size-adjusted power of the panel cointegration tests when time and cross-section dimensions andvarious parameters in the data generating process vary, e.g. the correlation parameter betweenthe disturbances to stationary and non-stationary part of the DGP for each cross-section. Theresults of the simulation study, which depend on a DGP with three variables, illustrate thatpanel-t statistic has the best size and power properties. The sizes of the panel-t and maximum-likelihood-based tests are around the nominal size of 5% when T and N increases. While thepower of panel-t statistic is near unity when N is large also when there is high correlation.The power of the standardized LR-bar statistic is higher than the standardized LR-max barstatistic when the cointegration rank is close to zero (ψ is near unity) and the correlation ishigh. Lastly, we test the Fisher relation among the OECD countries for different time spansusing the tests considered in this paper.

The second part of the paper covers the panel cointegration tests of Pedroni (1999),Larsson et al. (2001) and the maximum eigenvalue-based panel cointegration test. In thethird section we present the way how the DGP of Toda (1995) is modified for the paneldata, and the fourth section gives a description of the simulation study. The fifth section isdevoted to the interpretation of the simulation results. The validity of the Fisher relation isbe the subject of the sixth section. The seventh section concludes. The simulation results arepresented in Appendix A, B and C.

2

2 Panel Cointegration Tests

2.1 Pedroni (1999)

Pedroni (1999) extends the procedure of residual-based panel cointegration tests that heintroduced in Pedroni (1995) for the models, where there are more than one independentvariable. He proposes several residual-based null of no cointegration panel cointegration teststatistics. In this study two within-dimension-based1 (panel-ρ and panel-t) and two between-dimension-based2 (group-ρ and group-t) panel cointegration statistics of Pedroni (1999) will becompared with the maximum-likelihood-based panel cointegration statistics. Panel-ρ statis-tic is an extension of the non-parametric Phillips-Perron ρ-statistic, and parametric panel-tstatistic is an extension of the ADF t-statistic. Between-dimension-based statistics are justthe group mean approach extensions of the within-dimension-based ones. Group-ρ statisticis chosen, because Gutierrez (2002) has found out that this test statistic has the best poweramong the test statistics of Pedroni (1999), Larsson et al. (2001) and Kao (1999). Group-tstatistic is considered, because the data generating process is appropriate for parametric ADF-type tests. And the within-dimension versions of these statistics are considered in order to beable to compare.

The starting point of the residual-based panel cointegration test statistics of Pedroni(1999) is the computation of the residuals of the hypothesized cointegrating regression3,

yi,t = αi + β1ix1i,t + β2ix2i,t + . . . + βMixMi,t + ei,t (1)

t = 1, . . . , T ; i = 1, . . . , N ; m = 1, . . . , M

where T is the number of observations over time, N denotes the number of individual membersin the panel, and M is the number of independent variables. It is assumed here that the slopecoefficients β1i, . . . , βMi, and the member specific intercept αi can vary across each cross-section.

To compute the relevant panel cointegration test statistics the panel cointegration regres-sion in (1) should be estimated first. For the computation of the panel-ρ and panel-t statisticstake the first-difference of the original series and estimate the residuals of the following regres-sion:

∆yi,t = b1i∆x1i,t + b2i∆x2i,t + . . . + bMi∆xMi,t + πi,t

Using the residuals from the differenced regression, with a Newey-West (1987) estimator cal-culate the long run variance of πi,t, which is symbolized as L2

11i .

L211i =

1

T

T∑t=1

π2i,t +

2

T

ki∑s=1

(1− s

ki + 1

) T∑t=s+1

πi,tπi,t−s

1Within-dimension-based statistics are calculated by summing the numerator and the denominator over Ncross-sections separately.

2Between-dimension-based statistics are calculated by dividing the numerator and the denominator beforesumming over N cross-sections.

3In this study the regression equation with an heterogeneous intercept will be considered. Note that itcould also be estimated without a heterogeneous intercept, or with time trend and/or common time dummies.

3

For panel-ρ and group-ρ statistics estimate the regression ei,t = γiei,t−1 + ui,t using theresiduals ei,t from the cointegration regression (1). Then compute the long-run variance (σ2

i )and the contemporaneous variance (s2

i ) of ui,t, where

s2i =

1

T

T∑t=1

ui,t

σ2i =

1

T

T∑t=1

uit +2

T

ki∑s=1

(1− s

ki + 1

) T∑t=s+1

ui,tui,t−s

and ki4 is the lag length. is the lag length. In addition to this calculate also the term

λi =1

2

(σ2

i − s2i

).

On the other side for panel-t and group-t statistics using again the residuals ei,t of cointe-

gration regression (1) estimate ei,t = γiei,t−1 +∑Ki

t=1 γi,k∆ei,t−k + u∗i,t and compute the varianceof u∗i,t, which is denoted as s∗2i . In this study to determine the lag truncation order of the ADFt-statistics the step-down procedure and the Schwarz lag order selection criterion are used.

s∗2i =1

T

T∑t=1

u∗2i,t, s∗2N,T ≡1

N

N∑i=1

s∗2i .

The next step is the calculation of the relevant panel cointegration statistics using thefollowing expressions.

- Panel-ρ statistic

T√

NZρN,T−1 ≡ T√

N

(N∑

i=1

T∑t=1

L−211ie

2i,t−1

)−1 N∑i=1

T∑t=1

L−211i

(ei,t−1∆ei,t − λi

)(2)

- Panel-t statistic

Z∗tN,T

≡(

s∗2N,T

N∑i=1

T∑t=1

L−211ie

2i,t−1

)−1/2 N∑i=1

T∑t=1

L−211iei,t−1∆ei,t (3)

- Group-ρ statistic

TN−1/2ZρN,T−1 ≡ TN−1/2

N∑i=1

(T∑

t=1

e2i,t−1

)−1 T∑t=1

(ei,t−1∆ei,t − λi

)(4)

4Pedroni (1995) used ki = 4(

T100

)2/9as lag truncation function for the Newey-West kernel estimation

as recommended in Newey and West (1994). The nearest integer is taken as the lag length for different Tdimensions.

4

- Group-t statistic

N−1/2Z∗tN,T

≡ N−1/2

N∑i=1

(T∑

t=1

s∗2i e2i,t−1

)−1/2 T∑t=1

ei,t−1∆ei,t (5)

Lastly, apply the appropriate mean and variance adjustment terms to each panel cointe-gration test statistic, so that the test statistics are standard normally distributed.

κN,T − µ√

N√ν

=⇒ N(0, 1),

where κN,T is the appropriately standardized form of the test statistic, µ and ν are thefunctions of moments of the underlying Brownian motion functionals. The appropriate meanand variance adjustment terms for different number of regressors (m is the number of regressorswithout taking the intercept into account) and different panel cointegration test statistics aregiven in Table 2 in Pedroni (1999)5.

The null hypothesis of no cointegration for the panel cointegration test is the same foreach statistic,

H0 : γi = 1 for all i

whereas the alternative hypothesis for the between-dimension-based and within-dimension-based panel cointegration tests differs. The alternative hypothesis for the between-dimension-based statistics is

H1 : γi < 1 for all i,

where a common value for γi = γ is not required. For within-dimension-based statistics thealternative hypothesis

H1 : γ = γi < 1 for all i

assumes a common value for γi = γ.

Under the alternative hypothesis, all the panel cointegration test statistics considered inthis paper diverge to negative infinity. Thus, the left tail of the standard normal distributionis used to reject the null hypothesis.

2.2 Larsson, Lyhagen and Løthgren (2001)

Larsson et al. (2001) present a maximum-likelihood-based panel test for the cointegratingrank in heterogeneous panels. They propose a standardized LR-bar test based on the meanof the individual rank trace statistic of Johansen (1995) .

The panel data set consists of N cross-sections observed over T time periods, where i isthe index for the cross-section, t represents the index for the time dimension and j = 1, . . . , p

5This table contains the mean and variance values for the cases when there is no heterogeneous intercept,or when there is a heterogeneous intercept or/and a time trend in the heterogeneous regression equation.

5

is the number of variables in each cross-section. The following heterogeneous V AR(ki) model,

Yit =

ki∑

k=1

AikYi,t−k + εit i = 1, . . . , N, (6)

is considered for each cross-section under the assumptions that εit is Gaussian white noise witha nonsingular covariance matrix εit ∼ Np(0, Ωi), and the initial conditions Yi,−ki+1, . . . , Yi,0 arefixed. One shortcoming of this model is that it allows neither an intercept nor a time trendin the VAR model. The error correction representation for Equation (6) is

∆Yit = ΠiYi,t−1 +

ki−1∑

k=1

Γik∆Yi,t−k + εit i = 1, . . . , N (7)

Πi = −(Ip − Ai1 − Ai2 − . . .− Ai,ki)

Γik = −(Ai,k+1 + . . . + Ai,ki) k = 1, . . . , ki − 1,

where Πi is of order (p× p). In the reduced rank form it is possible to write Πi = αiβ′i, where

αi and βi are of order p× ri and have full column rank.

Larsson et al. (2001) consider the null hypothesis that all of the N cross-sections haveat most r cointegrating relationships among the p variables. Then the null hypothesis for thepanel cointegration test looks like

H0 : rank(Πi) = ri ≤ r for all i = 1, . . . , N,

whereH1 : rank(Πi) = p for all i = 1, . . . , N.

The starting point for the standardized LR-bar statistic of Larsson et al. (2001) is thecomputation of the trace statistic for each cross-section i, which is denoted as

LRiTH(r)|H(p) = −2 ln QiTH(r)|H(p) = −T

p∑j=ri+1

ln(1− λi,j), (8)

where λi,j is the jth eigenvalue of the ith cross-section to the eigenvalue problem given inJohansen (1995). The average of the N individual trace statistics is used in order to calculatethe standardized LR-bar statistic:

LRNTH(r)|H(p) =1

N

∑LRiTH(r)|H(p) (9)

The standardized LR-bar statistic for the panel cointegration rank test is defined as

γLRH(r)|H(p) =

√N

(LRNTH(r)|H(p) − E(Zk)

)√

V ar(Zk), (10)

6

where E(Zk) is the mean and V ar(Zk) is the variance of the asymptotic trace statistic Zk.

Zk ≡ tr

∫ 1

0

(dW )W ′(∫ 1

0

WW ′)−1 ∫ 1

0

W (dW )′

,

where−2 ln QiTH(r)|H(p) ω−→ Zk

and W is a k = (p−r) dimensional Brownian motion. Larsson et al. (2001) have simulated themean and variance of the asymptotic trace statistic using the simulation procedure describedin Johansen (1995) for different k values6.

Under the assumptions that

a.) one deals with variables that are integrated at most of order one;7

b.) E(εit) = 0, E(εitεjt) = Ωi for i = j and E(εitεjt) = 0 otherwise8;

c.) the first two moments of Zk exists;

d.) E(−2 ln QT ) = E(Zk) + O(T−1) and V ar(−2 ln QT ) = V ar(Zk) + O(T−1)9;

e.) the short term dynamics varies over individuals, but the long term dynamics is assumedto be constant10,

the standardized LR-bar statistic is standard normally distributed as N and T →∞ in sucha way that

√NT−1 → 0.

The panel cointegration rank test of Larsson et al. (2001) is a one-sided test H0 :rank(Πi) = ri ≤ r, which is rejected for all i, if the standardized LR-bar statistic is big-ger than the (1 − α) standard normal quantile, where α is the significance level of the test.The sequential procedure of Johansen (1988) is used as the testing procedure. First H0 : r = 0is tested, and if r = 0 is rejected H0 : r = 1 is tested. The procedure continues until the nullhypothesis is not rejected or the H0 : p− 1 is rejected.

2.3 Standardized LR-max Bar Statistic

In this subsection using the same procedure defined in Larsson et al. (2001), a panel coin-tegration test, which is an extension of the maximum eigenvalue statistic defined by Johansen(1991,1995), is proposed. The starting point of the standardized LR-max bar statistic is againthe heterogeneous VAR model defined in (6). VECM representation of this VAR model will

6Simulated mean and variance values of the asymptotic trace statistic can be found in Table 1 of Larssonet al. (2001)

7Assumption 1 in Larsson et al. (2001).8Necessary in order to establish the asymptotic distribution of the panel cointegration test.9Necessary in order to find the joint convergence rate of N and T .

10Assumption 3 and Assumption 3′ in Larsson et al. (2001). These assumptions are made in order to assurethe standard normal distribution of standardized LR-bar statistic as N →∞.

7

help us to calculate the maximum eigenvalue statistic for each cross-section. However, thistime the null hypothesis that all of the N groups in the panel have at most r cointegratingrelations among the p variables, against the alternative hypothesis that all of the N groupsin the panel have at most one rank greater than r is tested. The null hypothesis for thestandardized LR-max bar statistic can be formulated as,

H0 : rank(Πi) = ri ≤ r for all i = 1, . . . , N

and the alternative hypothesis is,

H1 : rank(Πi) = ri ≤ r + 1 for all i = 1, . . . , N.

The maximum eigenvalue statistic for each cross-section i is computed as

LRmaxiT H(r)|H(r + 1) = −2 ln QiTH(r)|H(r + 1) = −T ln(1− λi,r+1), (11)

where λi,r+1 is the (r + 1)th eigenvalue of the ith cross-section. Analogous to the LR-barstatistic, LR-max bar statistic is defined as the average of N individual maximum eigenvaluestatistics,

LRmaxNT H(r)|H(r + 1) =

1

N

N∑i=1

LRmaxiT H(r)|H(r + 1). (12)

And the standardized LR-max bar statistic will be

γLRmax H(r)|H(r + 1) =

√N

(LRmax

NT H(r)|H(r + 1)− E(Zmaxk

))√

V ar(Zmaxk )

. (13)

Zmaxk is the asymptotic maximum eigenvalue statistic, which is defined as

Zmaxk ≡ λmax

∫ 1

0

(dW )W ′(∫ 1

0

WW ′)−1 ∫ 1

0

W (dW )′

,

where W is again a k = (p− r) dimensional Brownian motion, E(Zmaxk ) and V ar(Zmax

k )are the simulated mean and variance of the asymptotic maximum eigenvalue statistic. Theappropriate mean and variance of the asymptotic maximum eigenvalue statistic for differentvalues of k are simulated using the procedure in Johansen (1995, Chapter 15) and presentedin Table I. As the simulation values of Larsson et al. (2001) will be slightly different fromthe ones in Table I, the mean and variance of the asymptotic trace statistic are simulated onemore time in order to sustain the equality of the first (k = 1) mean and variance values forZk and Zmax

k .

Under the same assumptions like the standardized LR-bar statistic and taking the proofof the asymptotic distribution of the standardized panel trace statistic analogous for thestandardized panel maximum eigenvalue statistic, standardized LR-max bar is also standardnormally distributed as N and T →∞11.

γLRmax =⇒ N(0, 1)

11This conclusion is valid, since Zk and Zmaxk have the same Brownian motion functionals.

8

The testing procedure of the standardized LR-max bar statistic is again the sequentialtesting procedure of Johansen (1988). The null hypothesis of standardized panel maximumeigenvalue test is rejected like the standardized panel trace statistic, when γLRmax > z1−α.

k = (p− r) E(Zk) V ar(Zk) k = (p− r) E(Zmaxk ) V ar(Zmax

k )1 1.139 2.180 1 1.139 2.1802 6.079 10.492 2 5.416 8.9893 14.946 24.836 3 10.363 15.3804 27.769 44.908 4 15.536 21.0245 44.433 70.720 5 20.748 26.1306 64.964 102.582 6 26.020 30.9187 89.331 140.616 7 31.315 35.4218 117.499 184.468 8 36.618 39.8239 149.495 232.116 9 41.957 43.87610 185.094 284.881 10 47.256 47.42311 224.629 345.084 11 52.582 51.15612 267.693 412.094 12 57.898 54.987

Table I: Simulated moments for Zmaxk and Zk.

3 Data Generating Process

In this research the Monte Carlo study is based on the Toda’s (1995) data generatingprocess, as it has been used in several papers in the literature12. The canonical form of theToda process allows us to see the dependence of the test performance on some key parameters.In the following lines you can find the modified data generating process of Toda for the paneldata.

Let yi,t be a p-dimensional vector, where i is again the index for the cross-section, t isthe index for the time dimension and p denotes the number of variables in the model. Datagenerating process has the form of a VAR(1) process. The general form of the modified Todaprocess for a system of three variables and in the absence of a linear trend in the data lookslike then,

yi,t =

ψa 0 00 ψb 00 0 ψc

yi,t−1 + ei,t (14)

t = 1, . . . , T ; i = 1, . . . , N,

where the initial values of yi,t, which can be represented as yi,0 are zero. The error terms for

12Lutkepohl and Saikkonen (2000), Saikkonen and Lutkepohl (1999,2000), Hubrich, Lutkepohl and Saikko-nen (2001) etc.

9

each cross-section has the following structure.

εi,t =

(ε1i,t

ε2i,t

)≡ i.i.d.N

(0,

(Ir ΘΘ′ Ip−r

))

In this framework the true cointegrating rank of the process is denoted by r, whereas ε1i,t andε2i,t are the disturbances to the stationary and non-stationary parts of the data generatingprocess, respectively, while Θ represents the vector of instantaneous correlation between thestationary and non-stationary components of the relevant cross-section.

Taking Equation (14) into account, when ψa = ψb = ψc = 1 a cointegrating rank of r = 0is obtained. Thus, the data generating process becomes,

yi,t = I3yi,t−1 + εi,t, (15)

where εi,t ≡ i.i.d.N(0, I3), which means that the process consists of three non-stationary com-ponents and these components are instantaneously uncorrelated. This can also be illustratedas,

∆yit = Πi,tyi,t−1 + εi,t

∆yit = εi,t

where Πi,t = −(I − Ai1) and Ai1 represents the coefficient matrix of the VAR(1) process.

With |ψa| < 1 and ψb = ψc = 1 the true cointegrating rank of the DGP is 1, and the it iscomposed of one stationary and two non-stationary components, which can be formulated as.

yi,t =

(ψa 00 I2

)yi,t−1 + εi,t (16)

with

εi,t ≡ i.i.d.N

(0,

(1 ΘΘ′ I2

))

where Θ = (θa, θb) and |θa| , |θb| < 1.

The cointegrating rank of the process is r = 2 when ψa and ψb are less than unity inabsolute value and ψc = 1. This can be represented in matrix form as,

yi,t =

ψa 0 00 ψb 00 0 1

yi,t−1 + εi,t (17)

with

εi,t ≡ i.i.d.N

(0,

(I2 Θ′

Θ 1

))

where Θ = (θa, θb) and θa, θb are less than unity in absolute value. This process consists of onenon-stationary and two stationary components and these components are correlated, when θa

and θb 6= 0.

10

When |ψa| , |ψb| and |ψc| < 1 the DGP is a I(0) process, and the cointegrating rank isr = 3, which can be represented as,

yi,t = Ψyi,t−1 + εi,t, (18)

where Ψ = diag(ψa, ψb, ψc) and εi,t ≡ i.i.d.N(0, I3).

4 Simulation Study

In order the see the performance of the tests to the changes in some key parameters,throughout the simulation study the time and cross-section dimensions, ψa, ψb, ψc parame-ters and also the correlation parameters between the disturbances to the non-stationary andstationary part of the DGP for each cross-section, which are denoted as θa and θb will vary.

The correlation parameters θa and θb take the values 0.0, 0.4, 0.7 and ψ parameters takethe values 0.5, 0.8, 0.95, 1. Mainly these values are similar to the values considered by Toda(1995). The only difference is Toda has taken 0.8 and 0.90 as the upper bound for θa, θb andψa, ψb, ψc, respectively. The value 0.95 for ψ parameters will help us to see how the tests reactwhen the cointegrating rank of the process is near zero. The lower bounds for θa, θb and ψa,ψb, ψc are assumed to be the same as the values considered by Toda (1995). The performanceof the tests under the assumption of no instantaneous correlation between the disturbances ischecked by θa = θb = 0.

In order to be able to compare the results with Larsson et al. (2001), for the cross sectiondimension N = 1, 5, 10, 25, 50 and for the time dimension T = 10, 25, 50, 100, 200 aretaken into account, The total number of replications is 1000. While generating the randomerror terms, seeded values are used and the first 100 observations are deleted, so that thestarting values are not anymore zero. The tests were programmed in GAUSS 5.0.

The maximum lag order for the panel-t and group-t statistics is limited to 3, becausethis was the maximum lag order allowing an efficient estimation for small time dimensioncase, e.g T = 10. In addition to this, a kernel estimator is used to select the lag orderfor the non-parametric panel-ρ and group-ρ statistics, as explained in Section 2. For themaximum-likelihood-based test statistics a VAR model lag order selection criterion was notused, because the data was generated under the assumption of a V AR(1) process. Only thenull of no cointegration hypothesis is tested for both residual-based and maximum-likelihood-based panel cointegration tests, because the residual-based tests cannot test for the order ofpanel cointegrating rank.

In the next sections the simulation results for the empirical size and power and the size-adjusted power of the panel cointegration tests will be discussed. The tables for the simulationresults are presented in the Appendix B and C.

11

5 Interpretation of the Simulation Results:

The importance of this simulation study lies in the DGP. In order to see the size andpower properties of his tests Pedroni (1995) based his Monte Carlo study on MA processesand the size and power results for the system consisting of more than one independent variablewas missing. In this study the DGP is based on AR process and study covers the small sampleproperties of the residual-based tests when there is more than one independent variable in theDGP. Size-adjusted power of the panel cointegration tests is presented for the first time in theliterature.

5.1 Empirical Size and Power Properties

The most interesting results for empirical size and power properties of the panel cointe-gration tests are presented in Appendix B 13. The graphs for the empirical size properties canbe found in Figures 1-5, on the other hand Figures 6-26 illustrate the empirical power results.

5.1.1 Size properties

Figures 1 and 2 show us that the empirical sizes of group-ρ and panel-ρ statistics are alwayszero for T = 10, 25 and N ≥ 1, which means that the true hypothesis of no cointegration cannever be rejected. The severe size distortions for the other test statistics when T is smalland N is large, can easily be recognized from Figures 1 and 2, e.g. the empirical sizes ofthe test statistics except for the panel-and group ρ statistics are unity when T = 10 andN ≥ 25. This points out the fact that these tests are not appropriate if the time dimensionis much smaller than the cross-section dimension. However, in Figures 4 and 5 it is obviousthat when T and N dimensions grow the empirical sizes of maximum-likelihood-based andpanel-t tests approach to the nominal size of 5% level, especially for T = 200 and N ≥ 5. Onthe other hand, the empirical size of the group-ρ statistics is around 5% when T = 100 andN ≥ 5, and for panel-ρ when T = 50 and N ≥ 5. The size distortions of group-t, panel-t andmaximum-likelihood-based test statistics also decrease for fixed N when T increases.

5.1.2 The power properties

As there is not much space to present all the results regarding the empirical power prop-erties, we discuss here only the cases where ψ parameters are 0.5, 0.95, 1 and where the distur-bances are either uncorrelated or strongly correlated. For those who are also interest in othercases, the results can be provided by the author.

First of all the empirical power results for ψa = ψb = ψc = 0.5, 1 without correlation are

13In the figures, “Larsson(CW)” represents the standardized LR-bar test statistic calculated with the sim-ulated mean and variance values in this paper. “Std. LR-bar” is for the standardized LR-bar test statisticcalculated with the simulated mean and variance values of Larsson et al. (2001). “sc” is the abbreviation forSchwarz lag selection Criterion, whereas “sd” denotes the step-down lag selection method.

12

examined. What Figures 6, 8, and 10 in Appendix B exhibit is, the rejection rate of the nullof no cointegration for panel-ρ and group-ρ statistics when the true rank is bigger than zero, isalways 0 for T = 10 and N ≥ 1. This is also valid when we alter the parameters for θ’s and/orψ’s. This outcome is represented partly in the Appendix B due to the lack of space. When Nis large the powers of the test statistics (except for panel-ρ and group-ρ statistics) approachunity even for small T dimensions, e.g. T = 10. For group-ρ and panel-ρ test statisticsthe power converges to unity first when T = 25, N ≥ 10 and N ≥ 25, respectively. Theresults for T = 50, 100 and 200 are neglected, because the powers of all the test statistics areunity afterwards. In general the power of choosing a higher rank than the hypothesized rank(H0 : ri = 0) when the true rank is bigger than zero, goes to unity for all the test statistics,when T and N increase.

If we increase ψ parameters to 0.8 from 0.5 for different rank assumptions there won’t besevere changes in the results, so we do not consider them here.

With the increase in ψ parameters to 0.95, some noticeable changes in the power propertiesof the panel cointegration tests occur. For example, large T and N dimensions are necessary,so that the test statistics have high power. With the comparison of Figures 7, 9 and 11 withFigures 12, 15 and 18, correspondingly it can be concluded that in particular the powers ofthe standardized LR-bar and LR-max bar statistics do not approach unity so fast. In additionto this, the powers of the group-ρ and panel-ρ statistics are around zero also for T = 25. Formoderate panel datas the powers of maximum-likelihood-based test statistics are distinct fromunity when the true cointegrating rank is 1, which is illustrated by Figure 13. If the true rankis bigger than 1, especially when r = 3, the difference between the powers of standardizedLR-bar and LR-max bar statistics increases. This difference can be observed in Figures 16,17 and 19. Last but not least, the powers of the panel-t statistics converge unity for r = 1, 2and 3 when cross-section dimension is large enough, e.g. 50.

The power results do not change drastically if the correlation between the error termsis not high or ψ parameters are low. So we just take the cases where θa = θb = 0.7 and ψparameters are near unity into account. Figures 21 and 24 demonstrate the convergence ofthe power of all the panel cointegration tests to unity for T = 10. For large T dimensionsonly the results for group-ρ, panel-ρ and group-t statistics are presented, because the powerof maximum-likelihood-based and the panel-t tests statistics are near unity for almost all Tand N dimensions. Form Figures 22 and 25 for r = 1 and 2, respectively, it is obvious thatthe powers of four test statistics are near zero. When the true cointegrating rank is r = 1 thepowers of the group-ρ and group-t statistics do not approach unity, even for high T and Ndimensions. This is not anymore valid for the true cointegrating rank of 2.

In general the power results of the panel-t and group-t statistics for Schwarz and step-down methods of lag selection converge to each other, if T and N increases. StandardizedLR-bar statistics do not exhibit different results in powerwhen the statistics are calculatedeither with the simulated mean and variance values from this paper or with the simulatedvalues from Larsson et al. (2001).

13

5.2 Size-Adjusted Power Properties

In section 5.1 it was pointed out that the empirical power results of the panel-ρ andgroup-ρ statistics are always zero and the power results of the other test statistics approachunity for small T dimensions, even if the θ and ψ parameters change. In order to have a betterview about the small sample properties of the tests statistics, it is a good idea to adjust thepower for size. The relevant graphs for the size-adjusted power results are demonstrated inAppendix C starting from Figure 27.

When ψa = ψb = ψc = 0.5, 1 and there is no correlation, just the graphs for T = 10will be discussed, because the powers of all the test statistics approach unity for T ≥ 25 andN ≥ 10. If T = 10, maximum-likelihood-based and group-t statistics have the lowest powerfor the true cointegrating ranks of 1 and 2. In Figures 27 and 28 it is clear that the panel-ρstatistic has the highest power, which reaches 0.891 and 0.681, respectively for r = 1 and 2.If the true cointegrating rank is 3 like in Figure 29, the power of rejecting the null-hypothesisof no cointegration is the highest for the standardized LR-bar statistics with 0.534, whereasthe group-t statistics have the lowest power with 0.04.

As there is not much difference in the size-adjusted power results when ψ increases to 0.8,we just do not discuss this case in this paper.

If the ψ parameters are near unity with 0.95 and T = 10, the powers of all the test statisticsare at most 0.074 for r = 1, 2 and 3, which is also obvious from Figures 30, 34 and 38. Figures31 and 32 show us that the maximum-likelihood-based test statistics have the lowest power andthe panel-ρ and panel-t test have the highest power. With the true cointegrating assumptionof r = 2, Figures 35 and 36 present that the maximum-likelihood-based test statistics havethe lowest power again. The powers of all the test statistics converge to unity for high T andN dimensions, which proves what the theory assumes. One interesting outcome of the MonteCarlo study belongs to the case where T = 100 and r = 3. For this case the standardizedLR-bar test statistics have the highest power among all the test statistics, whereas the powerof the standardized LR-max bar test statistic is the lowest. This eye-catching difference canbe observed in Figure 40.

In order to understand how the test statistics behave under the assumption of correlatederror terms, only the case with the highest correlation parameters is discussed because of thesame reasons we have stressed in section 5.1. For ψa = ψb = ψc = 0.95, 1 the powers of themaximum-likelihood-based test statistics and the panel-t statistics approach to one even ifT = 10. On the other hand, the powers of the other test statistics are near zero for small Tdimensions. (Figures 42 and 45) For T = 100 and 200 only the group-ρ, panel-ρ and group-tstatistics are illustrated again, because the other test statistics converge to unity faster. Thepower of rejecting the cointegrating rank of zero when the true rank is 1 for group-ρ andpanel-ρ statistics cannot go to unity even if T and N dimensions are high. The power canconverge to unity for the panel-ρ statistic if the true cointegrating rank is 2. (Figure 47)

The powers of group-t and panel-t test statistics are again not much different for Schwarzand step-down lag selection methods, when T and N increase. Also the powers of the stan-dardized LR-bar test statistics are the same when the statistics are either calculated with thesimulated mean and variance values in this paper or with the simulated values in Larsson et

14

al. (2001).

6 Empirical Example: Fisher Hypothesis

In this section we try to find out whether the panel cointegration analysis give differentresults than the results of the usual cointegration techniques to an empirical example. Forthis purpose Fisher Hypothesis is considered, which is a widely tested economic relation inthe macroeconomic literature.

There is a mixture of conclusions in the empirical literature for Fisher effect. The non-stationarity of the nominal interest and inflation rates made the application of the cointegra-tion techniques possible in order to test for the long-run relation between the nominal interestand inflation rates. The studies which find evidence for Fisher relation using the unit root andcointegration techniques are: Atkins (1989), Evans and Lewis (1995), Crowder and Hoffman(1996), Crowder (1997), whereas the studies of Rose (1988), MacDonald and Murphy (1989),Mishkin (1992) and Dutt and Ghosh (1995) cannot find any evidence for the Fisher effect.

As an alternative method to the cointegration techniques the structural VAR methodologyof King and Watson (1997) is applied by Koustas (1998) and Koustas and Serletis (1999), wherethey couldn’t find any evidence for Fisher relation either.

With the application of the panel unit root and cointegration tests the recent panel datastudy for 9 industrialized countries by Crowder (2003) concludes that the Fisher effect exists.

Before interpreting the results of the panel cointegration tests to the Fisher relation, usingthe data set considered in this paper, it maybe better to give a short description of the Fisherrelation. The Fisher Hypothesis states that the real interest rate (rt) is the difference betweenthe nominal interest rate (it) and the expected inflation rate (πe

t ),

rt = it − πet

which means that no one will lend at a nominal rate lower than the expected inflation, andthe nominal interest rate will be equal to the cost of borrowing plus the expected inflation.

it = rt + πet (19)

Another aspect of the Fisher relation is that the real interest rates are constant or showlittle trend in the long run. This can be explained with the phenomena that the nominalinterest rate absorbs all the changes in the expected inflation rate when the change in thegrowth rate of the money supply alter the inflation rate. If the real interest rate changeswith a change in the expected inflation, then the Fisher Hypothesis will not hold. Whenstationarity of the real interest rates with a positive constant (r∗) and a normally distributederror term (ut ∼ N(0, σ2

u)) is assumed, the equation for rt becomes,

rt = r∗ + ut. (20)

In addition to this, with the assumption that the agents do not make systematic errorsand the actual inflation rate (πt) differs from the expected inflation rate (πe

t ) with a stationary

15

process(ξt ∼ N(0, σ2

ξ )), the equation for πt is,

πt = πet + ξt. (21)

When we insert (20) and (21) into (19), the Fisher equation for the cointegration analysislooks like,

it = a + bπt + εt,

where a = r∗, εt = ut − ξt and according to the theory b = 1. We search for the existence ofa cointegrating relation between the nominal interest rate and the inflation rate in the paneldata, in order to see if the Fisher relation holds.

To test the Fisher Hypothesis two different data sets consisting of quarterly nominalinterest-and inflation rates are considered. The first data set is the monthly data for 19OECD countries14 from 1986:06 to 1998:12. The second data set consists of monthly data for11 OECD countries15 from 1991:02 to 2002:12. The results of the panel cointegration tests forthe Fisher Hypothesis are presented in Appendix A.

While testing the Fisher Hypothesis with the panel cointegration tests, we face a problem.Standardized LR-bar and LR-max bar tests cannot be applied to the VAR models with anintercept. Therefore we have to limit our attention to the case where there is no intercept inthe VAR model for the maximum-likelihood-based panel cointegration tests.

In order to standardize the test statistics of Pedroni, mean and variance values for thecase when there is one independent variable in the system (m = 1) are necessary. These valuescan be found in Pedroni (1995). The lag selection criterion for the maximum-likelihood-basedpanel cointegration tests is the Schwarz Criterion and the maximum lag order is set equal to 6.For the ADF t-statistic based panel cointegration tests of Pedroni, two lag selection methodswill be considered: Step-down method and Schwarz Criterion. The maximum lag order forthese methods is limited to 12, because the data sets consist of monthly data.

When the first data set (1989 : 06 − 1998 : 12) is considered, country-by-country tracetests point out the existence of the Fisher relation, except for Austria and UK. On the otherhand, when the whole panel data is tested, standardized LR-bar test cannot reject the nullhypothesis of all the countries having at most cointegrating rank of 2. This means that theunderlying heterogeneous VAR model is stable. This result is also valid for the cases whenthe standardized LR-bar statistic is applied to the data set without Austria or to the data setwithout Austria and UK.

The results of the standardized LR-max bar statistic are not different from the ones forthe standardized LR-bar statistic 16. Standardized panel maximum eigenvalue test also cannotreject the null hypothesis of the r = ri ≤ 2, which points out the stability of the variables.

Country-by-country test results for the second data set (1991 : 01 − 2002 : 12) cannotreject the null hypothesis of ri = 1, except for Japan. On the other hand, the results show that

14Germany, France, Italy, Netherlands, Spain, Finland, Austria, Ireland, Portugal, Belgium, US, Japan, UK,Denmark, Mexico, Norway, Iceland, Sweden, Canada.

15US, Korea, Japan, UK, Denmark, Mexico, Norway, Iceland, Hungary, Sweden, Canada.16Results are presented in Table II and Table III of Appendix A

16

the Fisher Hypothesis does not hold for Sweden. This outcome for Sweden is different from thecountry-by-country for the first data set. Standardized LR-bar test accepts the hypothesis ofcointegrating rank of two, even when we test the relation without taking Japan into account.The results for the standardized LR-max bar test do not give also a different outcome for thesecond data set17.

The residual-based panel cointegration tests allows us to consider two different cases, thecase where there is a heterogeneous intercept and the case where there is no heterogeneous in-tercept in the regression equation. The results of Pedroni’s tests are presented in the AppendixA starting with Table VI to Table IX. The outcomes of the residual-based panel cointegra-tion tests are different from the maximum-likelihood-based tests. For the residual-based panelcointegration tests the rank order of the cointegrating matrix cannot be tested. By testing thenull hypothesis of no cointegration it can just be determined whether there is a cointegrationrelation.

All of the residual-based panel cointegrating tests reject the null of no cointegration forboth data sets, when there is no heterogeneous intercept in the panel regression equation.This is also the same case when we exclude Austria and UK from the first data set, and Japanfrom the second data set. However, some of the test statistics give different results for bothdata sets with the assumption of a heterogeneous intercept in the panel regression equation.The panel-ρ test statistic cannot reject the null of no cointegration, if the first data set isconsidered as a whole, which means that the Fisher Hypothesis does not hold. This is alsovalid for the panel-t statistics if the tests are undertaken for the second data set excludingJapan.

7 Conclusion

With the extensive simulation study in Section 5, which covers the empirical size, powerand size-adjusted power of six panel cointegration tests, it can be concluded that the panel-ttest statistic has the best size and power properties. We found out that the power of thepanel-t statistic approaches unity for small T and N dimensions, even when there is strongcorrelation between the innovations to the non-stationary and stationary part of the datagenerating process, while the empirical size of it is around the empirical nominal size of5% when T = 200 and N ≥ 5. On the other hand, the other three residual-based panelcointegration test statistics; group-ρ, panel-ρ and group-t have really poor power results if thecorrelation parameters and ψ parameters are high (e.g. when θa = θb = 0.7 and ψa = ψb = 0.95respectively).

The second test statistics which have the best size and power properties are the maximum-likelihood-based panel cointegration test statistics namely, the standardized LR-bar and LR-max bar statistics. They have better power if the correlation parameter is high and the ψparameter is around unity. The empirical size of the standardized LR-bar and LR-max barstatistics is around 5% like the panel-t test statistic when both T and N grow, especially iftime dimension grows faster than the cross-section dimension just as the theory points out.

17Results are presented in Table IV and Table V of Appendix A, respectively.

17

The power difference between the standardized LR-bar and LR-max bar statistics is large, forsome combinations of θa = θb = 0.0, 0.4, 0.7 (e.g. θa = θb = 0.4), if the true cointegratingrank is r ≥ 2 and the ψ parameters are 0.95. For such cases it may be better to applystandardized LR-bar statistic, instead of standardized LR-max bar, because the standardizedLR-bar statistic has higher power, especially when T is large. It should be also emphasizedthat the size and power results of the residual-based panel cointegration tests can depend onthe choice of the dependent variable. In this paper the first variable of the DGP has beentaken as the dependent variable for the residual-based panel cointegration tests.

In Section 6 while we were testing the Fisher hypothesis with the panel cointegrationtest statistics, we were able to present the results of the residual-based panel cointegrationtests under the assumption of a heterogeneous intercept in the panel regression equation,whereas maximum-likelihood-based statistics had to be considered without a heterogeneousintercept in the VAR model. For a future study the procedure in Larsson et al. (2001) canbe extended for a maximum-likelihood-based panel cointegration test statistic with a constantand a linear trend in the data. Residual-based panel cointegration tests of Pedroni pointed outthe existence of the Fisher relation for two different data set consisting of OECD countries.However, the maximum-likelihood based test statistics failed to find any evidence.

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21

8 Appendix A

Table II: Empirical results of the trace test. Monthly data from 1986:06 through 1998:12 is used. Alltests are performed at 5% level. For country by country tests the critical values are 12.53 and 3.84for testing r = 0 and r = 1 respectively. The panel rank test has a critical value of 1.645. There isneither in the VAR model nor in the cointegrating equation an intercept.

Country by Country Tests

LRiT (H(r)|H(p))

Country lag r=0 r=1 rank

Germany 2 48.01 1.57 1

France 2 63.31 1.44 1

Italy 1 15.75 2.04 1

Netherlands 4 17.37 1.98 1

Spain 5 30.16 2.97 1

Finland 2 26.82 1.61 1

Austria 5 36.16 4.81 2

Ireland 2 37.45 1.91 1

Portugal 4 18.36 3.48 1

Belgium 2 75.39 1.43 1

US 2 36.48 2.14 1

Japan 2 94.15 2.01 1

UK 5 30.25 6.46 2

Denmark 4 13.98 1.61 1

Mexico 3 25.09 1.52 1

Norway 4 21.00 1.11 1

Iceland 4 17.94 1.99 1

Sweden 1 20.67 1.38 1

Canada 1 38.94 2.81 1

Panel Tests r=0 r=1

γLR 39.00 3.50 2

38.42 3.75 2∗

37.17 2.23 2∗∗

γLR∗

39.08 3.51 2

38.51 3.77 2∗

37.25 2.24 2∗∗

Notes:

∗indicates results for the panel cointegration test without Austria

∗∗indicates results for the panel cointegration test without Austria and UK.

γLR∗indicates standardized LR-bar statistic calculated with the mean and variance values simulated in this study.

22

Table III: Empirical results of the maximum eigenvalue test. Monthly data from 1986:06 through1998:12 is used. All tests are performed at 5% level. For country by country tests the critical valuesare 11.44 and 3.84 for testing r = 0 and r = 1 respectively. The panel rank test has a critical valueof 1.645. There is neither in the VAR model nor in the cointegrating equation an intercept.

Country by Country Tests

LRiT (H(r)|H(r+1))

Country lag r=0 r=1 rank

Germany 2 46.44 1.57 1

France 2 61.86 1.44 1

Italy 1 13.71 2.04 1

Netherlands 4 15.39 1.98 1

Spain 5 27.19 2.97 1

Finland 2 25.21 1.61 1

Austria 5 31.35 4.81 2

Ireland 2 35.54 1.91 1

Portugal 4 14.89 3.48 1

Belgium 2 73.97 1.43 1

US 2 34.34 2.14 1

Japan 2 92.14 2.01 1

UK 5 23.79 6.46 2

Denmark 4 12.38 1.61 1

Mexico 3 23.57 1.52 1

Norway 4 19.89 1.11 1

Iceland 4 15.95 1.99 1

Sweden 1 19.30 1.37 1

Canada 1 36.13 2.81 1

Panel Tests r = 0 r = 1

γLRmax 39.80 3.51 2

39.08 3.77 2∗

38.49 2.24 2∗∗

Notes:

∗indicates results for the panel cointegration test without Austria

∗∗indicates results for the panel cointegration test without Austria and UK.

23

Table IV: Empirical results of the trace test. Monthly data from 1991:01 through 2002:12 is used.All tests are performed at 5% level. For country by country tests the critical values are 12.53 and3.84 for testing r = 0 and r = 1 respectively. The panel rank test has a critical value of 1.645. Thereis neither in the VAR model nor in the cointegrating equation an intercept.

Country by Country Tests

LRiT (H(r)|H(p))

Country lag r=0 r=1 rank

US 2 52.05 1.53 1

Korea 2 91.14 1.67 1

Japan 2 130.53 27.34 2

UK 4 31.97 3.18 1

Denmark 4 14.94 2.66 1

Mexico 2 58.34 1.77 1

Norway 4 28.83 1.02 1

Iceland 5 20.14 3.66 1

Hungary 5 66.64 3.16 1

Sweden 1 11.06 3.61 0

Canada 4 18.20 0.54 1

Panel Tests r=0 r=1

γLR 42.44 7.63 2

33.39 2.43 2∗

γLR∗

42.54 7.68 2

32.46 2.45 2∗

Notes:

∗indicates results for the panel cointegration test without Japan.

γLR∗indicates standardized LR-bar statistic calculated with the mean and variance values simulated in this study.

24

Table V: Empirical results of the trace test. Monthly data from 1991:01 through 2002:12 is used. Alltests are performed at 5% level. For country by country tests the critical values are 11.44 and 3.84for testing r = 0 and r = 1 respectively. The panel rank test has a critical value of 1.645. There isneither in the VAR model nor in the cointegrating equation an intercept.

Country by Country Tests

LRiT (H(r)|H(r+1))

Country lag r=0 r=1 rank

US 2 50.52 1.53 1

Korea 2 89.46 1.67 1

Japan 2 103.19 27.34 2

UK 4 28.79 3.18 1

Denmark 4 12.28 2.67 1

Mexico 2 56.57 1.77 1

Norway 4 27.81 1.02 1

Iceland 5 16.47 3.67 1

Hungary 5 63.48 3.16 1

Sweden 1 7.45 3.61 0

Canada 4 17.67 0.54 1

Panel Tests r=0 r=1

γLRmax 41.65 7.68 2

33.37 2.45 2∗

Notes:

∗indicates results for the panel cointegration test without Japan

25

Table VI: Empirical results of Pedroni’s panel cointegration tests without an intercept in the regres-sion equation. Monthly data from 1986:06 through 1998:12 is used. All tests are performed at 5%level. The panel rank test has a critical value of 1.645.

Country Slope ρ-stat t-stat(sc) t-stat(sd) lag(sc) lag(sd)

Germany 596.56 -0.19 -5.05 -3.92 9 1

France 1024.76 -0.24 -6.14 -6.15 5 5

Italy 846.66 -0.23 -4.64 -3.80 9 1

Netherlands 493.55 -0.16 -10.98 -4.17 9 1

Spain 780.86 -0.34 -9.21 -5.48 9 2

Finland 867.21 -0.23 -5.22 -3.67 9 0

Austria 436.18 -0.14 -9.36 -5.72 9 3

Ireland 77.77 -0.04 -2.46 -2.51 1 2

Portugal 594.51 -0.22 -5.50 -3.94 9 1

Belgium 776.12 -0.22 -6.75 -6.38 9 8

US 606.97 -0.19 -5.59 -4.00 10 1

Japan 251.62 -0.08 -5.83 -3.38 11 2

UK 533.78 -0.22 -10.09 -4.29 10 1

Denmark 792.24 -0.19 -8.80 -5.27 9 2

Mexico 491.30 -0.27 -5.19 -4.98 2 1

Norway 826.55 -0.23 -5.64 -4.07 9 1

Iceland 616.94 -0.16 -3.86 -3.32 3 0

Sweden 414.31 -0.12 -4.16 -3.03 9 0

Canada 695.63 -0.21 -4.95 -3.75 9 0

Group Tests Panel Tests

Group-ρ -12.92∗

-12.851∗

-12.392∗

Panel-ρ -15.57∗

-16.001∗

-15.272∗

Group-t(sc) -24.16∗

-22.701∗

-20.972∗

Panel-t(sc) -31.92∗

-29.121∗

-26.602∗

Group-t(sd) -14.38∗

-13.621∗

-13.222∗

Panel-t(sd) -23.60∗

-21.571∗

-20.342∗

Notes:

1indicates results for the panel cointegration tests without Austria

2indicates results without Austria and UK

* indicates rejection of the null of no cointegration.

26

Table VII: Empirical results of Pedroni’s panel cointegration tests without an intercept in the regres-sion equation. Monthly data from 1991:01 through 2002:12 is used. All tests are performed at 5%level. The panel rank test has a critical value of 1.645.

Country Slope ρ-stat t-stat(sc) t-stat(sd) lag(sc) lag(sd)

US 581.63 -0.21 -6.16 -4.77 9 1

Korea 609.56 -0.18 -5.84 -4.43 9 2

Japan 131.37 -0.05 -6.02 -4.55 9 3

UK 503.50 -0.21 -11.35 -4.89 10 1

Denmark 562.19 -0.13 -8.62 -4.80 9 2

Mexico 482.24 -0.25 -4.44 -4.44 1 1

Norway 654.62 -0.19 -5.04 -4.00 9 1

Iceland 619.07 -0.15 -3.82 -3.56 3 1

Hungary 392.18 -0.24 -8.51 -4.88 12 4

Sweden 0.20 -0.02 -1.91 -1.91 0 0

Canada 0.00 -0.33 -7.35 -7.35 7 5

Group Tests Panel Tests

Group-ρ -11.87* -12.411∗

Panel-ρ -11.72* -12.821∗

Group-t(sc) -18.36* -17.601∗

Panel-t(sc) -15.43* -14.811∗

Group-t(sd) -11.70* -11.141∗

Panel-t(sd) -11.18* -10.731∗

Notes:

1indicates results for the panel cointegration tests without Japan.

* indicates rejection of the null of no cointegration.

27

Table VIII: Empirical results of Pedroni’s panel cointegration tests with an intercept in the regressionequation. Monthly data from 1986:06 through 1998:12 is used. All tests are performed at 5% level.The panel rank test has a critical value of 1.645.

Country Intercept Slope ρ-stat t-stat(sc) t-stat(sd) lag(sc) lag(sd)

Germany 4.86 202.11 -0.08 -2.69 -2.44 3 2

France 5.29 362.34 -0.08 -3.40 -3.40 5 5

Italy 5.97 381.25 -0.15 -2.71 -2.71 1 1

Netherlands 5.76 49.36 -0.09 -0.94 -0.69 6 1

Spain 6.34 349.90 -0.15 -5.37 -3.24 9 0

Finland 5.72 419.24 -0.14 -3.56 -2.82 6 0

Austria 5.68 91.19 -0.03 -3.09 -2.17 12 3

Ireland 8.46 -21.18 -0.32 -5.37 -5.53 1 3

Portugal 6.97 298.00 -0.13 -3.62 -2.75 9 1

Belgium 5.43 205.31 -0.05 -2.39 -2.12 3 2

US 4.30 163.63 -0.06 -2.76 -2.52 3 2

Japan 2.91 105.10 -0.02 -3.09 -1.85 11 2

UK 6.97 172.93 -0.09 -5.68 -2.86 10 1

Denmark 6.83 117.12 -0.03 -1.86 -1.55 5 0

Mexico 12.01 310.17 -0.28 -5.28 -5.15 2 1

Norway 6.83 140.90 -0.06 -1.84 -1.84 0 0

Iceland 5.81 419.83 -0.31 -5.20 -4.90 3 1

Sweden 7.45 151.48 -0.13 -2.97 -2.98 0 0

Canada 5.32 279.66 -0.13 -3.44 -3.02 3 0

Group Tests Panel Tests

Group-ρ -3.23* -3.551∗

-3.622∗

Panel-ρ -1.50 -1.991∗

-1.862∗

Group-t(sc) -7.53* -7.431∗

-6.562∗

Panel-t(sc) -6.10* -6.021∗

-4.982∗

Group-t(sd) -4.50* -4.591∗

-4.472∗

Panel-t(sd) -5.00* -4.981∗

-4.172∗

Notes:

1indicates results for the panel cointegration tests without Austria

2indicates results without Austria and UK.

* indicates rejection of the null of no cointegration.

28

Table IX: Empirical results of Pedroni’s panel cointegration tests with an intercept in the regressionequation. Monthly data from 1991:01 through 2002:12 is used. All tests are performed at 5% level.The panel rank test has a critical value of 1.645.

Country Intercept Slope ρ-stat t-stat(sc) t-stat(sd) lag(sc) lag(sd)

US 4.29 71.99 -0.03 -1.11 -1.11 1 1

Korea 9.53 189.63 -0.08 -2.65 -2.43 3 2

Japan 1.64 94.53 -0.05 -5.59 -4.31 11 3

UK 6.28 59.24 -0.08 -6.10 -4.10 12 1

Denmark 5.75 38.84 -0.03 -1.75 -1.75 0 0

Mexico 9.00 332.18 -0.28 -4.77 -4.77 1 1

Norway 6.84 10.97 -0.06 -2.33 -2.33 0 0

Iceland 7.15 195.55 -0.11 -3.58 -3.01 8 1

Hungary 12.97 178.22 -0.13 -4.62 -3.08 12 4

Sweden 6.69 -1.26 -0.05 -2.14 -2.14 0 0

Canada 0.0005 0.0009 -0.33 -7.41 -7.41 5 5

Group Tests Panel Tests

Group-ρ -3.84* -4.111∗

Panel-ρ -2.22* -2.821∗

Group-t(sc) -7.32* -6.291∗

Panel-t(sc) -2.98* -1.171

Group-t(sd) -5.24* -4.611∗

Panel-t(sd) -2.29* -0.651

Notes:1indicates results for the panel cointegration tests without Japan.

* indicates rejection of the null of no cointegration.

29

9A

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30

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10

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41

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43