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Testing for Panel Cointegration with Multiple

Structural Breaks*

Joakim Westerlund

Department of Economics, Lund University, Lund, Sweden(e-mail: [email protected])

Abstract

This paper proposes a Lagrange multiplier (LM) test for the null hypothesis of

cointegration that allows for the possibility of multiple structural breaks in

both the level and trend of a cointegrated panel regression. The test is general

enough to allow for endogenous regressors, serial correlation and an unknown

number of breaks that may be located at different dates for differentindividuals. We derive the limiting distribution of the test and conduct a

small Monte Carlo study to investigate its finite sample properties. In our

empirical application to the solvency of the current account, we find evidence

of cointegration between saving and investment once a level break is

accommodated.

I. Introduction

There is a vast literature concerned with extending time-series cointegrationor, rather, non-cointegration tests, to panel data with both a time-series

dimension T and a cross-sectional dimension N. However, there is only a

handful of studies that have attempted to develop panel data tests under the

maintained hypothesis of cointegration. The most noticeable study in this field

is that of McCoskey and Kao (1998). In their study, the authors extend to

*The author thanks Anindya Banerjee and two anonymous referees for many valuable commentsand suggestions. The author would also like to thank David Edgerton, Johan Lyhagen, MichaelBergman and seminar participants at Lund University. Financial support from the Jan Wallander and

Tom Hedelius Foundation, research grant number P2005-0117:1, is gratefully acknowledged. Theusual disclaimer applies.

JEL Classification numbers: C12, C32, C33, F21.

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 68, 1 (2006) 0305-9049

101 Blackwell Publishing Ltd, 2006. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK

and 350 Main Street, Malden, MA 02148, USA.


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panel data the univariate LM test for cointegration proposed by Harris and

Inder (1994) and Shin (1994), which is the locally best unbiased invariant test

of the error covariance matrix of a linear regression model. McCoskey andKao (1998) show that the proposed test has a limiting normal distribution

under the null hypothesis of cointegration as T tends to infinity and then N

sequentially. Because of its good finite sample properties and the attractive-

ness of the null hypothesis of cointegration rather than the opposite, the test

has become very popular in empirical research (e.g. see, McCoskey and Kao,

2001).

Having been derived using sequential limit arguments wherebyTis passed

to infinity prior to N, the LM test may be motivated in research situations

involvingTwhich is substantially larger thanN. However, researchers need to

be aware that the probability of a structural break in the cointegration relation

increases with the length of the time-series dimension of the panel. As shown

by Hao (1996), this alters the limiting distribution of the test as the

deterministic component of the cointegration regression needs to be modified

to collect for the presence of the structural break. Erroneous omission of

structural breaks is therefore likely to lead to size distortions and deceptive

inference when testing for cointegration. One study that explicitly addresses

this concern when testing the hypothesis of a unit root in panel data is that of

Im, Lee and Tieslau (2005), which generalizes the univariate LM unit root test

of Amsler and Lee (1995). However, to our knowledge there are no such studythat deals with the hypothesis of cointegration in the presence of structural

change in panel data.

In this paper, we propose a simple test for the null hypothesis of

cointegration that accommodate for structural change in the deterministic

component of a cointegrated panel regression. The test is based on the LM

test of McCoskey and Kao (1998) and it is able to accommodate for an

unknown number of breaks in both the constant and trend of the individual

regressions, which may be located at different dates for different individuals.

Test statistics are derived when the locations of the breaks are knowna priori and when they are determined endogenously from the data. Test

statistics are also derived when there is no break but the deterministic

component includes individual-specific constant and trend terms.

Using sequential limit arguments, it is shown that the test has a limiting

normal distribution, i.e. free of nuisance parameters under the null hypothesis.

In particular, it is shown that the limiting distribution is invariant with respect

to both the number and the locations of the breaks and that there is no need to

compute different critical values for all possible patterns of break points as in,

e.g. Bartley, Lee and Strazicich (2001). This invariance property makes the

test computationally very convenient. We also evaluate the small-sample

performance of the test via Monte Carlo simulations. The results suggest that

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the test has small size distortions and reasonable power. In our empirical

application, we reexamine the data set employed by Ho (2002) and Taylor

(2002) to assess the solvency of the current account. Contrary to much of thefindings presented in the earlier literature, we find evidence suggesting that

saving and investment are cointegrated once a level break in the cointegration

vector is accommodated.

The paper proceeds as follows. In section II, we present the LM test

statistic under the assumption that the locations of the structural breaks are

known. Sections III and IV describes the asymptotic distribution of the test,

while section V relaxes the assumption of known breaks. Section VI is then

devoted to the Monte Carlo study, whereas section VII relates itself with the

empirical application. Section VIII concludes the paper. For notational

convenience, the Bownian motion Bi(r) defined on the unit interval

r2 [0, 1] will be written as only Bi and integrals such asR1

0Wirdr will

be written asR1

0Wi,

Rr0

Wisds asRr

0Wi and

R10

WirdWir asR1

0 WidWi. The

symbol is used to signify weak convergence of the associated probability

measure, [z] is used to denote the largest integer less than zandkzk denotesthe Euclidian norm tr(z0z)1/2 ofz.

II. The panel LM test with breaksConsider the multidimensional time-series variableyit, which is observable for

i 1, . . . ,N cross-sectional and t 1, . . . , T time-series observations. Thedata generating process (DGP) for yit is given by the following system of

equations

yitz0itcij x

0itbi eit; 1

eitrit uit; 2

ritrit1 /iuit; 3

where xit xit)1 + vit is a K-dimensional vector of regressors and zit is avector of deterministic components. The corresponding vectors of param-

eters are denoted bi and cij respectively. The index j 1, . . . , Mi + 1 isused to denote the structural breaks. There can be at most Mi such breaks,

or Mi + 1 regimes, that are located at the dates Ti1, . . . , TiMi, where Ti0 1and TiMi+1 T. Furthermore, the initial value of rit is assumed to be zero,which entails no loss of generality as long as zit includes an individual-

specific intercept. For convenience in constructing the test and in deriving

its asymptotic distribution, we assume that the vector wit uit; v0it0 iscross-sectionally independent and that it follows a general linear process

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whose parameters satisfy the summability conditions of the following

assumption.

Assumption 1(Error process). (i) The vectorswijandwktare independent forallj,tandi 6 k; (ii) The vectorwitsatisfies wit Ci(L)it, where L is the lagoperator, CiL

P1j0CijL

j;P1

j0 kCijC0ijk< 1 and it is distributed

independent and identically distributed (i.i.d.) (0, IK+1); (iii) the lower right

K Ksubmatrix ofXi Ci(1)Ci(1)0 is positive definite.

In addition to this, we make the following assumptions regarding the

structural break model.

Assumption 2 (Break model). (i) The locations of the structural breaks are

given as a fixed fraction kij2 (0, 1) ofTsuch thatTij [kijT] and kij)1 < kijfor j 1, . . . , Mi; (ii) Both Mi and kij are assumed to be known.

Assumption 1 (i) states that the members of the panel are independent

of each other. This assumption is convenient as it will allow us to

apply standard central limit theory in a relatively straightforward manner.

For many applications, such as the one considered for this paper, however,

the independence assumption may be quite restrictive and section VII

therefore suggests some possibilities for dealing with the case of cross-

sectionally correlated data. However, for the present we maintain Assump-

tion 1 (i).As for the time-series dimension, Assumption 1 (ii) states that the standard

invariance principle is assumed to hold for each individuali as Tgrows large,

which ensures that the partial sum process constructed from witconverges to

the vector Brownian motion Bi. Thus, if SiT PTr

t1wit, then T)1/2SiT

Bi Ci(1)Wi as T! 1 for a fixed N, where Bi (Bi1, Bi20)0 is a vector

Brownian motion andWi (Wi1, Wi20)0 is a vector standard Brownian motion

with covariance matrix equal to identity. The covariance matrix of Bi is

defined as

Xi limT!1

T1E SiTS0iT

x2i11 x0i21

xi21 Xi22

:

This matrix can be regarded as the long-run covariance matrix of wit in the

sense that it captures both the contemporaneous variances and covariances

as well as the covariances at all lags and leads. For later discussion, we

also define the long-run variance of uit conditional on vit as x2i1:2

x2i11 x0i21X

1i22xi21. Assumption 1 (iii) requires Xi22being a positive definite

matrix, which precludes the possibility of cointegration within xitin the event

that we have multiple regressors. The off-diagonal elements ofXicaptures the

dependence between the first differentiated regressors and the equilibrium

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non-empty. Thus, we require that N1/N! d as N! 1, where d 2 (0, 1].The panel LM test statistic for this hypothesis is given as follows.

Definition 1 (The panel LM test statistic). Letx2i1:2 x2i11 x0i21X

1i22xi21

and SitPt

kTij11eik; where e

it is any efficient estimate of eit. The panel

LM test statistic is defined as follows

ZM XNi1

XMi1j1

XTijtTij11

Tij Tij12x2i1:2S

2it: 4

Remark 1. The statistic is written as an explicit function of M

(M1,. . .

, MN)0

to denote that it has been constructed for a certain number ofbreaks for each cross-section and that its asymptotic distribution depend on it.

Remark 2. To compute the statistic, we need to obtain an efficient estimator

eitof the error term eit. Under Assumption 1 this can be achieved using either

the dynamic ordinary least square (DOLS) estimator of Saikkonen (1991) or

the fully modified OLS (FMOLS) estimator of Phillips and Hansen (1990).

The estimation is carried out separately for each subsample ranging from Tij)1toTijtime-series observations. For Case 1, the regression is fitted without any

deterministic components. For Cases 2 and 4, the regression is fitted with an

individual-specific intercept as the deterministic component, while, for Cases3 and 5, it is fitted with both individual-specific intercepts and trends. In

addition, for Cases 13, we have Mi 0 for all i suggesting that theestimation is carried out using the entire sample.

Remark 3. To be able to construct x2i1:2, we need to obtain a consistent

estimatorXi ofXi. To this end, because we are allowing for general forms of

serial dependence within the individual regressions over time, consistency

may be achieved using any semiparametrical kernel estimator of the following

form

XiT1

Xkjk

1 j

k 1

XTtj1

witw0itj:

As OLS is consistent even with endogenous regressors, Xi may be

constructed using wit eit; v0it

0, where eitdenotes the OLS estimate of eit.

The weight function 1 ) j/(1 + k) is the Bartlett kernel, which ensures the

positive semidefiniteness ofXi. For consistency of the test, it is necessary

that the bandwidth parameter k satisfies the property that k! 1 and k/T! 0 as T! 1. The choice k [T1/3] is sufficient. As the long-runconditional variance of eit is assumed to be constant, the estimation of x2i1:2may be carried out using the full length of the time-series dimension with

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dummy variables to account for the breaks. However, in practice, it may be

more convenient to estimate not only the parameters but also the variances

separately for each segment, which does not alter the limit distribution of thetest. Besides being easy to implement, this approach has the additional

advantage that it makes the test robust to shifts in the variance if we assume

that they are located at the same dates as the shifts in the deterministic

component.

III. Asymptotic distribution

In this section, we derive the asymptotic distribution of the test statistic under

the null hypothesis. To this effect, we positQ and R, respectively, to be the

expected value and the variance of the following vector standard Brownian

motion functional

Fi

Z 10

V2i where Vi Wi1:2

Z r0

Wi2

Z 10

Wi2 W0i2

1Z 10

Wi2dWi1:2:

The process Wi1:2 x1i1:2Bi1:2 with Bi1:2 Bi1 x

0i21X

1i22Bi2 is a scalar

standard Brownian motion, that is independent of Wi2. The vector standard

Brownian motion Wi2 may be written as Wi2 z0; W0i2

0, where z is the

appropriately scaled limit of the deterministic componentzit. Specifically,z{[} in Case 1,z 1 in Cases 2 and 4, and z (1, r)0 in Cases 3 and 5. Thefunctional Fi is referred to as a generalized Cramer-von Mises distribution

with 1 d.f. This functional has the additive property that the sum of Mi + 1

independent generalized Cramer-von Mises distributions with 1 d.f. each is

Cramer-von Mises with Mi + 1 d.f. The expected value of such a sum is equal

to (Mi + 1)Q and the variance is equal to (Mi + 1)2R. Let us define H and R as

the cross-sectional averages of these expectations and variances for each

individual. As the following theorem indicates, when the test statistic is

normalized by the appropriate order ofN, then the asymptotic distribution will

depend only on the known values of H and R.

Theorem 1(Asymptotic distribution). Under Assumptions 1 and 2, and the

null hypothesis of cointegration, as T! 1and then N! 1

N1=2ZM N1=2H )N0; R: 5

Remark 4. The proof of Theorem 1 is provided in the Appendix. It uses the

sequential limit theory developed by Phillips and Moon (1999), which means

that in deriving the asymptotic distribution of the test, we first take the limit as

T! 1 followed sequentially by the limit as N! 1. The sequential limittheory greatly simplifies the derivation of the limit distribution of the test but it




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is not the most general method. In fact, by using the method of joint limits, it

is possible allow for both Tand Nto approach infinity concurrently rather in

sequence. As pointed out by Phillips and Moon (1999), this implies that thejoint limit distribution characterizes the limit distribution for any monotonic

expansion rate ofTrelative to N. The authors also provide a set of conditions

that are required for sequential convergence to imply joint convergence.

Remark 5. In our case, the sequential limit method substantially simplifies the

derivation of the limit distribution for at least two reasons. One reason is that it

allows us to control the effects of the nuisance parameters associated with the

locations of the structural breaks and the serial correlation properties of the

data in the first step as T! 1by virtue of the standard invariance principle.

To get an intuition on this, note that each of the subsample quantitiesPTijtTij11

Tij Tij12x2i1:2S

2it comprised of Z(M) can be treated as a test

statistic that is not subject to structural change. As shown in the Appendix,

each of these subsample statistics converges to a generalized Cramer-von

Mises distribution with 1 d.f. as T! 1. Hence, by the additive property ofFi, the intermediate limit distribution of Z(M) for each individual can be

described entirely in terms of a Cramer-von Mises distribution with Mi + 1

d.f. This means that the limit distribution is invariant not only with respect to

the serial correlation properties of the data but also with respect to the

locations of the structural breaks and that it only depends onMi, the number of

breaks for each cross-section, which is assumed to be known. Another reason

why sequential limits simplify the derivation is that, as the members of the

panel are assumed to be independent of each other, the statistic may be treated

as a sum ofNindependent drawings from a set of underlying distributions to

which standard LindbergFeller central limit arguments may be applied to

obtain a limiting normal distribution.

Remark 6. To be able to compute the standardized statistic in equation (5),

we need to obtain numerical values of the moments Q and R. One standard

way in which moments have been obtained in situations where no closed formexpressions exist is to use Monte Carlo simulation of the limiting distribution

of the test. Table 1 presents the simulated values of Q and R using this

method. To obtain the moments, we make 10,000 draws ofK+ 1 independent

random walks of lengthT 1,000. By using these random walks as simulatedBrownian motions, we can construct approximations of the functional Fi,

which are then used to compute the moments. The moments apply in general

for any deterministic specification and for any number of regressors when the

slope parameters are estimated separately for each member of the panel.

However, the specific values ofQ and R depend on the particular model, such

as whether individual-specific intercepts or trends have been included in the

estimation, and onK, the number of regressors. Accordingly, Table 1 gives the

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moments for each of the five deterministic specifications and for up to five

regressors. To obtain H based on the simulated expectations in Table 1,

simply multiply the appropriate expected value byMi + 1 for each individual

and then take the cross-sectional average. Similarly, to obtain R, multiply the

appropriate variance by the square ofMi + 1 and then take the cross-sectional

average. Once the standardized test statistic in equation (5) has been

computed, it should be compared with the right tail of the normal distribution.Thus, large positive values indicate that the null hypothesis should be rejected.

Remark 7. The fact that the moments of Table 1 does not depend on the

locations of the structural breaks is a great advantage. For example, the

univariate cointegration test proposed by Bartley et al. (2001) allows for a single

break to affect the deterministic component of the cointegration vector.

However, this allowance makes the test asymptotically dependent on the

location of the break. To accommodate for this, the authors simulate critical

values for a few equally spaced break points, which can be used to perform the

test. There are at least two problems with this approach. The first problem is thatassuming equally spaced breaks is not appropriate in general in which case

critical values for a possible continuum of breakpoints may be required. The

second problem is that, even with discrete and equally spaced breaks, one faces

the complication factor of having to consult a table for the critical values for

every break location, which may be a very tedious undertaking in itself. With

multiple breaks, this approach becomes practically infeasible as constructing

tables for every possible pattern ofMibreaks would be extremely cumbersome.

Remark 8. Note that the breaks are allowed under both the null and alternative

hypotheses. This is very convenient as it facilitates a straightforward

interpretation of the test result. To illustrate this, let us consider the univariate

TABLE 1

Simulated asymptotic moments

Moment Case K 1 K 2 K 3 K 4 K 5

Q 1 0.35500 0.26963 0.20584 0.16688 0.13770

2/4 0.11601 0.08464 0.06539 0.05295 0.04442

3/5 0.05530 0.04686 0.04063 0.03532 0.03133

R 1 0.18017 0.11260 0.06310 0.04072 0.02670

2/4 0.01151 0.00559 0.00266 0.00144 0.00086

3/5 0.00115 0.00078 0.00056 0.00036 0.00027

Notes: Case 1 refers to the regression without any deterministic intercepts or trends, Case 2 refersto the regression with an individual-specific intercept, Case 3 refers to the regression with individual-specific intercepts and trends, Case 4 refers to the regression with breaks in the intercept and Case 5

refers to the regression with breaks in both intercept and trend. The value Krefers to the number ofregressors excluding any fitted deterministic intercepts or trends.




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unit root test of Zivot and Andrews (1992), which allows for a single unknown

break to affect the level and trend of the series. The problem with this test is that

the break is only permitted under the alternative hypothesis of stationarity. Thus,a rejection of the null does not necessarily imply a rejection of a unit rootper se

but rather a rejection of a unit root without breaks. This outcome calls for a

careful interpretation of the test result in applied work. Particularly, in the

presence of breaks under the null, researchers might incorrectly conclude that a

rejection of the null indicates evidence of stationarity with a break, when in fact

the series non-stationary with breaks.

IV. Response surface regressionsPreliminary results suggest that the simulated asymptotic moments may not be

borne out fully in realistically small samples and that moments for small

values ofTmay be required in order to allow for accurate testing. To this end,

in order to present the results more succinctly, we use response surface

regressions. We experimented with a variety of specifications and we opted for

the following linear regression model

yi d1 d2T1=2

i d3T1

i d4T2

i ei; 6

where yi is the simulated moment based on a sample of Ti time-seriesobservations and ei is a heteroskedastic error term that reflects the

simulation uncertainty and the approximation of the true regression form

by using the quadratic one in equation (6). The intercept is an estimate of

the asymptotic moment of the test statistic and the remaining para-

meters determine the shape of the response surface for finite values of T.

Hence, by including T1=2

i , T1

i and T2

i as explanatory variables, we are

in effect allowing the small-sample moments of the test statistic to differ

from the asymptotic ones. The choice of regressors to include was dictated

by the size of the difference between the estimated intercept and theappropriate asymptotic moment, by the overall fit of the regressions and by

the significance of the parameters themselves. Adding powers larger than

two of Ti never seemed to be necessary so the same specification applies to

all moments.

The estimation proceeds as follows. For each combination ofKand T, we

generate 1,000 test statistics under the null hypothesis with all regression

parameters set equal to one. We use both FMOLS and DOLS estimation. The

FMOLS is performed using the Bartlett kernel with the bandwidth parameter

set equal to [T1/3] and the DOLS is performed using [0.5T1/3] lags and

leads of the first differences of the regressors. We generate moments

for T2 {20, 50, 60, 70, 100, 200, 500, 1,000} time-series observations and

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up to five regressors. Most samples are relatively small as these seem to

provide more information about the shape of the response surfaces. However,

a few large values of T are also included to ensure that the estimates ofthe asymptotic moments are sufficiently accurate. Moments for all three

deterministic cases are extracted and stored. We then perform 100 replications

of each experiment, which means that there is a total of 800 observations

available for each of the response surface regressions.

As the regression error is heteroskedastic by construction, we follow

MacKinnon (1996) and apply the generalized method of moments estimator

proposed by Cragg (1983). To this effect, letZdenote the matrix of right-hand

side variables and let Y denote the vector of simulated moments for each

experiment. Also, suppose that e denotes the residual obtained from the OLS

fit ofYonZ, then Vis the diagonal matrix with the principal diagonal equal to

that of ee0. The Cragg (1983) estimator may now be written as

d Z0WW0 V W1W0 Z1Z0WW0 V W1W0 Y;

where d is the vector of estimated response surface parameters and W is a

matrix comprised of one dummy variable for each value ofT. The results are

presented in Table 2 for the FMOLS-based test and in Table 3 for the DOLS-

based test. These permit a quick computation of the approximate moments for

a particular value of T. As a measure of regression uncertainty, the tablereports the estimated standard error for each regression. These reflect both the

simulation uncertainty from estimating the moment rather than knowing it and

the approximation error of using a quadratic regression form rather than the

true one. The standard error is always smaller than 0.03, assuring reasonably

high precision in the estimates and low regression uncertainty. Another

measure of the quality of the estimated response surfaces is the size of the

differences between the estimated intercepts and the moments reported in

Table 1 based on simulating the asymptotic distribution of the test. It was

observed that the intercepts appear to be remarkably precise estimates of theasymptotic moments.

V. Unknown break points

In the previous sections, we have assumed that both the number and the

locations of the structural breaks are known. This is not necessary. In fact, if

there is no a priori knowledge about the breaks, one may prefer to treat them

as endogenous variables that need to be estimated from the data. For this

purpose, we suggest using the proposal of Bai and Perron (1998, 2003), which

obtains the location of the breaks by globally minimizing the sum of squared

residuals as follows




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TABLE 2

FMOLS-based response surface moments

Case K

Expected value Variance

Intercept T)1/2 T)1 T)2 SE Intercept T )1/2 T)

1 1 0.35264 0.05048 )0.61249 12.82076 0.00407 0.18995 )0.42167 0

1 2 0.26144 0.03296 )0.91888 17.62065 0.00383 0.09720 0.11888 )2

1 3 0.20695 )0.12335 )0.20455 15.76128 0.00576 0.06441 )0.12325 )0

1 4 0.16916 )0.26228 0.82432 8.03949 0.00679 0.04111 )0.13058 0

1 5 0.13943 )0.24630 0.75407 11.69389 0.00890 0.02693 )0.10471 0

2 1 0.11708 )

0.05532 0.31233 3.14865 0.00421 0.01130 )

0.02838 )

02 2 0.08557 )0.06494 0.24917 6.69821 0.00571 0.00581 )0.02252 0

2 3 0.06730 )0.11746 0.60610 6.52534 0.00806 0.00309 )0.02008 0

2 4 0.05391 )0.09801 0.49008 11.09135 0.01095 0.00161 )0.00995 0

2 5 0.04451 )0.09041 0.53038 13.72616 0.01394 0.00095 )0.00646 0

3 1 0.05515 0.01680 0.21483 4.98491 0.00794 0.00125 )0.00527 0

3 2 0.04749 )0.02300 0.39897 6.46768 0.00968 0.00085 )0.00385 0

3 3 0.04103 )0.03676 0.45948 9.02230 0.01186 0.00057 )0.00232 )0

3 4 0.03558 )0.04270 0.53224 11.23719 0.01436 0.00036 )0.00087 )0

3 5 0.03073 )0.02145 0.41501 15.37253 0.01708 0.00025 )0.00023 )0

Notes: See Table 1 for an explanation of the different cases. The table gives simulated finite and asymptotic mome

appropriate moment is obtained from the table by calculating the fitted value of the corresponding regression.

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TABLE 3DOLS-based response surface moments

Case K

Expected value Variance

Intercept T)1/2 T)1 T)2 SE Intercept T )1/2 T)

1 1 0.36132 )0.22961 1.57112 )16.31658 0.00211 0.19043 )0.44026 0.

1 2 0.27401 )0.31577 1.99793 )15.55645 0.00149 0.11030 )0.24182 0.

1 3 0.22226 )0.63793 4.13834 )30.95020 0.00373 0.07280 )0.41392 2.

1 4 0.18901 )0.82065 5.10891 )33.87714 0.00639 0.05053 )0.39326 2.

1 5 0.14815 )0.40345 1.59114 29.82585 0.00997 0.03162 )0.22870 1.

2 1 0.11625 0.00344 0.25486 10.42508 0.00626 0.01159 )0.03922 0.

2 2 0.08775 )0.06769 0.69281 9.15342 0.00706 0.00642 )0.03858 0.

2 3 0.06583 0.02019 )0.05966 22.41660 0.00804 0.00301 )0.01180 0.

2 4 0.05238 0.03925 )0.27469 30.42694 0.00951 0.00177 )0.01178 0.

2 5 0.04255 0.08621 )0.72910 41.76848 0.01105 0.00109 )0.00809 0.

3 1 0.05347 0.10835 )0.31741 19.11540 0.00821 0.00124 )0.00398 0.

3 2 0.04542 0.10271 )0.39739 23.86645 0.00878 0.00091 )0.00446 0.

3 3 0.03855 0.11087 )0.54675 30.28444 0.00987 0.00062 )0.00303 0.

3 4 0.03248 0.14376 )0.88995 39.98488 0.01128 0.00044 )0.00217 0.

3 5 0.02710 0.19392 )1.38020 52.16569 0.01289 0.00027 )0.00027 )0.

Notes: See Table 1 for an explanation of the different cases. The table gives simulated finite and asymptotic mom

appropriate moment is obtained from the table by calculating the fitted value of the corresponding regression.

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Tiarg minTi X

Mi1

j1 XTij

tTij1

1

yit z0itcij x

0itbi

2; 7

where Ti Ti1; . . . ; TiMi 0

is the vector of estimated break points, cij andbiare

the estimates of the cointegration parameters based on the partition Ti (Ti1, . . . , TiMi)

0 and s is a trimming parameter such thatkij) kij)1 > s, which

imposes a minimum length for each subsample. Because the minimization is

taken over all possible partitions of permissable length, the break-point

estimators are said to be global minimizers. The estimation is performed in

two separate steps. In the first step, the global minimizers of the sum of

squared residuals are estimated and stored together with the associated optimal

break partitions for each possible number of breaksMi 1, . . . , J, whereJissome predetermined upper boundary. In the second step, the number of breaks

are estimated using an information criterion.

The purpose of the first step is to estimate the unknown regression parameters

together with the unknown break points when Tobservations are available for

each individual. This can be achieved using the dynamic programming

algorithm developed by Bai and Perron (2003). However, in our case, as the

slope parameters bi are not subject to shift and are estimated using the full length

of the time-series dimension, the algorithm cannot be applied directly. This is so

because the estimate ofbiassociated with the minimum of the sum of squared

residuals depends on the optimal break partition Ti that we are trying to estimate,which means that we cannot concentrate outbi from the objective function.

Therefore, the minimization of the sum of squared residuals cannot be performed

with respect toTidirectly but must be carried out iteratively.

The iterative procedure suggested by Bai and Perron (2003) proceeds in the

following fashion. Given a starting value for bi, initiate the procedure by

minimizing the objective function with respect to cijand Ti while keeping bifixed. This requires an evaluation of the optimal break partition for all

permissable subsamples that admits the possibility ofMibreaks. Becausebiis

held fixed, this stage amounts to minimize the objective function of a purestructural change model to which the dynamic programming algorithm apply.

The next stage is to minimize with respect to cijand bi simultaneously while

keeping Ti fixed. Then, iterate until the marginal decrease in the objective

function converges or until the number of iterations reaches some predetermined

upper boundary. Each iteration assures a decrease in the objective function.

The first step yields estimated break partitions and sum of squared

residuals for each number of breaks that lies in the interval [1, J]. The second

step uses these sum of squared residuals to estimate the number of

breaks to include in the cointegrated regression. To this end, we follow the

recommendation of Bai and Perron (2003) and use the Schwarz Bayesian

information criterion. Also, as the case with no breaks is permissable, the sum

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of squared residuals obtained from the first step are compared with those

obtained for the corresponding model configuration without any break. The

estimated break vector ^

Ti is then obtained as the partition associated with theparticular number of breaks that minimizes the criterion.

The two steps described above are repeatedNtimes, which produces a vector

of estimated break points for each individual in the sample. The LM statistic can

then be constructed using Tiin place ofTifor each individual. As shown by Bai

and Perron (1998), the above procedure yields consistent estimators of both kijandMifor each individual as Tgrows large. This is very convenient because it

implies that the sequential limit distribution of the statistic derived in section III

will be unaltered even if we treat the locations of the breaks as unknown.

VI. Monte Carlo simulations

To examine the small-sample properties of the test, we conduct Monte Carlo

experiments similar to the design of McCoskey and Kao (1998). The DGP can

be summarized using the following system of equations

yitz0itcij xitbi rit uit;

xitxit1 vit;

ritrit1 /uit:

The error vector wit (uit, vit)0 is generated as an MA(1) process satisfyingwit eit+ heit)1, whereh is the moving average parameter, eit N(0, V) andVis a positive definite matrix with V11 V22 1.

1 For the initiation ofxit,ritandeit, we use the value zero. The data were generated for 1,000 panels with

N cross-sectional and T + 50 time-series observations. The first 50 observa-

tions for each cross-section is then disregarded in order to reduce the effect of

the initial condition.

The parametrization of the DGP is as follows. For simplicity, make the

assumption that there is a single break such thatki1 k1 for all i and k1

(0.3, 0.5, 0.7).2

We also assume that there is a single regressor with slopebi N(1, 1). For the deterministic component, we have five differentconfigurations, each of which correspond to one of our five model specifica-

tions. For Cases 13, we have M 0 so cij ci1 is constant for all i.Specifically, ci1 0 in Case 1, ci1 N(1, 1) in Case 2 and ci1 N(1, I2) inCase 3. For Case 4, we have cij N(lj, 1), whilecij N(lj, I2) for Case 5. In

both cases, the break parameters are generated withl11 andd l2) l1,where the parameterd (1, 2, 3) is used to control the size of the break.

1For some simulation results for the panel LM test with autoregressive errors, see Westerlund

(2005).2Some simulations results for the case with two structural breaks are available from the authorupon request.




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The parameter/ determines whether the null hypothesis is true or not. For

brevity, we make the assumption that this parameter takes on a common value

for all individuals. Therefore, under the null hypothesis, we have / 0,while/ (0.05, 0.1) under the alternative hypothesis. The parameters h andV introduces nuisance in the DGP. The effect of the MA(1) component is

governed by h ()0.5, 0, 0.5), while the degree of endogeneity is capturedby the off-diagonal elementsV12and V21ofV. For brevity, we only present the

results for the case when V12 V21 0.4.Simulations have been carried out using both DOLS and FMOLS estimation.

However, the results were very similar and we therefore only present the results

for the FMOLS estimator, which employs a semiparametric correction to

account for the presence of endogenous regressors. To this end, we use the

Bartlett kernel with the bandwidth parameter chosen as a fixed function ofTsuch

that [T1/3]. In estimating the unknown break points, we follow the recommen-

dation of Bai and Perron (2003) and use the trimming parameters 0.15. Theconvergence criterion in the iterative procedure is set to 0.0001 and the

maximum number of iterations allowed is 50. The maximum number of breaks

considered isJ 5. The test results are based on the moments obtained usingthe estimated response surfaces in Table 2. For brevity, we only report the size

and size-adjusted power of a nominal 5% level test.

We begin by considering the performance of the test for the case with no

structural breaks. The results are presented in Table 4. Turning first to theresults on the empirical size we see that the test generally maintains the

nominal level well when h 0 but it is distorted when h 6 0. For positivevalues of h, the test has a size above the nominal level. Conversely, for

negative values ofh, the test has a size below the nominal level. The larger the

absolute value of h, the larger the distortion. Yet, negative values of h in

general led to more severe distortions than positive. In addition, while

decreasing in T, we see that the distortions have a tendency of accumulating

and to become more serious as Nincreases.

The results on the size-adjusted power suggest that the test performs wellwith rejection frequencies that are close to one in most experiments. The test

of Case 1 has the highest power and the test of Case 3 has the lowest power.

Thus, the introduction of deterministic intercept and trend terms that need to

be estimated affects the test by reducing its power. Moreover, the power

increases as both N and T grows, which is presumably a reflection of

consistency of the test. As expected, the rate at which this happens depend

strongly on the influence of the unit root component in the errors as indicated

by the value taken by /.

For Cases 4 and 5 with known breaks, the simulations were carried out

using d 1 for the size of the shifts. The results are reported in Table 5. Inagreement with the results for the models with no break, we see that the test is

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TABLE 4

Size and size-adjusted power for Cases 13

Case /

N 10, T 100 N 20, T 100 N 10, T 200

h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.5

1 0.00 0.078 0.108 0.002 0.050 0.124 0.000 0.070 0.094 0.000

1 0.05 0.836 0.870 0.910 0.988 0.982 0.992 0.994 0.992 0.994

1 0.10 0.980 0.978 0.998 1.000 1.000 1.000 1.000 1.000 1.000

2 0.00 0.074 0.120 0.002 0.066 0.118 0.000 0.076 0.108 0.002

2 0.05 0.598 0.602 0.556 0.804 0.786 0.814 0.962 0.978 0.982

2 0.10 0.952 0.946 0.946 0.998 0.994 1.000 1.000 1.000 1.000

3 0.00 0.058 0.148 0.000 0.066 0.138 0.000 0.080 0.146 0.000

3 0.05 0.244 0.218 0.258 0.334 0.388 0.414 0.782 0.814 0.784

3 0.10 0.716 0.694 0.768 0.920 0.934 0.950 0.996 0.998 0.996

Notes: The value/ refers to the weight being attached to the unit root component in the regression errors and h refers toTable 1 for an explanation of the different cases.



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correctly sized when h 0 but there is a problem with size distortions whenh6 0. As expected from the asymptotic theory, we see that the performance

under the null is unaffected by the location of the structural break. One notableexception is for Case 5 where the results suggest that the test can be severely

distorted unless the time-series dimension of the individual subsamples before

and after each break is sufficiently large. In fact, based on the results obtained

from the simulations, we recommend not using the test in this case unless the

size of each subsample is T 100 or greater.As for the power of the test, the results indicate that the performance is good

in general. Notably, the loss of power caused by estimating the parameters of the

model for each subsample separately rather than over the entire length of the

time-series dimension as in Cases 2 and 3 need not be large and is effectively

non-existing in some experiments. These results appear to be rather robust and

applies regardless of the location of the structural break. As in the no break case,

we also observe large gains in power as N,Tand/ increases.

The results on the size and power for the DGP with unknown breaks are

presented in Table 6. In this case, the data were generated with k1 0.5 fixedand we instead vary the magnitude of the breaks. The results under the null

hypothesis indicate that the test maintain the nominal size well and that there are

no large differences in the performance depending on whether the breaks are

treated as known or not. As for the performance under the alternative

hypothesis, we find that the power of the test decreases considerably in someexperiments when compared with the case with known breaks, which accords

with the results reported by Kurozumi (2002) in the context of stationarity

testing in time-series data. One reason for this might be that the estimated

number of breaks tend to lie above its true value, which means that the

alternative model becomes close to the null hypothesis, as is the case when more

breaks are being estimated, thus causing a loss of power. However, given the

smallness of the weight attached to the random walk component in the errors,

the power still appears to be reasonable in most experiments. In addition, it was

observed that the power increases as the size of the break becomes larger. Aswill be argued momentarily, a larger break is easier to detect, which suggests

that the power increases as the precision of the estimated breaks improves.

By treating the locations of the breaks as unknown, the test results can also

be analysed in terms of the accuracy of the break estimates. To this end, Bai

and Perron (1998) show that, although consistent as T increases, the rate at

which the estimated break fractions converge to their true values depend on

the magnitude of the breaks, which seems reasonable as a smaller break is

more difficult to discern. Bartley et al. (2001) analyses this issue in small

samples and found that Ti can be estimated fairly well when the size of the

break is larger than two in absolute value. To examine the accuracy of the

estimated breaks in our case, Table 7 presents the correct selection frequencies

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TABLE 5

Size and size-adjusted power for Cases 4 and 5 with known breaks

Case k1 /

N 10, T 100 N 20, T 100 N 10, T 200

h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.5 h 0 h )0.5 h 0

4 0.3 0.00 0.022 0.044 0.002 0.018 0.066 0.000 0.020 0.066 0.000

4 0.3 0.05 0.246 0.246 0.256 0.356 0.316 0.344 0.784 0.766 0.822

4 0.3 0.10 0.720 0.758 0.710 0.926 0.904 0.934 0.994 0.994 0.998

4 0.5 0.00 0.016 0.054 0.000 0.018 0.060 0.000 0.030 0.052 0.000

4 0.5 0.05 0.244 0.258 0.234 0.340 0.354 0.334 0.760 0.796 0.826

4 0.5 0.10 0.716 0.706 0.706 0.922 0.910 0.896 0.996 0.994 0.998

4 0.7 0.00 0.012 0.044 0.000 0.020 0.056 0.000 0.010 0.042 0.000

4 0.7 0.05 0.276 0.262 0.230 0.438 0.442 0.462 0.858 0.818 0.854 4 0.7 0.10 0.768 0.730 0.704 0.930 0.936 0.952 0.996 1.000 1.000

5 0.3 0.00 0.200 0.230 0.042 0.454 0.534 0.110 0.026 0.074 0.000

5 0.3 0.05 0.090 0.170 0.098 0.132 0.144 0.110 0.372 0.410 0.480

5 0.3 0.10 0.256 0.388 0.286 0.510 0.504 0.408 0.946 0.944 0.954

5 0.5 0.00 0.042 0.124 0.000 0.072 0.186 0.000 0.012 0.074 0.000

5 0.5 0.05 0.240 0.234 0.210 0.364 0.340 0.338 0.776 0.772 0.798

5 0.5 0.10 0.718 0.702 0.668 0.920 0.910 0.920 1.000 1.000 0.998

5 0.7 0.00 0.154 0.222 0.044 0.430 0.510 0.130 0.030 0.088 0.000

5 0.7 0.05 0.298 0.304 0.234 0.460 0.428 0.456 0.816 0.792 0.846

5 0.7 0.10 0.752 0.706 0.752 0.940 0.932 0.936 0.996 0.996 1.000

Notes: The valuek1refers to the location of the structural break, the value / refers to the weight being attached to the uerrors and h refers to the moving average parameter. See Table 1 for an explanation of the different cases.



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TABLE 6

Size and size-adjusted power for Cases 4 and 5 with unknown breaks

Case d /

N 10, T 100 N 20, T 100 N 10, T 200

h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.

4 1 0.00 0.018 0.000 0.146 0.012 0.006 0.142 0.020 0.000 0.150

4 1 0.05 0.083 0.073 0.147 0.173 0.103 0.153 0.280 0.187 0.170

4 1 0.10 0.127 0.157 0.153 0.253 0.163 0.220 0.507 0.600 0.560

4 2 0.00 0.007 0.003 0.030 0.007 0.007 0.015 0.018 0.010 0.037

4 2 0.05 0.093 0.087 0.110 0.133 0.073 0.110 0.273 0.267 0.320

4 2 0.10 0.197 0.173 0.330 0.390 0.207 0.437 0.643 0.700 0.880

4 3 0.00 0.013 0.003 0.007 0.010 0.007 0.010 0.013 0.007 0.013

4 3 0.05 0.173 0.117 0.150 0.207 0.127 0.100 0.427 0.297 0.493

4 3 0.10 0.310 0.233 0.427 0.533 0.383 0.550 0.747 0.780 0.943

5 1 0.00 0.040 0.070 0.000 0.070 0.140 0.000 0.017 0.020 0.000

5 1 0.05 0.120 0.083 0.093 0.097 0.160 0.110 0.300 0.207 0.347

5 1 0.10 0.267 0.140 0.210 0.387 0.327 0.380 0.750 0.523 0.943

5 2 0.00 0.037 0.047 0.000 0.080 0.133 0.000 0.020 0.007 0.000

5 2 0.05 0.120 0.120 0.087 0.103 0.163 0.107 0.263 0.210 0.303

5 2 0.10 0.260 0.207 0.250 0.310 0.263 0.353 0.763 0.533 0.900

5 3 0.00 0.057 0.080 0.000 0.070 0.153 0.000 0.007 0.017 0.000

5 3 0.05 0.087 0.083 0.107 0.107 0.100 0.100 0.293 0.200 0.383

5 3 0.10 0.167 0.153 0.307 0.353 0.297 0.390 0.713 0.660 0.960

Notes: See Table 1 for an explanation of the different cases and Table 5 for an explanation of the various parameters. structural break.

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for both the number and the locations of the estimated break fractions obtained

under the null hypothesis with k1 0.5 held fixed. The results indicate that

the breaks can be estimated with high accuracy in most cases and that theprecision of the estimates increase as both T and d increases, which is

consistent with the asymptotic results presented by Bai and Perron (1998).

Overall, the simulations leads us to the conclusion that the test performs

well in general with reasonable power and small size distortions in most

experiments. The results also suggest that the performance of the test in Cases

4 and 5 is unaffected by the locations of the breaks but the performance of the

test with shifts in both intercept and trend can be poor unless the time series of

each subsample is sufficiently long. We have also examined the effects of a

misspecified model where the true DGP includes a structural break that is not

considered by the researcher. In this case, the rejection frequencies are

effectively one in all experiments and the results are therefore not presented.

Thus, serious size distortions are likely to arise when an existing structural

breaks are omitted.

VII. An application to the current account

In this section, we make an attempt to demonstrate empirically the usefulness

of our test by revisiting the data set used by Taylor (2002) and Ho (2002) to

assess the solvency of the current account. To this end, let Iitbe the ratio ofinvestment to gross domestic product (GDP) at current local prices and letSit

be the corresponding saving rate. The starting point of our investigation is the

following regression

Iitai biSit eit: 8

Recent theoretical and empirical work suggests that saving and investment as a

fraction of GDP are non-stationary variables (e.g. see Coakley, Kulasi and

Smith, 1996; Levy, 2000). The implication being that the current account must

be stationary as debt cannot explode. Because the current account is identicallythe difference between saving and investment, this suggests that saving and

investment as a fraction of GDP should be cointegrated. Yet, for a theory so

straightforward and widely accepted, the solvency of the current account has

proven extremely difficult to establish empirically (e.g. see Leachman, 1991;

de Haan and Siermann 1994; Lemmen and Eijffinger, 1995; Ho, 2002).

Although there are many explanations for these empirical findings, this

section focuses on two explanations that has attracted much attention recently.

The first explanation is that conventional time-series cointegration tests may

have low power against persistent alternatives because of the short-sample

periods usually employed (e.g. see de Haan and Siermann, 1994; Ho, 2002).

The second explanation is that the empirical relationship between saving and




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TABLE 7

Correct selection frequencies for Cases 4 and 5 with unknown breaks

Case d Frequency

N 10, T 100 N 20, T 100 N 10, T 200

h 0 h )0.5 h 0.5 h 0 h )0.5 h 0.5 h 0 h )0.5 h 0

4 1 Number 0.647 0.567 0.613 0.641 0.578 0.609 0.725 0.658 0.686

4 1 Location 0.292 0.177 0.327 0.281 0.171 0.324 0.305 0.197 0.363

4 2 Number 0.786 0.669 0.810 0.791 0.663 0.798 0.856 0.734 0.845

4 2 Location 0.549 0.380 0.579 0.541 0.382 0.580 0.570 0.424 0.605

4 3 Number 0.846 0.733 0.874 0.856 0.706 0.868 0.901 0.757 0.8934 3 Location 0.669 0.522 0.716 0.672 0.511 0.710 0.709 0.534 0.726

5 1 Number 0.954 0.762 0.987 0.960 0.741 0.987 0.979 0.828 0.993

5 1 Location 0.944 0.749 0.979 0.950 0.729 0.979 0.974 0.823 0.989

5 2 Number 0.956 0.730 0.996 0.961 0.747 0.996 0.987 0.822 0.999

5 2 Location 0.949 0.724 0.990 0.956 0.740 0.990 0.981 0.818 0.994

5 3 Number 0.961 0.747 0.997 0.963 0.745 0.997 0.987 0.818 0.999

5 3 Location 0.957 0.742 0.994 0.959 0.741 0.993 0.981 0.817 0.998

Notes: The number frequency specifies the frequency count of correctly chosen number of breaks. The location frequencorrectly chosen break locations when the estimated number of breaks is equal to its true value. See Table 1 for an explanatifor an explanation of the various parameters. The value drefers to the size of the structural break.

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investment that is being estimated need not be invariant to policy regime

changes, which may lead to structural shifts in both saving and investment and

hence in the relationship between them (e.g. see Alexakis and Apergis, 1994;Sarno and Taylor, 1998; Corbin, 2004; Ozmen and Parmaksiz, 2004).

However, while very reasonable and potentially appealing, these explan-

ations have been found to be empirically inadequate and far from convincing,

and this section therefore offers an alternative route. The idea is that, to

provide any robust evidence on the cointegration between saving and

investment, one needs to consider not the low power and the presence of

structural change separately but simultaneously. The intuition is simple. First,

while increasing the length of the sample may be justified from a power point

of view, it also increases the probability of a break occurring somewhere in the

sample. On the other hand, modifying existing tests so as to accommodate for

structural change is typically very costly in terms of power. Therefore, it is the

joint consideration of both these aspects that is likely to be key when testing

the solvency of the current account. One way to accomplish this is to use the

panel LM test developed in the previous sections.

The data that we use consists of cross-sectional observations on saving and

investment as fractions of GDP sampled at annual frequency between 1883

and 1992 for 15 Organization for Economic Co-operation and Development

(OECD) countries plus Argentina.3 Before applying our test, we examine

whether the variables are stationary or not and whether there exist any structuralbreaks or not. To test if the variables are non-stationary, we employ the panel data

unit root test recently proposed by Im et al. (2005), which allows for the

possibility of a single heterogeneous shift in the level of the series.4 The

calculated values of the statistic for the saving and investment variables are

)0.560 and)1.330 respectively. Thus, as the critical region of the test is given by

the left tail of the normal distribution, the null hypothesis of a unit root test cannot

be rejected. This conclusion is supported by the individual Schmidt and Phillips

(1992) and Amsler and Lee (1995) unit root test statistics presented in Table 8.

The first does not allow for any break, while the second allows for a single breakto affect the level of each series. The results suggest that we cannot reject the null

of a unit root at the 10% level of significance for any of the countries.

To test the presence of structural change in the estimated regressions, we

employ for each individual the B(s) statistic of Hao and Inder (1996), and

the MeanF and SupF statistics of Hansen (1992). The results presented in

Table 8 suggest that at least six of the individual cointegration regressions

3For a detailed description of the data, see Taylor (2002).4In calculating the value of the test statistic, we use a lag augmentation of three to account for the

effects of serial correlation. The location of the break is determined endogenously via grid search atthe minimum of the statistics. To this end, we use a trimming parameter of 0.15 to eliminate theendpoints.




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TABLE 8

Country specific tests

Country

Unit root saving Unit root investment Cointegration tests Stability tests

Z(s) ~s Z(s) ~s Zt Zt B(s) MeanF Sup

Argentina )1.942 )1.935 )3.349 )3.320 )4.657 )3.900 0.748 10.349 13

Australia )2.806 )2.854 )2.511 )2.068 )6.877 )6.397 0.667 1.741 4

Canada )2.374 )2.408 )2.144 )2.093 )3.991 )3.041 1.363 3.434 14

Finland )

2.266 )

2.959 )

2.117 )

2.166 )

5.576 )

4.580 1.033 7.939 14Spain )1.957 )2.353 )2.143 )2.682 )5.386 )4.838 1.091 2.298 11

Sweden )1.671 )1.855 )2.487 )2.588 )5.462 )4.393 0.920 7.622 15

UK )2.178 )2.238 )2.446 )2.942 )4.457 )2.261 1.990 12.636 17

USA )2.157 )2.895 )2.462 )3.273 )3.971 )3.364 0.798 3.266 6

Notes: In constructing the test statistics, we use three lags, a bandwidth of [ T1/3] and a trimming parameter of 0.15. The the estimated break points used in computing the panel LM statistic. The maximum number of breaks allowed is five. Tfollows: )3.72 for the ~s statistic of Amsler and Lee (1995) and the Z(s) statistic of Schmidt and Phillips (1992); )3.065Ouliaris (1990); )4.34 for theZt statistic of Gregory and Hansen (1996); 1.0477 for theB(s) statistic of Hao and Inder (199SupF statistics of Hansen (1992) respectively.

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cannot be regarded as stable and there is a need to allow for structural change. To

this effect, we now employ the panel LM statistic. The calculated value for Case

2 with only an individual-specific constant term is 9.025 for the FMOLS-basedtest and 8.173 for the DOLS-based test. Thus, if we ignore the possibility of

structural change, then we find no evidence of cointegration. By contrast, if

we allow for a level shift in each regression, then the calculated values of

the statistics are 2.138 and 1.414 for the FMOLS- and DOLS-based test

respectively. Thus, in this case we cannot reject the null hypothesis on the 1%

level, which suggests that the variables are cointegrated around a broken

intercept. To reinforce this assertion, Table 8 presents some confirmatory

evidence for each country. The Ztstatistic of Phillips and Ouliaris (1990) do not

allow for any break, while theZ

t

statistic of Gregory and Hansen (1996) is able

to accommodate for a single break in the intercept of each regression. As seen

from the table, the null of no cointegration is rejected on six occasions at the 10%

level for both statistics.

Table 8 also reports the estimated break points obtained by using the

procedure described in section V. At least two breaks are found for each

country with all breaks occurring during the period 190675. From an

historical point of view, this seems very reasonable. First, there are essentially

no breaks prior to the advent of the First World War, which agrees with

the stability of the classical gold standard regime. Secondly, there is a

preponderance of breaks occurring between 1913 and 1949. This accordsapproximately with the interwar period and seems consistent with the findings

of Levy (2000), Hoffmann (2004) and Corbin (2004). The break in 1913 for

Argentina is also expected given the Bearing Crisis. Thirdly, there are several

breaks occurring between the years 1957 and 1975. This coincides with the oil

price shocks of that period, the breakdown of the Bretton Woods system and

the establishment of the Exchange Rate Mechanism.

An important caveat worth noting is that so far all forms of cross-sectional

dependency has been ignored. If this assumption is violated, then the panel

LM test depends on various nuisance parameters associated with the cross-sectional correlation properties of the data, which means that the test no longer

has a limiting normal distribution. A limited form of cross-sectional

dependence can be permitted by using data that has been demeaned with

respect to a common time effects, which does not affect the distribution of the

test. Therefore, to accommodate some form of cross-sectional dependency, we

applied the test to the cross-sectionally demeaned data. Using this approach

the calculated values of the FMOLS- and DOLS-based test statistics are 1.783

and 1.632 respectively. Thus, in agreement with the earlier results, we cannot

reject the null on the 1% level of significance.

However, in general, with dynamic feedback effects that runs from one

cross-section to another, and which are not common across the members of the




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panel, then a common time effects will not account for all dependencies. One

possible solution to this problem is to employ the bootstrap approach, which

makes inference possible even under very general forms of cross-sectionaldependence. The bootstrap opted for this section proceeds as follows.

1 Obtain Ti for each i using the procedure outlined in section V.

2 Estimate cij and bi for each i by FMOLS or DOLS using Tobserva-

tions and dummy variables to account for the breaks. Obtain

et e1t; . . . ; e

Nt

0.

3 Compute the centred residual

~et et T

1 XT

t1

et:

4 Let wt ~e0t; v

0t

0, where vt v

01t; . . . ; v

0Nt

0. Generate the bootstrap

innovations wt ~e0t ; v

0t

0by resampling the vectorwt with replace-

ment.5

5 Generate the bootstrap sample yit and xit with x

i0 0 recursively as

xitxit1 v

it;

yitz0itcij x

0itbi e

it:

6 Compute the standardized panel LM test statistic for each bootstrap

sample while treating Ti as known.

We apply the bootstrap approach to the saving and investment data using

5,000 bootstrap replications. The 1% critical value obtained from the

bootstrap distribution is 2.218 for the FMOLS-based test and 1.647 for the

DOLS-based test. Hence, the null is not rejected on the 1% level.

Nevertheless, it is tempting, particularly given the results from the more

straightforward setting of the previous sections, to conclude that the effects of

cross-sectional correlation are small and that the critical value from the normal

distribution in fact apply. To check whether this is indeed the case, we

performed a small set of Monte Carlo simulations based on the DGP of

section VI but now with the error uit having the factor specification uitpft+ eit, where (eit, vit, ft)

0 N(0, I3). With this modification, the correlationbetweenuitand uktfor i 6 kis given by p

2/(1 + p2). Thus, a non-zero value

on the parameterp induces correlation between the members of the panel.

Table 9 presents some results on the size of a nominal 5% level test for this

DGP using both the bootstrapped critical value and the critical value obtained

from the normal distribution. It is observed that the bootstrap test performs

5By resampling wtrather than ~et, we can preserve not only the cross-sectional correlation structureofeitbut also any endogenous effects that may run across the individual regressions of the system.

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well and the test based on the normal distribution appears to be remark-

able robust to moderate degrees of cross-sectional correlation. To get an

appreciation of the size of the problem in the saving and investment data, weestimate the correlation matrix of the equilibrium errors. The largest

correlation is 0.402, suggesting that the simulation results for the DGP with

p 1 should be relevant in which case Table 9 indicates that the effects ofcross-sectional dependence on the test should be small and that inference

based on the normal distribution should not be misleading.

The bootstrap approach can accommodate for all types of cross-sectional

dependencies that are absorbed in the long-run covariance matrix of the

equilibrium errors and the first differences of the regressors. Another source of

cross-sectional dependence is cointegration that runs between the membersof the panel. Banerjee, Marcellino and Osbat (2004) uses Monte Carlo

simulations to study this issue in small samples and found that the

consequences of cross-sectional cointegration on existing panel cointegration

tests can be quite severe. The authors suggest testing for the presence of cross-

sectional cointegration by using the procedure developed by Gonzalo and

Granger (1995), which involves first extracting the common trends from each

cross-section and then testing for cointegration among these trends. When we

employ this approach to the saving and investment data, we end up marginally

rejecting the null of no cointegration on the 1% level on one occasion. Hence,

there appears to be no severe violation of the assumption of no cross-sectional

cointegration.

TABLE 9

Size for Case 4 with cross-sectional dependence

p N T

FMOLS DOLS

ZB(M) Z(M) Z B(M) Z(M)

1 10 100 0.024 0.028 0.038 0.030

2 10 100 0.036 0.074 0.040 0.040

6 10 100 0.024 0.076 0.046 0.068

1 20 100 0.038 0.062 0.044 0.046

2 20 100 0.026 0.120 0.046 0.082

6 20 100 0.028 0.146 0.036 0.090

1 10 200 0.024 0.030 0.032 0.030

2 10 200 0.034 0.072 0.032 0.050

6 10 200 0.026 0.080 0.036 0.0801 20 200 0.038 0.064 0.036 0.048

2 20 200 0.032 0.108 0.040 0.118

6 20 200 0.024 0.116 0.024 0.122

Notes: FMOLS, fully modified ordinary least square; DOLS, dynamic ordinary least square.The valuep refers to the loading parameter in the common time effects specification. We use Z(M)

and ZB(M) to refer to the test based on the normal and bootstrapped 5% critical value.




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VIII. Conclusions

In this paper, we extend the panel LM cointegration test proposed by

McCoskey and Kao (1998) to allow for multiple structural breaks in both level

and trend of the cointegration regression. Test statistics are derived when both

the number and the locations of the breaks are known and when they are

determined endogenously from the data. Test statistics are also derived when

the cointegration regression is not subject to structural change but the

deterministic component includes individual-specific intercepts and trends.

Using sequential limit theory, we are able to show that the test has a limiting

normal distribution that is free of nuisance parameters under the null

hypothesis. In particular, we show that the limiting distribution is invariant

with respect to both the number and the locations of the breaks and that thereis no need to compute different critical values for all possible patterns of break

points, which makes the test computationally very convenient. Results form

Monte Carlo experiments suggest that the test has small size distortions and

reasonable power. To demonstrate the practical usefulness of the new test, we

reexamine the data set employed by Ho (2002) and Taylor (2002) to assess the

solvency of the current account. Contrary to most of the findings presented in

the earlier literature, we find evidence suggesting that saving and investment

are cointegrated once a level break in the cointegration regression is

accommodated.

Final Manuscript Received: May 2005

References

Alexakis, P. and Apergis, N. (1994). The Feldstein-Horioka puzzle and exchange rate regimes:

evidence from cointegration tests, Journal of Policy Modeling, Vol. 16, pp. 459472.

Amsler, C. and Lee, J. (1995). An LM test for a unit root in the presence of a structural break,

Econometric Theory, Vol. 11, pp. 359368.

Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structuralchanges,Econometrica, Vol. 66, pp. 4778.

Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models,

Journal of Applied Econometrics, Vol. 18, pp. 122.

Banerjee, A., Marcellino, M. and Osbat, C. (2004). Some cautions on the use of panel

methods for integrated series of macroeconomic data, Econometrics Journal, Vol. 7,

pp. 322340.

Bartley, W. A., Lee, J. and Strazicich, M. C. (2001). Testing the null of cointegration in the

presence of a structural break, Economics Letters, Vol. 73, pp. 315323.

Coakley, J., Kulasi, F. and Smith, R. (1996). Current account solvency and the Feldstein-

Horioka puzzle, The Economic Journal, Vol. 106, pp. 620627.

Corbin, A. (2004). Capital mobility and adjustment of the current account imbalances: abounds testing approach to cointegration in 12 countries, International Journal of Finance

and Economics, Vol. 9, pp. 257276.

128 Bulletin



30/33

Cragg, J. G. (1983). More efficient estimation in the presence of heteroskedasticity of

unknown form, Econometrica, Vol. 51, pp. 751763.

Gonzalo, J. and Granger, C. (1995). Estimation of common long-memory components in

cointegrated systems, Journal of Business and Economic Statistics, Vol. 13, pp. 2735.

Gregory, A. M. and Hansen, B. E. (1996). Residual-based tests for cointegration in models

with regime shifts, Journal of Econometrics, Vol. 70, pp. 99126.

de Haan, J. and Siermann, C. L. J. (1994). Saving, investment and capital mobility: a comment

on Leachman, Open Economies Review, Vol. 5, pp. 517.

Hansen, B. E. (1992). Tests for parameter instability in regressions with I(1) processes,

Journal of Business and Economic Statistics, Vol. 10, pp. 4559.

Hao, K. (1996). Testing for structural change in cointegrated regression models: some com-

parisons and generalizations, Econometric Reviews, Vol. 15, pp. 401429.

Hao, K. and Inder, B. (1996). Diagnostic test for structural change in cointegrated regression

models, Economics Letters, Vol. 50, pp. 179187.

Harris, D. and Inder, B. (1994). A Test of the Null Hypothesis of Cointegration, NonstationaryTime Series Analysis and Cointegration, in Hargreaves C. P. (ed.),Nonstationary Time Series

Analysis and Cointegration, Oxford University Press, New York, NY, pp. 133152.

Ho, T. (2002). A Panel cointegration approach to the saving-investment correlation, Empirical

Economics, Vol. 27, pp. 91100.

Hoffmann, M. (2004). International capital mobility in the long run and the short run: can we

still learn from saving-investment data?, Journal of International Money and Finance,

Vol. 23, pp. 113131.

Im, K. S., Lee, J. and Tieslau, M. (2005). Panel LM unit root tests with level shifts, Oxford

Bulletin of Economics and Statistics, Vol. 67, pp. 393419.

Kurozumi, E. (2002). Testing for stationarity with a break,Journal of Econometrics, Vol. 108,

pp. 6399.Leachman, L. (1991). Saving, investment and capital mobility among OECD countries, Open

Economies Review, Vol. 2, pp. 137163.

Lemmen, J. J. G. and Eijffinger, S. C. W. (1995). The quantity approach to financial integ-

ration: the Feldstein-Horioka criterion revisited, Open Economies Review, Vol. 6, pp. 145

165.

Levy, D. (2000). Investment-saving comovement and capital mobility: evidence from century

long U.S. time series, Review of Economic Dynamics, Vol. 3, pp. 100136.

MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration

tests, Journal of Applied Econometrics, Vol. 11, pp. 601618.

McCoskey, S. and Kao, C. (1998). A residual-based test of the null of cointegration in panel

data,Econometric Reviews, Vol. 17, pp. 5784.McCoskey, S. and Kao, C. (2001). A Monte Carlo comparison of tests for cointegration in

panel data, Journal of Propagation in Probability and Statistics, Vol. 1, pp. 165198.

Park, J. P. and Phillips, P. C. B. (1988). Statistical inference in regressions with integrated

regressors: part I, Econometric Theory, Vol. 4, pp. 468497.

Phillips, P. C. B. and Hansen, B. E. (1990). Statistical inference in instrumental variables

regression with I(1) process, Review of Economics Studies, Vol. 57, pp. 99125.

Phillips, P. C. B. and Moon, H. R. (1999). Linear regression limit theory of nonstationary panel

data,Econometrica, Vol. 67, pp. 10571111.

Phillips, P. C. B. and Ouliaris, S. (1990). Asymptotic properties of residual based tests for

cointegration,Econometrica, Vol. 58, pp. 165193.

Saikkonen, P. (1991). Asymptotic efficient estimation of cointegration regressions, Econo-metric Theory, Vol. 7, pp. 121.




31/33

Sarno, L. and Taylor, M. P. (1998). Exchange controls, international capital flows and saving-

investment correlations in the UK: an empirical investigation, Weltwirtschaftliches Archiv,

Vol. 134, pp. 6997.

Schmidt, P. and Phillips, P. C. B. (1992). LM tests for a unit root in the presence of deter-

ministic trends, Oxford Bulletin of Economics and Statistics, Vol. 54, pp. 257287.

Shin, Y. (1994). A residual-based test of the null of cointegration against the alternative of no

cointegration,Econometric Theory, Vol. 10, pp. 91115.

Taylor, A. M. (2002). A century of current account dynamics, Journal of International Money

and Finance, Vol. 21, pp. 725748.

Ozmen, E. and Parmaksiz, K. (2004). Policy regime change and the Feldstein-Horioka puzzle:

the UK evidence, Journal of Policy Modeling, Vol. 25, pp. 137149.

Westerlund, J. (2005). A panel CUSUM test of the null of cointegration, Oxford Bulletin of

Economics and Statistics, Vol. 62, pp. 231262.

Zivot, E. and Andrews, D. W. K. (1992). Further evidence on the great crash, the oil-price

shock and the unit root hypothesis, Journal of Business and Economic Statistics, Vol. 10,pp. 251270.

Appendix

Mathematical proofs

In this appendix, we prove the asymptotic distribution for the DOLS estimator.

The proof uses the methods of Shin (1994) so only essential details will be

given.

Proof of Theorem 1. Under the null hypothesis, the DGP for the most general

case with endogenous regressors and breaks in both the level and trend may be

written as the following truncated regression

yitX0

itdijXH

kH

v0itk/ik uit; A1

where Xit z0it;x

0it

0, dij (cij, bi)

0 and uit Pjkj>Hv0itk/ik u

it is a

stationary error term comprised of the projection of uit

onto all the lags and

leads above the truncation point H and an orthogonal term uit. The DOLS

estimator is obtained by performing OLS on (A1). Consider the DOLS

estimator of segmentjusing the subsamplet Tij)1, . . . , Tijand letdi denotethe OLS estimate ofdijbased on this subsample. The fitted regression may be

written in the following fashion

yitX0

itdiXH

kH

v0itk/ik uit: A2

Assumption 2 (i) ensures that each subsample approaches to infinity when

T! 1. Hence, there will be no loss of generality by analysing each subsampleas a sample of length T that is not subject to structural change. To this end,

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consider the partial sum process SiT PTr

t1 uitobtained from equation (A2).

If we let D1 diag(T)1/2, T)3/2, T)1IK) and D2 diag(T

)1, T)2, T)3/2IK),

then this sum may be written as

T1=2SiT T1=2

XTrt1

yit T1=2

XTrt1

X0itdi T1=2

XTrt1

XHkH

v0itk/ik

T1=2XTrt1

uit T1=2

XTrt1

X0itdi di

T1=2 XTr

t1X

H

kH

v0itk/ik /ik

T1=2XTrt1

uit T1=2

XTrt1

Xjkj>H

v0itk/ij

D12XTrt1

X0itD11 di di T

1=2XTrt1

XHkH

v0itk/ik /ik:

Under the regulatory conditions of Saikkonen (1991), the second and

fourth terms are op(1) and can be disregarded. Moreover, by the multi-variate invariance principle, we have D12

PTrt1Xit)

Rr0B0i2 and

T1=2PTr

t1uit)Bi1:2. Also, by Theorem 3.3 of Park and Phillips (1988),

as T! 1

D11 di di )

Z 10

Bi2B0i2

1Z 10

Bi2dBi1:2:

Using these weak convergence results, and as Wi1:2 x1i1:2Bi1:2 and Wi2

X1=2

i22 B

i2, the limit of T)1/2S

iTas T! 1 may now be readily obtained as

follows

T1=2SiT )Bi1:2

Z r0

B0i2

Z 10

Bi2B0i2

1Z 10

Bi2dBi1:2

xi1:2Wi1:2 xi1:2

Z r0

W0i2

Z 10

Wi2 W0i2

1Z 10

Wi2dWi1:2

xi1:2Vi: A3

Given that the restriction placed on the bandwidth expansion rate is satisfied,

then x2i1:2 is consistent forx2i1:2 as T! 1. Hence, we obtain




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T2x2i1:2 XT

t1

S2iT )Fi Z 1

0

V2i : A4

Consequently, if we define Rij PTij

tTij11Tij Tij1

2x2i1:2S

2it, then

it follows that Rij Fij, where Fi1, . . . , FiMi+1 are Mi + 1 independent

generalized Cramer-von Mises distributions. Using this result, we can prove

the intermediate limit ofZ(M) as follows

ZM XNi1

XMi1j1

XTijtTij11

Tij Tij12x2i1:2S

2it

XNi1

XMi1j1 R

ij

)XNi1

XMi1j1

Fij: A5

Thus, as the limiting distribution passing T! 1 is i.i.d. over the cross-section,it follows thatE(Fij) Q andEF

2ij R for alliandj. Consequently, if we let

FiPMi1

j1 Fij, thenE(Fi) (Mi + 1)Q andEF2

i Mi 12R. Let H and

R be the cross-sectional averages of the expected value and the variance ofFi

respectively. To derive the sequential limiting distribution ofZ(M), first defineRi PMi1

j1 Rij and note thatN1PN

i1Ri converges to H in probability as

T! 1 and then N! 1 sequentially by the law of large numbers. Next,expand the statistic in the following manner

N1=2ZM N1=2H N1=2 N1XNi1

Ri H

!: A6

Assume that the Lindeberg condition is satisfied, then N1=2ZM N1=2 H )N0; R as T! 1 prior to N by direct application of the

Lindeberg-Feller central limit theorem. This establishes the limit distributionof the panel LM statistic as required for the proof. n

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