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A more powerful modification of Johansen'scointegration testsSteve Leybourne a , Tae-Hwan Kim a b & Paul Newbold aa School of Economics, University of Nottingham , Nottingham, NG7 2RD, UKb Department of Economics , Yonsei University , 134 Shinchon-dong, Seodaemun-gu, Seoul,120-749, KoreaPublished online: 11 Apr 2011.

To cite this article: Steve Leybourne , Tae-Hwan Kim & Paul Newbold (2008) A more powerful modification of Johansen'scointegration tests, Applied Economics, 40:6, 725-729, DOI: 10.1080/00036840600749714

To link to this article: http://dx.doi.org/10.1080/00036840600749714

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Applied Economics, 2008, 40, 725–729

A more powerful modification of

Johansen’s cointegration tests

Steve Leybournea, Tae-Hwan Kima,b,* andPaul NewboldaaSchool of Economics, University of Nottingham, Nottingham, NG7 2RD, UKbDepartment of Economics, Yonsei University, 134 Shinchon-dong,

Seodaemun-gu, Seoul, 120-749, Korea

We apply the idea of using reversed time series to improve the power of

Johansen tests. We suggest computationally simple variants of the trace

and maximum eigenvalue statistics and establish their limit distributions.

Both are shown, via simulation, to yield nontrivial power gains.

I. Introduction

Testing for cointegration is now a core component ofthe analysis of multiple integrated time series, withthe trace and maximum eigenvalue statistics proposedby Johansen (1988, 1991), being by some way themost frequently applied. As such, any modification ofJohansen’s tests that yields superior power to theoriginals should be of some interest to the practi-tioner. In this article, we suggest a means ofimproving the power of the Johansen tests thatinvolves near-negligible additional computationalcost to the user. This is based on combining statisticscomputed from both the forward and reversed timeseries, a methodology adopted in Leybourne et al.(2005) in the context of increasing the power ofDickey–Fuller (DF) unit root tests. We provideasymptotic critical values of the new tests anddemonstrate the power gains that can be obtainedvia Monte Carlo simulation.

II. The Model

Consider the following multivariate data generatingprocess:

�yt ¼ �þ�yt�1 þ �t; t ¼ 1, 2, . . . ,T ð1Þ

where yt¼ (y1t, y2t, . . . , ypt)0 is a p� 1 vector and "t is

IID(0, �). Let r(0� r� p) be the rank of the matrix�. The matrix � is decomposed as

� ¼ ��0

where � and � are p� r matrices of rank r. The nullhypothesis of interest is H0: rank(�)¼ 0 and thealternative hypothesis isH1: rank(�)¼ 1. For later usewe denote by Hr the hypothesis that rank(�)¼ r. Thetime series yt is a multivariate pure random walkwithout any common stochastic trend under the nullhypothesis, while it is cointegrated with the cointe-grating vector � under the alternative hypothesis.

III. The Test and Its Asymptotic NullDistribution

We briefly review Johansen’s (1988, 1991) procedureto test the null hypothesis. The quasi log-likelihoodfunction (QLLF), conditional on y0, is given by

LFð�,�,�Þ ¼ �Tp

2

� �lnð2�Þ �

T

2

� �ln j�j

�1

2

� �XTt¼1

h�yt � ���yt�1ð Þ

0��1

� �yt � ���yt�1ð Þ

i

*Corresponding author. E-mail: tae-hwan.kim@yonsei.ac.kr

Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online � 2008 Taylor & Francis 725http://www.tandf.co.uk/journalsDOI: 10.1080/00036840600749714

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Let L*FðH0Þ and L*FðH1Þ be the maximized value of

the QLLF under H0 and H1 respectively. Then, themaximum eigenvalue statistic testing H0 against H1,denoted as LRF(0, 1), is given by

LRFð0, 1Þ ¼ 2ðL*FðH1Þ � L*FðH0ÞÞ

¼ �T 1� �1

� �

where �i is the ith-largest eigenvalue (i¼ 1, 2, . . . , p)calculated from the following characteristic equation:

�S11 � S10S�100 S01

��� ��� ¼ 0 ð2Þ

where S11 ¼ T�1PT

t¼1 vtv0t, S10 ¼ T�1

PTt¼1 vtu

0t,

S00 ¼ T�1PT

t¼1 utu0t, S01 ¼ S010. Here, vt and ut are

the residual vectors from the ordinary least squares(OLS) regressions of yt�1 and �yt, respectively, on aconstant vector.

It follows from Johansen (1991) that

LRFð0, 1Þ ) �max Q*F

n o

where �max{A} denotes the maximum eigenvalue ofthe matrix A and the p� p matrix Q*

f is given by

Q*F ¼

Z 1

0

dW*ðsÞW*ðsÞ0Z 1

0

W*ðsÞW*ðsÞ0ds

� ��1

�

Z 1

0

W*ðsÞdW*ðsÞ0

and W*(s) is a demeaned Weiner process obtained byprojecting a p-dimensional Weiner process W(s) on aconstant: i.e. W*ðsÞ ¼WðsÞ �

R 10 WðsÞds.

Following Leybourne et al. (2005), we nowconsider a modification of the maximum eigenvaluetest discussed earlier designed to achieve better powerproperties. The reversed time series of yt is given byzt¼ yTþ1� t. The corresponding QLLF is nowconstructed based on zt:

LRð�,�,�Þ ¼ �Tp

2

� �lnð2�Þ �

T

2

� �ln j�j

�1

2

� �XTt¼1

h�zt � ���zt�1ð Þ

0��1

� �zt � ���zt�1ð Þ

i

The subscript R indicates that QLLF is based on thereverse time series. Let L*RðH0Þ and L

*RðH1Þ and be

the maximized value of the QLLF under H0 and H1,respectively. Then, the maximum eigenvalue statisticLRR(0, 1) based on the reversed series is given by

LRRð0, 1Þ ¼ 2 L*RðH1Þ � L*RðH0Þ

� ¼ �T 1� ~�1

� ð3Þ

where �i is the ith-largest eigenvalue (i¼ 1, 2, . . . , p)calculated from the following characteristic equation:

� ~S11 � ~S10~S�100

~S01

��� ��� ¼ 0

where ~Sij are obtained exactly the same way (i.e.including only a constant in the auxiliary regressions)as in (2), but now using the reversed series zt�1 and�zt in place of yt�1 and �yt. Then, our proposedstatistic, denoted Minmax(0, 1), is given by

Minmaxð0, 1Þ ¼ minðLRFð0, 1Þ,LRRð0, 1ÞÞ

As can be seen earlier, the only additional step theusers of this new test need to carry out is to computethe LRR(0, 1) statistic based on the reverse time seriesexactly the same way as for the forward time seriesand then to take the minimum value of the two.Given the recent development in computing technol-ogy, this sort of additional computing burden isnegligible (e.g. if EVIEW is used, it almost amountsto pushing the appropriate button one more time).The following theorem provides the limiting nulldistribution of the new statistic. The proof is given inthe Appendix.

Theorem 1: If yt is generated by (1), then under thenull hypothesis

MinLRð0, 1Þ ) min �max Q*F

�, �max Q*

R

�� where

Q*R ¼ �

Z 1

0

dW*ðsÞW*ðsÞ0 � Ip

� � Z 1

0

W*ðsÞW*ðsÞ0dr

� ��1

� �

Z 1

0

W*ðsÞdW*ðsÞ0 � Ip

� �

The general form of the null and alternativehypotheses of the maximum eigenvalue statistic isgiven by Hr: rank(�)¼ r against Hrþ1:rank(�)¼ rþ 1 where r can take any integer valuebetween 0 and p� 1 inclusively. For this general case,our proposed maximum eigenvalue statistic is

Minmaxðr, rþ 1Þ ¼ minðLRFðr, rþ 1Þ,LRRðr, rþ 1ÞÞ

where LRF(r, rþ 1) and LRR(r, rþ 1) are the usuallog-likelihood ratio statistics testing Hr against Hrþ1

based on forward and reverse time series, respec-tively. It can be shown that the null limitingdistribution of Minmax(r, rþ 1) is the same as theone given in Theorem 1. The only difference is thatW(s) is now a (p� r)-dimensional standard Weinerprocess.

The usual Johansen trace statistic is for testing Hr

against Hp where r can again take any integer valuebetween 0 and p� 1 inclusively and it is given by

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LRF (r, p) for which the following holds fromJohansen (1991):

LRFðr, pÞ ) tracefQ*Fg

Our modification for this trace statistic is carried outin exactly the same fashion as for the maximumeigenvalue statistic. The modified trace statistic,denoted by Mintrace(r, p), is given by

Mintraceðr, pÞ ¼ minðLRFðr, pÞ,LRRðr, pÞÞ

where LRR(r, p) is the corresponding trace statisticbased on the reverse time series zt. It can be shownthat

Mintraceðr, pÞ ) minðtracefQ*Fg, tracefQ

*RgÞ

where W(s) is again a (p� r)� 1 standard vectorWeiner process.

So far, we have focused on the constant only casein detail as all the auxiliary regressions in (2) and (4)include only a constant for the deterministic compo-nent. The trend case is a straightforward extension.First, all the auxiliary regressions in (2) and (4) nowinclude a linear time trend as well as a constant.Second, the null limiting distribution for the trendcase is the same as in Theorem 1, but where W*(s) isobtained by projecting a p-dimensional Weinerprocess W(r) on (1, r); i.e. it is now a demeaned anddetrended Weiner process.

IV. Monte Carlo Simulations

We first report in Table 1, asymptotic null criticalvalues (1, 5 and 10%) for some selected values for thenumber of random walks under the null hypothesis.If compared with the corresponding critical values ofthe Johansen’s maximum eigenvalue and trace andtests, it can be seen that the critical values of our twotests are slightly smaller.

Next, we investigate the power gains achievable infinite samples through our proposed modification.

We generate random variates using the followingDGP:

y1t þ by2t ¼ u1t; u1t ¼ u1t�1 þ �1t,

y1t þ ay2t ¼ u2t; u2t ¼ �u2t�1 þ �2t,

t¼�100, . . . ,�1, 0, 1, . . . ,T and discard the first 101observations in the simulations. It is assumed thata 6¼ b. This DGP has been frequently used in the studyof cointegration as in Banerjee et al. (1986), Engleand Granger (1987) among others. In the earliermentioned DGP, � is the key parameter for control-ling cointegration relationship between y1t and y2t.If �¼ 1, then y1t and y2t are both I(1), but notcointegrated. On the other hand, if |�|< 1, then y1tand y2t are both I(1) and cointegrated with thecointegrating vector of (1, a).

The earlier-mentioned Data Generation Process(DGP) can be expressed as in (1) as follows;

�yt ¼ �yt�1 þ �t

where yt ¼ ðy1t, y2tÞ0, �¼ ��0 with

� ¼�

b

a� b�� 1ð Þ

1

a� b�� 1ð Þ

0B@

1CA

� ¼1

a

� �

and

�t ¼

1

a� ba�1t � b�2tð Þ

1

a� b��1t þ �2tð Þ

0B@

1CA

We set a¼�1 and b¼ 1 and vary the key parameter �between 1 and 0.5 with 0.05 increments. For eachvalue of �, we compute the probability of rejectingthe null of no cointegration at the 5% significancelevel for the two new tests along with the two originalJohansen tests. Since the significance level is 5% andthe number of random walks is two under the null,from Table 1 we use 13.12 (constant case) and 16.83

Table 1. Asymptotic null critical values

Min max Eigen test Minmax (r, rþ 1) Min trace test Mintrace (r, p)

Constant Trend Constant Trend

Random walks (p� r) 0.10 0.05 0.01 0.10 0.05 0.01 0.10 0.05 0.01 0.10 0.05 0.01

2 11.29 13.12 17.10 14.77 16.83 21.28 13.38 15.47 19.90 18.91 21.36 26.573 17.35 19.60 23.47 20.89 23.33 28.12 25.87 28.45 33.34 33.86 36.71 42.664 23.75 26.02 31.28 26.84 29.29 34.65 43.38 46.09 52.36 52.44 56.02 63.265 29.64 32.12 37.62 33.02 35.99 41.04 64.07 67.93 75.63 75.75 80.31 88.77

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(trend case) for our modified maximum eigenvaluetest and 15.47 (constant case) and 21.36 (trend case)for our modified trace test. The empirical powerfunctions for these two tests are shown in Table 2(constant case) and Table 3 (trend case).

We first discuss the results for the constant onlycase. When �¼ 1, there is no cointegrating relation-ship between y1t and y2t so that the rejectionprobabilities are fairly close to the nominal size. As� moves away from 1, the rejection probability of thetwo Johansen’s tests, as expected, increases. By thetime � is near 0.6, the test rejects the null most ofthe time. Our modified maximum eigenvalue andtrace tests deliver larger powers than the correspond-ing Johansen’s tests. The maximum gain occurs when� is around 0.80; in that case there is 11–13%

improvement in power. As shown in Table 3,the results for the trend case are qualitatively similarto the constant case.

V. Summary

The idea of using reversed time series has been used invarious situations to improve the power of DF unitroot tests. In this article, we apply the same principleto Johansen’s maximum eigenvalue and trace tests toimprove the power of the two tests. Using a simpledata generation process, we illustrate the implemen-tation procedure of our new tests, derive the limitingdistribution of the new tests and demonstrate byMonte Carlo simulations that the gain in powerinduced by our modification is appreciable despiteits computational simplicity. In the presence of serialcorrelation in the error term, the usual procedure ofincluding lagged first differences is applied.

Acknowledgement

Tae-Hwan Kim is grateful to the College of Businessand Economics at Yonsei University for financialsupport.

References

Banerjee, A., Dolado, J. J., Hendry, D. F. andSmith, G. W. (1986) Exploring equilibrium relation-ships in econometrics through static models: someMonte Carlo evidence, Oxford Bulletin of Economicsand Statistics, 48, 253–77.

Engle, R. F. and Granger, C. W. J. (1987) Cointegrationand error correction: representation, estimation andtesting, Econometrica, 55, 251–76.

Johansen, S. (1988) Statistical analysis of cointegrationvectors, Journal of Economic Dynamics and Control,12, 231–54.

Johansen, S. (1991) Estimation and hypothesis testing ofcointegration vectors in Gaussian vector autoregres-sive models, Econometrica, 59, 1551–80.

Leybourne, S. J., Kim, T.-H. and Newbold, P. (2005)Examination of some more powerful modifications ofthe Dickey–Fuller test, Journal of Time Series Analysis,26, 355–69.

Table 2. Rejection probabilities for constant case

Max Eigen test Trace test

� Minmax(0, 1) Johansen Mintrace(0, 2) Johansen

1.00 0.05 0.05 0.05 0.050.95 0.07 0.06 0.09 0.080.90 0.15 0.12 0.18 0.140.85 0.33 0.24 0.34 0.270.80 0.56 0.43 0.55 0.440.75 0.78 0.67 0.76 0.640.70 0.92 0.85 0.90 0.810.65 0.98 0.95 0.97 0.930.60 1.00 0.99 0.99 0.980.55 1.00 1.00 1.00 0.990.50 1.00 1.00 1.00 1.00

Table 3. Rejection probabilities for trend case

Max Eigen test Trace test

� Minmax(0, 1) Johansen Mintrace(0, 2) Johansen

1.00 0.05 0.04 0.05 0.050.95 0.06 0.06 0.08 0.070.90 0.11 0.09 0.14 0.120.85 0.21 0.16 0.23 0.200.80 0.37 0.29 0.39 0.320.75 0.58 0.49 0.57 0.490.70 0.78 0.68 0.74 0.660.65 0.91 0.85 0.88 0.800.60 0.98 0.95 0.95 0.920.55 0.99 0.98 0.99 0.970.50 1.00 1.00 1.00 0.99

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Appendix

Proof of Theorem 1: Since the weak convergenceresult: LRFð0, 1Þ ) �maxfQ

*Fg has been proved in the

literature, we only prove the other case:LRRð0, 1Þ ) �maxfQ

*Rg. Once we prove this claim,

then the result in Theorem 1 follows by thecontinuous mapping theorem because min(x1, x2) isa continuous function in both arguments.

First we note that under the null hypothesis thereverse time series zt is characterized by

zt ¼ zt�1 þ �t

where �t ¼ ��Tþ2�t. Next we consider each term inthe characteristic equation in (4):

T�1 ~S11¼T�2XTt¼1

zt�1�zt�1ð Þ zt�1�zt�1ð Þ0

¼T�2XTþ1t¼2

yt�1y0t�1�T

�3=2XTþ1t¼2

yt�1T�3=2

XTþ1t¼2

y0t�1

)P

Z 1

0

WðrÞWðrÞ0dr�

Z 1

0

WðrÞdr

Z 1

0

WðrÞ0dr

� �P0

¼P

Z 1

0

W*ðrÞW*ðrÞ0dr

� �P0�PB11P

0

~S10¼T�1XTt¼1

zt�1� zt�1ð Þ �zt��zt� 0

¼T�1XTþ1t¼2

zt�1�0t�1�T3=2

XTþ1t¼2

zt�1T�1=2

XTþ1t¼2

�0t�1

¼�T�1XTþ1t¼2

yt�1�0t�1�T�1

XTþ1t¼2

�t�0tþT�3=2

�XTþ1t¼2

yt�1T�1=2

XTþ1t¼2

�0t�1

) P �

Z 1

0

WðrÞdWðrÞ0 � Ip þ

Z 1

0

WðrÞdrWð1Þ0� �

P0

¼ P �

Z 1

0

dWðrÞW*ðrÞ0 � Ip

� �P0 � PB10P

0

~S01 ¼ ~S010 ) PB010P0 � PB01P

0

~S00 ¼ T�1XTt¼1

�zt ��zt�

�zt ��zt� 0

¼ T�1XTt¼1

1�t1�0t þ opð1Þ ) �

where P is the square root matrix of �, i.e. �¼PP0.The characteristic equation in (4) is written as

T�T�1 ~S11 � ~S10~S�100

~S01

��� ��� ¼ 0

Combining the earlier asymptotic results with the factthat the determinant function is continuous, it can beeasily seen that T ~�i converges to i, the ith-largestroot of the following equation:

PB11P0 � PB10P

0ðPP0Þ�1PB01P0

�� �� ¼ 0

which can be equivalently written as

jIp � B10B�111 B01j ¼ jIp �Q*

Rj ¼ 0

Hence, we have T ~�i ) i implying that

LRRð0, 1Þ ¼ �T 1� ~�1�

¼ T ~�1 þ opð1Þ

) 1 ¼ �max Q*R

�

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