testing for cointegration using the johansen approach: are we using the correct critical values?

JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 24: 825–831 (2009)Published online 6 April 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/jae.1073

TESTING FOR COINTEGRATION USING THE JOHANSENAPPROACH: ARE WE USING THE CORRECT CRITICAL

VALUES?PAUL TURNER*

Department of Economics, Loughborough University, UK

SUMMARYThis paper presents Monte Carlo simulations for the Johansen cointegration test which indicate that the criticalvalues applied in a number of econometrics software packages are inappropriate. This is due to confusionin the specification of the deterministic terms included in the vector error correction model (VECM). Theresult is a tendency to reject the null of no cointegration too often. Copyright 2009 John Wiley & Sons,Ltd.

1. INTRODUCTION

Johansen’s (1988) test for cointegration has become a standard part of the toolkit of many appliedeconometricians. This is partly due to the perception that it has higher power than alternative tests.This paper argues that, to some extent, the tendency of the Johansen test to reject a null of nocointegration may be due to the use of inappropriate critical values.

The vector error correction model used to construct Johansen tests can be written in the followinggeneral form:

xt D ˛�ˇ0xt�1 � ˇ0 � ˇ1t� � �0 � �1t Ck∑

jD1

jxt�j C εt �1�

where xt is a p ð 1 vector of variables observed at date t, ˛ is a p ð r matrix of coefficients(where r is the cointegrating rank of the system), ˇ is a p ð r matrix of coefficients which definesthe r cointegrating vectors in the system, ˇ0 is an r ð 1 vector of intercepts for the cointegratingvectors, ˇ1 is an r ð 1 vector of coefficients which allow for linear deterministic trends in thecointegrating vectors, �0 is a p ð 1 vector of intercepts in the equations which make up the VECM,�1 is a p ð 1 vector of linear trend coefficients for the VECM and j, j D 1, . . . , k, are sequencesof p ð p matrices which define the lag structure of the VECM. For simplicity, we omit otherpossible deterministic variables such as seasonal dummies.

One of the main problems which arises in the application of the Johansen method is that thespecification of the deterministic terms which enter the VECM relationship used is often leftunclear. There are two classifications in common use. The first is that associated with Johansen’s

Ł Correspondence to: Paul Turner, Department of Economics, Loughborough University, Loughborough LE11 3TU, UK.E-mail: [email protected]

Copyright 2009 John Wiley & Sons, Ltd.

826 P. TURNER

original work, while the second is associated with the work of Pesaran et al. (2000) (PSS).1 Threeof the five cases are identical in the two classification systems. Case 0 of Johansen is identicalto Case I of PSS since in both cases there are no deterministic terms included in the VECM(ˇ0 D ˇ1 D �0 D �1 D 0). Case 1Ł of Johansen is identical to Case II of PSS. Here both casesinclude an intercept in the VECM but this is restricted so that it is included in the levels part only,i.e., the cointegrating vector (ˇ1 D �0 D �1 D 0). Since the intercept in differences is restricted tozero, this is appropriate when there are no trends evident in the individual series. Finally, Case 2Łof Johansen is identical to Case IV of PSS. Here both cases include an unrestricted intercept inthe VECM and a linear trend which is restricted so that it is included in the levels part only, i.e.,as part of the cointegrating vector (�1 D 0).

The differences between these classification systems come in the nature of the data generationprocesses used to construct the critical values for Case III of PSS and Case 1 of Johansen andbetween Case V of PSS and Case 2 of Johansen. Pesaran et al. describe it as follows:

In Case III, Johansen computes critical values including a linear trend in the VAR, but wedo not. Similarly, in Case V, the critical values reported by Johansen (and Osterwald-Lenum(1992)) are based on a VAR model containing both linear and quadratic trends, but in ourcomputations we only include linear trends with unrestricted coefficients. (PSS, p. 342)

Case 1 of Johansen and Case III of PSS both allow an unrestricted intercept in the VECM(�0 6D 0, ˇ0 6D 0). However, Case 1 of Johansen also allows for a restricted linear trend (though itis not clear what exactly is the nature of the restrictions applied). Similarly, Case 2 of Johansenand Case V of PSS both allow for unrestricted intercepts and linear trends in the VECM (�0 6D 0,�1 6D 0, ˇ0 6D 0, ˇ1 6D 0) but Johansen also allows for a restricted quadratic trend. These differencesare crucial when it comes to the appropriate choice of critical values for the tests.

2. COMPARISON OF FOUR ECONOMETRICS PACKAGES

Following the discussion in the previous section, it is useful to compare the output for cointegrationtests from four popular econometrics software packages. The packages examined are Microfitversion 4.0, EViews version 5.1, PcGive version 11.1 and STATA version 9.2 The results in Table I3

are based on cointegration tests carried out on an artificial dataset comprising two independentrandom walk variables. In each case the specification adopted was to conduct a cointegration testwith an unrestricted intercept but no trend in the VECM (Case 1 of Johansen and Case III of PSS).For the sake of brevity, only the trace statistic has been presented. However, where available, theoutput for the maximum eigenvalue statistic produces identical patterns.

A striking feature of Table I is that, although each software package produces identical teststatistics, they differ in terms of the critical values and/or the p-values reported. The fact that

1 The most reliable empirically determined critical values are those provided by MacKinnon et al. (1999) (MHM). Theyconsider two classifications: that corresponding to Johansen’s original work, which we describe as the Johansen case;and that corresponding to Pesaran et al. (2000), which we describe as PSS. The MHM critical values are used in all theMonte Carlo simulations reported in this paper.2 The versions used were the latest release versions at the time that the simulations were carried out. EViews 6.0 andSTATA 10 have recently become available but the relevant test statistics and critical values have not changed.3 Table I reproduces the output of each package exactly as reported so that the reader can verify that the test statistics areidentical.

Copyright 2009 John Wiley & Sons, Ltd. J. Appl. Econ. 24: 825–831 (2009)DOI: 10.1002/jae

TESTING FOR COINTEGRATION USING THE JOHANSEN APPROACH 827

Table I. Trace test output from four econometrics software packagesMicrofit output

Null Alternative Statistic 95% critical value 90% critical value

r D 0 r ½ 1 6.3626 17.8600 15.7500r � 1 r D 2 1.8508 8.0700 6.5000

EViews output

Hypothesized no.of CE(s)

Eigenvalue Trace statistic 0.05 critical value Prob.

None 0.008983 6.362608 15.49471 0.6527At most 1 0.003695 1.850791 3.841466 0.1737

PcGive output

H0 : rank � Trace test [Prob.]

0 6.3626 [0.657]1 1.8508 [0.174]

STATA output

Maximum rank Parms LL Eigenvalue Trace statistic 5% critical value

0 2 �1435.8 . 6.3626 15.411 5 �1433.55 0.00898 1.8508 3.762 6 �1432.62 0.00369

Additional Monte Carlo simulations for the case of an unrestricted trend in the VECM can be seen in an earlier work-ing paper version of this paper at: http://www.lboro.ac.uk/departments/ec/RePEc/lbo/lbowps/Turner Johansen Simulations %20May 2007.pdf

the test statistics are identical presumably means that the same VECM is being used for theirconstruction.4 The same pattern was observed when we looked at the fifth option correspondingto an unrestricted linear trend in the VECM.

To determine the assumptions being made about the nature of the VECM it is necessary togo back to the source of the critical values and/or p-values reported. Microfit uses the tables ofcritical values reported in PSS. In particular, the 5% critical value of 17.86 for H0 : r D 0 canbe found in PSS (Table VI(c), p. 339) and is close to that of Case III of PSS, which assumesunrestricted intercepts but no trends. The 5% critical value of 15.49 for H0 : r D 0 reported byEViews can be found in the computer program associated with MacKinnon et al. (1999) (MHM)and matches that of Case 1 of Johansen. The p-value of 0.657 for H0 : r D 0 reported in thePcGive output is derived from the response surfaces estimated by Doornik (1998). The fact thatthis is very close to the p-value reported in the EViews output suggests strongly that this too isappropriate for Case 1 of the Johansen classification. For STATA, the critical values are those ofOsterwald-Lenum for Johansen Case 1. Overall, therefore, it appears that the critical values givenby Microfit are noticeably different from those given (or implied) by the other three packages.

4 In fact, it is possible to replicate these test statistics exactly using the procedure described in Davidson and MacKinnon(2004, pp. 640–642) and assuming a VECM with an unrestricted intercept only.


828 P. TURNER

3. IMPLICATIONS FOR REJECTION FREQUENCIES

Given these results, the natural question which arises is which, if any, of these software packagesis using the ‘correct’ set of critical values.5 A possible method for tackling this question is tomake use of Monte Carlo methods to establish whether the critical values used generate rejectionfrequencies which are consistent with the size of the test assumed. The Monte Carlo design inthis paper is one of the standard data generation processes (DGPs) adopted by Engle and Granger(1987). The form of the DGP is given by equations (2) and (3).

yt C xt D u1,t u1,t D ε1,t �2�

yt � 2xt D u2,t �1 � �L�u2,t D ε2,t �3�

where t D 1, . . . , T and 0 < � � 1. ε1,t and ε2,t, t D 1, . . . , T, are independent identically dis-tributed N(0, 1) variables. The variables y and x can be written as linear combinations ofthe u terms: yt D 2u1,t � u2,t and xt D �u1,t C u2,t. This formulation ensures that both y andx individually contain a unit root but there is a cointegrating relationship between y andx, with cointegrating parameter �2, providing � < 1. Thus each variable has both a unitroot component and a transitory autoregressive component when � < 1. Simulations of thisDGP were carried out using an EViews program, which is available from the author onrequest.

We proceed in two stages. First, we present Monte Carlo results for rejection frequen-cies of H0 : r D 0 against H1 : r ½ 1 using the PSS Case III and V critical values andJohansen Case 1 and 2 classification. These are given in Table II, which indicates that theuse of the Johansen classification produces rejection frequencies considerably in excess ofthe size of the test which is assumed. For the unrestricted intercept–no linear trend case(EViews Case 3), the rejection frequency is more than double the correct size of the trace

Table II. Rejection frequencies for trace test using different sets of critical values

Rejection frequencies for H0 : r D 0 against H1 : r ½ 1

EViews Case 3 specification and Johansen Case 1 critical values 12.01EViews Case 3 specification and PSS Case III critical values 5.59EViews Case 5 specification and Johansen Case 2 critical values 21.86EViews Case 5 specification and PSS Case V critical values 5.25

Rejection frequencies for H0 : r � 1 against H1 : r D 2

EViews Case 3 specification and Johansen Case 1 critical values 31.87EViews Case 3 specification and PSS Case III critical values 5.03EViews Case 5 specification and Johansen Case 2 critical values 63.67EViews Case 5 specification and PSS Case V critical values 5.04

These rejection frequencies were calculated using 10,000 replications of the DGP obtained using the EViews randomnumber generator. All critical values used for these simulations are taken from the computer program supplied by MHMfrom www.econ.queensu.ca/jae. The terms ‘Johansen critical value’ and ‘PSS critical value’ refer to the classificationsystem, not to the actual numerical values used.

5 By ‘correct’ we mean that the critical values match the specification of the deterministic terms included in the VECM.Some of the critical values given by regression packages are dated in the sense that they do not use the most up-to-dateMonte Carlo results (e.g., the values reported by STATA). However, this is a relatively minor problem when comparedwith a possible mismatch in the assumptions relating to the deterministic terms of the VECM.



test. When the PSS Case III critical values are used, the rejection frequency is close to thecorrect size. If we compare the rejection frequencies for the unrestricted intercept–unrestrictedtrend case (EViews Case 5), then the rejection frequency when the Johansen classifica-tion is used is even higher relative to the correct size of the test. Again, when the PSSCase V critical values are used, the rejection frequency is close to the correct size of thetest.

A potentially more serious problem emerges when we consider H0 : r ½ 1 against H1 : r D 2in systems with a single cointegrating vector. In this case the simulation model consisted ofequations (2) and (3) with � D 0.85. The results of simulations of this model are given in Table II.These show that the rejection frequencies are well above the correct size of the test in both cases.For EViews Case 3 the rejection frequency is over 30% and for Case 5 over 60%. Thus there is areal danger of detecting spurious ‘second’ cointegrating vectors when we use the incorrect criticalvalues.

Another interpretation of these results can be made using the relationship between the Johansenclassification and the p-values for the distribution of the PSS statistic. If the PSS distribu-tion is correct then the percentile corresponding to the Johansen classification should give therejection frequency when these are used in place of the PSS values. To examine this hypoth-esis we compare the percentiles of the PSS distribution with empirical rejection frequenciesbased on 10,000 replications for H0 : r D 0 against H1; r > 0 using the trace test with sys-tems of dimension 1, 2, 3, 4 and 5. The DGP used assumes that the variables entering theVECM are independent random walks and we calculate trace test statistics assuming (a) anunrestricted intercept in the VECM and (b) an unrestricted intercept and an unrestricted trendin the VECM. The results, given in Table III, show a close match between the predictedand actual rejection frequencies and act as further confirmation that the PSS distribution isappropriate.

4. CONCLUSIONS

The specification of the deterministic terms in the VECM used to construct Johansen cointegrationtests is of crucial importance. This is particularly the case when the VECM includes unrestrictedintercepts and/or trends. Case 1 of Johansen and Case III of PSS have very different associatedcritical values, and mismatching the critical values to the test being employed can lead to incorrectconclusions. In particular, the use of the critical values for the Johansen Case 1 classificationwhen the PSS Case III VECM is estimated means that there is a serious probability of detectingspurious cointegrating vectors. The same result holds when we use critical values for JohansenCase 2 and the test statistics are constructed using the PSS Case V VECM. Based on thisanalysis we recommend using MHM’s estimates of the PSS classification when assessing thesignificance of the test statistics produced by all the regression packages considered in thispaper.

ACKNOWLEDGEMENTS

Thanks are due to Terry Mills and three anonymous referees for comments on an earlier draft.The usual disclaimer applies and all errors are solely the author’s responsibility.


830 P. TURNER

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Surveys 12(5): 573–593.Engle RF, Granger CWJ. 1987. Cointegration and error-correction: representation, estimation and testing.

Econometrica 55: 251–276.Johansen S. 1988. Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control

12: 231–254.MacKinnon JG, Haug AA, Michelis L. 1999. Numerical distribution functions of likelihood ratio tests for

cointegration. Journal of Applied Econometrics 14: 563–577.Osterwald-Lenum M. 1992. A note with quantiles of the asymptotic distribution of the maximum likelihood

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testing for cointegration using the johansen approach: are we using the correct critical values?

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