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Spurious rejections by cointegration tests induced bystructural breaksStephen J. Leybourne a & Paul Newbold aa School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UKPublished online: 05 Oct 2010.

To cite this article: Stephen J. Leybourne & Paul Newbold (2003) Spurious rejections by cointegration tests induced bystructural breaks, Applied Economics, 35:9, 1117-1121, DOI: 10.1080/0203684032000082068

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Spurious rejections by cointegration tests

induced by structural breaks

STEPHEN J. LEYBOURNE and PAUL NEWBOLD*

School of Economics, University of Nottingham, University Park, NottinghamNG7 2RD, UK

The effects on three cointegration tests are examined when the series analysed areindependent integrated processes, each with a structural break. Although there aredifferences in detail among the tests, the results indicate in all cases that, whenstructural breaks are neglected in the analysis, spurious rejections, indicating thepresence of cointegration, can occur.

I . INTRODUCTION

The spurious regression phenomenon (Granger andNewbold, 1974; Phillips, 1986) highlights the possibilityof ‘discovering’ an apparently strong relationship betweenindependent integrated time series when inference is basedon ordinary least squares methodology. The various testsof cointegration in common use all avoid the spuriousregression phenomenon under the normal circumstancesfor which they were designed. This study demonstrates thatthis is no longer the case in the presence of neglectedstructural breaks. This is shown through simulationexperiments.

The generating model is of two independent randomwalks, but with each containing a single structural break.Both breaks in level and breaks in slope are separatelyconsidered. In the simulations, for simplicity, the breaks inthe two series are taken to be of the same type, and of thesame magnitude, but possibly occurring at different times.The effects of these breaks on inference for three cointe-gration tests are considered. These are the Engle–Grangertest (Engle and Granger, 1987), the Johansen trace test1

(Johansen, 1988, 1991; Johansen and Juselius, 1990), and atest based on a single equation embodying an error–correction specification (Banerjee et al., 1986). In all cases,the specification of the testing equation was assumedknown, apart from neglected structural breaks, so that, asthe generating models were independent random walks, noadditional dynamics were incorporated. A linear trend

term was incorporated in these equations, to allow forexample for the possibility that some linear combination ofthe two series is stationary around a linear trend.

Compos et al. (1996) also consider the possibility ofstructural breaks when analysing tests of cointegration.However, their analysis concentrates on the power of thetests. The results in the present paper, on the other hand,are best viewed as an extension of Leybourne et al. (1998).These authors demonstrated that, in the presence ofneglected structural breaks, Dickey-Fuller tests can gen-erate a spurious appearance of trend-stationarity when infact the true generating process is difference-stationary –that is, integrated of order one. Here it shall be shown thatthe spurious appearance of cointegration between twoindependent difference-stationary series can arise. Theprecise nature of this spurious cointegration problemdepends both on the nature of the structural breaks andon which particular cointegration test is employed.

II . GENERATING PROCESSES ANDCOINTEGRATION TESTS

The simulations are based on independent random walks,each embodying a structural break. Specifically, series( yt, xt) were generated as

yt ¼ bdtð�yÞ þ Syt; xt ¼ bdtð�xÞ þ Sxt ð1Þ

*Corresponding author. E-mail: paul.newbold@nottingham.ac.uk1 The maximal eigenvalue variant of the Johansen test was also investigated, but the results were very similar to those of the trace test.

Applied Economics, 2003, 35, 1117–1121

Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online � 2003 Taylor & Francis Ltd 1117http://www.tandf.co.uk/journals

DOI: 10.1080/0203684032000082068

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where

Syt ¼ Syt�1 þ "yt; Sxt ¼ Sxt�1 þ "xt

with "yt and "xt mutually independent standard normallydistributed white noise processes. For series of T observa-tions, �y and �x represent the fractions of the way throughthe series that the breaks occur. Breaks in level are specifiedin Equation 1 as

dtð�yÞ ¼ 1 if t > �yT; and 0 otherwise

dtð�xÞ ¼ 1 if t > �xT; and 0 otherwise

and breaks in slope are specified as

dtð�yÞ ¼ t� �yT if t > �yT; and 0 otherwise

dtð�xÞ ¼ t� �xT if t > �xT; and 0 otherwise

Let et denote the residuals from ordinary least squaresestimation of a regression of yt on xt and a linear trend.Then the Engle–Granger test statistic is the t-ratioassociated with the least squares estimate of � in

�et ¼ �et�1 þ ut

where ut is treated as a white noise error term.Let wt ¼ ðyt; xtÞ

0. The Johansen test is then derived fromthe vector autoregressive error–correction model

�wt ¼ �þ �tþ �ðyt�1 � cxt�1Þ þ t

where c is a scalar and �, �, and � are vectors of parametersand t is taken to be vector white noise. Finally, the error–correction test of Banerjee et al. is the t-ratio associatedwith the least squares estimate of k in

�yt ¼ aþ btþ d�xt þ kðyt�1 � xt�1Þ þ fxt�1 þ t ð2Þ

where t is treated as a white noise error term.

III . SIMULATION RESULTS

To illustrate the spurious cointegration phenomenon thatcan be induced by structural breaks, series of T¼ 100observations were simulated for combinations of breakfractions ð�y; �xÞ. In the case of a break in level, b ¼ 10 inEquation 1 was set, while for a break in slope b ¼ 2 was

used for purposes of illustration.2 All simulations are basedon 10 000 replications, employing tests of nominal 5%significance levels, critical values being obtained bysimulation when the series do not contain breaks.

The simulation results are most effectively displayedgraphically. The impact of breaks in level on the three testsis illustrated in Figs 1(a)–(c). To a very large extent, theresults in Fig. 1(a) for the Engle–Granger test parallel thoseof Leybourne et al. (1998), who found very severe cases ofspurious rejections by Dickey-Fuller tests when there is avery early break in the level of a difference-stationary timeseries. Very severe spurious rejections by Engle–Grangertests occur only for very low values of �y (not for �x) – thatis, for very early breaks in level of the series that is usedas the ‘dependent variable’ in the initial Engle–Grangerregression. This is the case whatever the value of �x. Thiscould be one explanation for the frequently observedempirical phenomenon that reversing the roles of the twovariables can radically change the inference following fromEngle–Granger tests. Much less serious cases of spuriousrejections arise when �y is either a little bigger than, or alittle less than, �x.

By contrast with the Engle–Granger test, the Johansentest treats the two time series symmetrically, so that theresults of Fig. 1(b) must necessarily differ from those ofFig. 1(a). In fact, the Johansen trace test generates spuriousrejections of a serious sort if either �y or �x is very small. Inaddition, moderately serious cases occur when either �y is alittle bigger than, or a little less than, �x. Certainly, bycontrast with Fig. 1(a) these spurious rejections are a gooddeal stronger than those in the same region for the Engle–Granger test.

Figure 1(c) shows results for the error–correction testbased on fitting the regression Equation 2. In common withthe Engle–Granger test, this test also leads to severespurious rejections when �y is very low. There are also quiteserious cases when �y ¼ �x, as opposed to the other testswhere problems were found when the break fractions wereclose but not precisely equal.

The results of Fig. 1 indicate the occurrences of spuriousrejections for all three tests for cointegration, but quitenoticeable differences among the tests in both the regionswhere these rejections are strongest, and in the strength ofthe rejections. This disparity is even more pronounced inFigs 2(a)–(c), which display the results when there arebreaks in slope. Leybourne et al. (1998) found most seriousspurious rejections by Dickey-Fuller tests when a break inslope is relatively, but not extremely, early. A parallel resultis exhibited in Fig. 2(a) by the Engle–Granger test. Therejection frequency increases from �y ¼ 0, quickly peaking,

2 These values of break sizes coincide with those that led to very strong cases of spurious rejections by Dickey-Fuller tests in Leybourneet al. (1998). The apparently much ‘smaller’ break size in the trend case is in line with Leybourne and Newbold (2000), who show thatlimiting null distributions for Dickey-Fuller tests statistics arise for breaks in level proportional to T1=2 and breaks in slope proportionalto T�1=2.

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Fig. 1. (a) Engle–Granger test, T¼ 100, b¼ 10 (break in level). Empirical size at nominal 5% significance level. (b) Johansen trace test,T¼ 100, b¼ 10 (break in level). Empirical size at nominal 5% significance level. (c) Error–Correction test, T¼ 100, b¼ 10 (break in level).Empirical size at nominal 5% significance level

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Fig. 2. (a) Engle–Granger test, T¼ 100, b¼ 2 (break in slope). Empirical size at nominal 5% significance level. (b) Johansen trace test,T¼ 100, b¼ 2 (break in slope). Empirical size at nominal 5% significance level. (c) Error–Correction test, T¼ 100, b¼ 2 (break in slope).Empirical size at nominal 5% significance level.

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and then declining rapidly, so that by �y ¼ 0:3 the problemhas evaporated. Note, however, that for given values of �y,the rejection frequencies are far from uniform in �x.Nevertheless, it is primarily the value of �y that determineswhether serious spurious rejections will occur.

The picture for the Johansen test, displayed in Fig. 2(b),is considerably more complex, showing many regions ofserious-to-severe spurious rejections. Of necessity thepicture is symmetric in ð�y; �xÞ. There are serious problemswhen either break fraction is moderately small, and alsowhen the break fractions are quite close to each other. Themost severe cases occur when both �y and �x aremoderately small, though not necessarily equal.

Finally, Fig. 2(c) shows results for the error–correctiontest based on Equation 2. The region of strong spuriousrejections for relatively low �y is quite similar to that for theEngle–Granger test. However, comparing Figs 2(a) and (c)reveals an additional region of serious spurious rejectionsfor the error–correction test when �y and �x are equal, orapproximately so. Severity in this region increases as thesebreak fractions become smaller, though peaking beforeeither is close to zero.

If, on the basis of our findings, these three tests ofcointegration were to be compared only on their suscept-ibility to spuriously indicate cointegration in the presence ofstructural breaks, it might be claimed that the Engle–Granger test emerges as the ‘winner’, on the grounds that itis likely to generate the problem over a smaller region ofbreak fractions than the other two tests. Indeed, the prudentanalyst could further substantially reduce that region bycarrying out a second test in which the roles of yt and xtwere reversed. However, we believe the main message is thatany test can produce badly misleading results unless propercare is taken in the analysis to allow for structural breaks.

IV. SUMMARY

It is well known that statistical procedures based onincompletely specified models can generate spurious

inference. For example, economists have long been awarethat the appearance of a strong relationship betweenintegrated processes can arise from the fitting of regressionmodels with misspecified dynamics or error structures.We have demonstrated in this paper that spuriouscointegration can appear when breaks in level or slope ofintegrated series are ignored in the analysis. This is the casefor the three commonly used test we have investigated,though there are large differences among the results forthese tests.

REFERENCES

Banerjee, A., Dolado, J. J., Hendry, D. F. and Smith, G. W.(1986) Exploring equilibrium relationships in econometricsthrough static models: Some Monte Carlo evidence, OxfordBulletin of Economics and Statistics, 48, 253–77.

Compos, J., Ericsson, N. R. and Hendry, D. F. (1996)Cointegration tests in the presence of structural breaks,Journal of Econometrics, 70, 187–220.

Engle, R. F. and Granger, C. W. J. (1987) Cointegration anderror correction: Representation, estimation and testing,Econometrica, 55, 251–76.

Granger, C. W. J. and Newbold, P. (1974) Spurious regressions ineconometrics, Journal of Econometrics, 2, 111–20.

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

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

Johansen, S. and Juselius, K. (1990) Maximum likelihoodestimation and inference on cointegration – with applicationsto the demand for money, Oxford Bulletin of Economics andStatistics, 52, 169–210.

Leybourne, S. J., Mills, T. C. and Newbold, P. (1998) Spuriousrejections by Dickey-Fuller tests in the presence of a breakunder the null, Journal of Econometrics, 87, 191–203.

Leybourne, S. J. and Newbold, P. (2000) Behaviour of thestandard and symmetric Dickey-Fuller type tests when thereis a break under the null hypothesis, The EconometricsJournal, 3, 1–15.

Phillips, P. C. B. (1986) Understanding spurious regressions ineconometrics, Journal of Econometrics, 33, 311–40.

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