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Testing Linear Restrictions on Cointegration Vectors: Sizes and Powers of Wald Tests in Finite Samples

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Testing Linear Restrictions on Cointegration Vectors: Sizes and Powers of Wald Tests in Finite Samples A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Haug, Alfred A. Working Paper Testing linear restrictions on cointegration vectors: Sizes and powers of Wald tests in finite samples Technical Report, No. 1999,04 Provided in Cooperation with: Collaborative Research Center 'Reduction of Complexity in Multivariate Data Structures' (SFB 475), University of Dortmund Suggested Citation: Haug, Alfred A. (1999) : Testing linear restrictions on cointegration vectors: Sizes and powers of Wald tests in finite samples, Technical Report, No. 1999,04, Universität Dortmund, Sonderforschungsbereich 475 - Komplexitätsreduktion in Multivariaten Datenstrukturen, Dortmund This Version is available at: http://hdl.handle.net/10419/77134 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Testing linear restrictions on cointegrating vectors Sizes and powers of Wald tests in nite samples Alfred A Haug University of Canterbury and York University The Wald test for linear restrictions on cointegrating vectors is compared in nite samples using the Monte Carlo metho d The Wald test within the vector error correction based metho ds of Bewley et al and of Johansen the canon ical cointegration metho d of Park the dynamic ordinary least squares metho d of Phillips and Loretan Saikkonen and Sto ck and Watson the fully mo died ordinary least squares metho d of Phillips and Hansen and the band sp ectral techniques of Phillips are considered In terms of test size Jo hansens metho d seems to be preferred and in terms of test power it is Parks and Phillips However the relatively poor results in the context of cointegrating regres sions suggest that improvements on the p erformance of the Wald tests considered here are needed Running title Performance of Wald tests in nite samples Address Department of Economics UniversityofCanterbury Private Bag Christchurch New Zealand Email ahaugeconcanterburyacnz Phone Financial supp ort from the So cial Sciences and Humanities Research Council of Canada under grant is gratefully acknowledged The author thanks without implicating participants at the Canadian Economics Asso ciation Meetings the Econometric So ciety Australasian Meetings the New Zealand Econometrics Study Group Meetings and the Workshop on Fractional Intergration and Cointegration at the UniversityofDortmund for helpful comments on an earlier draft Intro duction Cointegration techniques have b een applied widely in empirical economics in recent years Numerous tests for cointegration and estimation metho ds for cointegrating vectors have b een suggested in the literature Almost all results are based on asymp totic theory and the p erformance in nite samples can dier substantially across tests and estimation metho ds even though metho ds might be asymptotically equivalent and ecient Cheung and Lai Gregory To da and Haug among others provided Monte Carlo comparisons of size distortions and of p owers for various tests for cointegration Sto ck and Watson Gonzalo Kitamura and Phillips and Ho and Sorensen among others compared with the Monte Carlo metho d the p erformance of estimators in terms of eg bias in median by the interquartile range and disp ersion as measured The purp ose of this pap er is to study the p erformance in nite samples of tests for parameter restrictions on cointegrating vectors The Monte Carlo metho d is employed for these purp oses Testing hyp otheses suggested by economic theory is a central concern of econometrics and testing hyp otheses ab out restrictions on parameters in cointegrating vectors is no exception The goal is to apply tests that have close to correct size and high p ower Wald tests have b een prop osed for testing linear restrictions on coin tegrating vectors for dierent though asymptotically equivalent estimation metho ds This Monte Carlo analysis studies the eects of varying the estimation technique on cal culating the Wald test The Wald test statistics are distributed as under the null hyp othesis and reduce to a t statistic when only one cointegrating vector is present and only a single parameter is involved The t statistic is then distributed asymptot ically as normal The asymptotically ecient estimation metho ds considered for the Wald or t tests in this pap er are in alphab etical order of the chosen abbreviations Bewley et als BoxTiao canonical variates based metho d BWLY Parks canon ical cointegration regression metho d CCR Phillips and Loretan Saikko nen and Sto ck and Watsons dynamic ordinary least squares metho d DOLS Phillips and Hansens fully mo died ordinary least squares metho d FM Johansens maximum likeliho o d metho d JOH and Phillips band sp ectral regression metho ds PH The most p opular metho d in empir ical applications seems to be JOH Less often used are CCR DOLS FM and PH Other metho ds have been suggested in the literature BWLY has been prop osed it may outp erform JOH point more recently and is included in this study b ecause estimates in certain cases as demonstrated by Bewley et al The ab ove metho ds are applied to several data generating pro cesses DGPs of practical relevance The Wald or t statistic for a linear restriction on the cointegrating vector is computed from the parameter and variance estimates of each metho d Then empirical sizes and powers of these tests are calculated and compared The Monte Carlo metho d is used in connection with a DGP that allows for endogenous weakly exogenous and sense of Engle et al strongly exogenous regressors in the In previous research Sto ckandWatson compared nite sample critical values of the t statistic for parameter restrictions on cointegrating vectors of ve of the six metho ds considered ab ove Their DGP revealed relatively mo dest size distortions Further Li and Maddala suggested to use the moving blo ck b o otstrap to correct size distortions for the t statistic for three of the ab ove six metho ds However these studies did not rep ort results on test powers of the t tests On the other hand Inder rep orted results for p owers of t tests for one of the ab ove metho ds FM and other metho ds not considered in my pap er His preferred choice was atwostage metho d combining an errorcorrection regression with the FM metho d arious estimation metho ds used in the Monte Section briey outlines the v Carlo study In Section the Monte Carlo design is explained and results are dis cussed Section concludes The Wald test in cointegrated systems The BoxTiao Metho d of Bewley et al BWLY Bossaerts and Bewley in several pap ers suggested a metho d for coin tegrated systems of equations based on the levels canonical correlation analysis sug gested byBox and Tiao This is in contrast to Johansens well known metho d which relates levels to rst dierences and do es therefore incorp orate information on the presence of unit ro ots into the estimation Bewley et al used the Monte Carlo metho d to compare Johansens estimators to theirs and found for a bivari ant cases less disp ersed ate rstorder mo del that their estimator is in several relev and lepto curtic in small samples than Johansens Gonzalo derived for the bivariate rst order mo del the asymptotic distribution of Bewleys BoxTiao estimator The distribution is nonsymmetric and nonstandard Also it includes terms that lead to nitesample bias in the median Despite these asymptotic problems hyp othesis tests on cointegrating vectors in small samples with this metho d may outp erform those with Johansens metho d parallel to the ndings of Bewley et al for prop erties of the two estimators Following Yang and Bewley consider a pdimensional vector autoregressive representation of order k for the cointegrating relationship Y Y v t T Y Y t k tk t t t with v distributed I IN Y is a p vector of variables integrated of order one t t denoted by I and is a vector of constants is the rst dierence op erator and is a p p matrix It is assumed that r p Then is a full rank p r matrix i of errorcorrection vectors and is a full rank p r matrix of r cointegrating vectors such that Y is integrated of order zero denoted I t See Bewley and Orden Bewley et al Bewley and Yang and Yang and Bewley The last two pap ers describ e cointegration tests within this system See Phillips and Sto ck and Watson for a theoretical and an empirical study resp ectively for Johansens metho d Phillips results also apply to Bewleys BoxTiao estimator The mo died BoxTiao pro cedure
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