Testing for Cointegration When Some of The

EconometricTheory, 11, 1995, 984-1014. Printed in the United States of America. TESTINGFOR COINTEGRATION WHEN SOME OF THE COINTEGRATINGVECTORS ARE PRESPECIFIED MICHAELT.K. HORVATH Stanford University MARKW. WATSON Princeton University Manyeconomic models imply that ratios, simpledifferences, or "spreads"of variablesare I(O).In these models, cointegratingvectors are composedof l's, O's,and - l's and containno unknownparameters. In this paper,we develop tests for cointegrationthat can be appliedwhen some of the cointegratingvectors are prespecifiedunder the null or underthe alternativehypotheses. These tests are constructedin a vectorerror correction model and are motivatedas Waldtests from a Gaussianversion of the model. Whenall of the cointegrating vectorsare prespecifiedunder the alternative,the tests correspondto the standardWald tests for the inclusionof errorcorrection terms in the VAR. Modificationsof this basictest are developedwhen a subsetof the cointegrating vectorscontain unknown parameters. The asymptoticnull distributionsof the statisticsare derived,critical values are determined,and the local power propertiesof the test are studied.Finally, the test is appliedto data on for- eign exchangefuture and spot pricesto test the stabilityof the forward-spot premium. 1. INTRODUCTION Economic models often imply that variables are cointegrated with simple and known cointegrating vectors. Examples include the neoclassical growth model, which implies that income, consumption, investment, and the capi- tal stock will grow in a balanced way, so that any stochastic growth in one of the series must be matched by corresponding growth in the others. Asset This paper has benefited from helpful comments by Neil Ericsson, Gordon Kemp, Andrew Levin, Soren Johansen, John McDermott, Pierre Perron, Peter Phillips, James Stock, and three referees. This research was supported by grants from the National Science Foundation (SES-91-22463 and SBR-94-09629) and the Sloan Foundation. Address correspondence to: Professor Mark Watson, Woodrow Wilson School, Prince- ton University, Princeton, NJ 08540, USA. 984 ? 1995 Cambridge University Press 0266-4666/95 $9.00 + .10 TESTINGWITH PRESPECIFIED COINTEGRATING VECTORS 985 pricingmodels with stablerisk premia imply corresponding stable differences in spot and forwardprices, long- and short-terminterest rates, and the logarithmsof stock pricesand dividends.Most theoriesof internationaltrade implylong-run purchasing power parity, so that long-runmovements in nom- inal exchangerates are matchedby countries'relative price levels. Certain monetaristpropositions are centeredaround the stabilityof velocity,imply- ing cointegrationamong the logarithmsof money, prices,and income.Each of thesetheories has two distinctimplications for the propertiesof economic time series under study: first, the series are cointegrated,and second, the cointegratingvector takes on a specificvalue. For example,balanced growth impliesthat the logarithmsof incomeand consumptionare cointegratedand that the cointegratingvector takes on the value of (1 -1). The most widelyused approachto testingthese cointegration propositions is articulatedand implementedin Johansenand Juselius(1992), who inves- tigate the empiricalsupport for long-run purchasingpower parity. They implementa two-stagetesting procedure. In the first stage, the null hypothesis of no cointegrationis testedagainst the alternativethat the data arecoin- tegratedwith an unknowncointegrating vector using Johansen's(1988) test for cointegration.If the null hypothesisis rejected,a second stage test is implementedwith cointegrationmaintained under both the null and alternative. The null hypothesisis that the data are cointegratedwith the specific cointegratingvector implied by the relevanteconomic theory ([1 -1] in the consumption-incomeexample), and the alternativeis that data are cointegrated with anotherunspecified cointegrating vector. Becausea consistent test for cointegrationis used in the first stage, potentialcointegration in the data is found with probabilityapproaching 1 in large samples. Thus, the probabilityof rejectingthe cointegrationconstraints on the data imposedby the economicmodel are given by the size of the test in the second step, at least in largesamples. An importantstrength of this procedureis that it can uncovercointegration in the data with a cointegratingvector different from the cointegratingvector imposed by the theory.The disadvantageis that the samplesizes used in economicsare often relativelysmall, so that the first- stage tests may have low power. This paperdiscusses an alternativeprocedure in whichthe null of no cointegrationis testedagainst the compositealternative of cointegrationusing a prespecifiedcointegrating vector. This approachhas two advantages.First, and most important,the resultingtest for cointegrationis significantlymore powerfulthan the test that does not impose the cointegratingvector. For example,in the bivariateexample analyzed in Section 3, these power gains correspondto samplesize increasesranging from 40 to 70Wofor a test with power equal to 5Oo. The second advantageis that the test statisticis very easy to calculate:it is the standardWald test for the presenceof the candi- date errorcorrection terms in the first differencevector autoregression. The countervailingdisadvantage of the testing approachis that it does not sep- 986 MICHAELT.K. HORVATHAND MARK W. WATSON aratethe two componentsof the alternativehypothesis and, thus, may fail to rejectthe null of no cointegrationwhen the data are cointegratedwith a cointegratingvector different from that used to constructthe test. We inves- tigate this in Section 3, whereit is shown that in situationswith weak cointegration(represented by a local-to-unityerror correction term) even inexact informationon the valueof the cointegratingvector often leadsto powerim- provementsover the test that uses no information.If the null hypothesisof noncointegrationis rejected,one can then determinewhether the prespecified cointegratingvector differs significantlyfrom the true cointegratingvector. The plan of this paperis as follows. In Section2, we considerthe general problemof testingfor cointegrationin a modelin whichsome of the potential cointegratingvectors are known,and some are unknown,under both the null and the alternative.In particular,we presentWald and likelihoodratio tests for the hypothesisthat the data are cointegratedwith rOk known and r0u unknown cointegratingvectors under the null. Under the alternative, there are rak and rauadditional known and unknowncointegrating vectors, respectively.The tests are constructedin the contextof a finite-orderGauss- ian vector error correctionmodel (VECM) and generalizethe procedures of Johansen (1988), who consideredthe hypothesistesting problemwith rOk = rak = 0. In Section 2, we also derive the asymptotic null distributions of the test statisticsand tabulate criticalvalues. Section 3 focuses on the powerproperties of the test. First, we presentcomparisons of the powerof likelihood-basedtests that do and do not use informationabout the value of the cointegratingvector. Next, becauseinformation about the potential cointegratingvector mightbe inexact, we investigatethe powerloss associ- ated with usingan incorrectvalue of the cointegratingvector. Finally, when thereare no cointegratingvectors under the null and only one cointegrating vectorunder the alternative,simple univariate unit root testsprovide an alter- nativeto the multivariateVECM-based tests. Section3 comparesthe power of theseunivariate unit root tests to the multivariateVECM-based tests. Sec- tion 4 containsan empiricalapplication that investigatesthe forwardpremia in foreignexchange markets by examiningthe cointegrationproperties of forward and spot prices. Section 5 containssome concludingremarks. 2. TESTING FOR COINTEGRATIONIN THE GAUSSIAN VAR MODEL As in Johansen(1988), we derivetests for cointegrationin the contextof the reducedrank GaussianVAR: Yt= dt + Xt, (2.1a) p Xt = E HiXt-i + Et, (2.1b) i-l1 TESTING WITH PRESPECIFIEDCOINTEGRATING VECTORS 987 where Y, is an n x 1 data vector from a sample of size T, dt represents deter- ministic drift in Y,, Xt is an n x 1 random vector generated by (2. lb), (, is NIID(0,Ee), and, for convenience, the initial conditions X_i, i = 0,. ..p, are assumed to equal 0. To focus attention on the long-run behavior of the process, it is useful to rewrite (2.lb) as p-i AXt = llX,_1 + E 4AXt-i + ,t, (2.1c) i=l where 1 = -In + I'1Hi. Our interest is focused on r = rank(IH), and we consider tests of the hypothesis Ho: rank(I) = r = r, Ha:rank(II) = r = rO + r, with r, > 0. The alternative is written so that ra represents the number of additional cointegrating vectors that are present under the alternative. We assume that ro = rOk + ro, where rOk is the number of cointegrating vectors that are known under the null and ro, represents the number of cointegrating vectors that are unknown (or, alternatively, unrestricted) under the null. Similarly, ra = rak+ ra, where the subscripts k and u denote known and unknown, respectively. The rOk prespecified vectors are thought to be cointegrating vectors under the alternative; under the null, they do not cointegrate the series. In spite of this, for expositional ease, they will be referred to as cointegrating vectors. As in Engle and Granger (1987), Johansen (1988), and Ahn and Reinsel (1990), it is convenient to write the model in vector error correction form by factoring the matrix HIas

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