Econometrica,Vol. 58, No. 1 (January,1990), 113-144 INFERENCE IN LINEAR TIME SERIES MODELS WITH SOME UNIT ROOTS BY CHRISTOPHERA. SIMS,JAMWS H. STOCK,AND MARK W. WATSON1 This paperconsiders estimation and hypothesistesting in lineartime seriesmodels when some or all of the variableshave unit roots.Our motivating example is a vectorautoregres- sion with some unit roots in the companionmatrix, which might includepolynomials in time as regressors.In the general formulation,the variable might be integratedor cointegratedof arbitraryorders, and mighthave drifts as well. We show that parameters that can be written as coefficientson mean zero, nonintegratedregressors have jointly normal asymptotic distributions,converging at the rate T'/2. In general, the other coefficients (including the coefficientson polynomialsin time) will have nonnormal asymptoticdistributions. The resultsprovide a formal characterizationof which t or F tests-such as Grangercausality tests-will be asymptoticallyvalid, and which will have nonstandardlimiting distributions. KEYwoRDs:Cointegration, error correction models, vector autoregressions. 1. INTRODUCTION VECTOR AUTOREGRESSIONShave been used in an increasinglywide variety of econometricapplications. In this paper,we investigatethe distributionsof least squaresparameter estimators and Wald test statisticsin lineartime seriesmodels that might have unit roots. The generalmodel includesseveral important special cases. For example, all the variables could be integrated of order zero (be "stationary"),possibly around a polynomial time trend. Alternatively,all the variablescould be integratedof orderone, with the numberof unit roots in the multivariaterepresentation equaling the number of variables,so that the vari- ables have a VAR representationin first differences.Another special case is that all the variablesare integratedof the same order,but there are linear combina- tions of these variablesthat exhibit reduced orders of integration,so that the system is cointegratedin the sense of Engle and Granger(1987). In addition to VAR's, this model contains as special cases linear univariatetime series models with unit roots as studied by White (1958), Fuller (1976), Dickey and Fuller (1979), Solo (1984), Phillips(1987), and others. The model and notation are presentedin Section 2. Section 3 provides an asymptotic representationof the ordinaryleast squares(OLS) estimatorof the coefficients in a regression model with "canonical" regressors that are a linear transformation of Y, the original regressors. An implication of this result is that the OLS estimator is consistent whether or not the VAR contains integrated components,as long as the innovationsin the VAR have enoughmoments and a 1 The authors thank Lars Peter Hansen, Hal White, and an anonymousreferee for helpful comments on an earlier draft. This researchwas supportedin part by the National Science Foundationthrough Grants SES-83-09329, SES-84-08797, SES-85-10289, and SES-86-18984.This is a revisedversion of two earlierpapers, "Asymptotic Normality of Coefficientsin a VectorAutoregres- sion with Unit Roots," March,1986, by Sims, and "WaldTests of LinearRestrictions in a Vector Autoregressionwith Unit Roots,"June, 1986, by Stockand Watson. 113 114 CHRISTOPHER A. SIMS, JAMES H. STOCK AND MARK W. WATSON zero mean, conditionalon past values of Y,. When an interceptis includedin a regressionbased on the canonicalvariables, the distributionof coefficientson the stationary canonical variates with mean zero is asymptoticallynormal with the usual covariancematrix, converging to its limit at the rate T1V2.In contrast, the estimated coefficients on nonstationarystochastic canonical variates are nonnormally distributed,converging at a faster rate. These results imply that estimators of coefficientsin the original untransformedmodel have a joint nondegenerateasymptotic normal distribution if the model can be rewrittenso that these original coefficientscorrespond in the transformedmodel to coeffi- cients on mean zero stationarycanonical regressors. The limiting distributionof the Wald F statisticis obtainedin Section 4. In general, the distributionof this statisticdoes not have a simple form. When all the restrictionsbeing tested in the untransformedmodel correspondto restric- tions on the coefficientsof mean zero stationarycanonical regressorsin the transformedmodel, then the test statistichas the usual limiting x2 distribution. In contrast,when the restrictionscannot be writtensolely in termsof coefficients on mean zero stationarycanonical regressors and at least one of the canonical variatesis dominatedby a stochastictrend, then the test statistichas a limiting representationinvolving functionals of a multivariateWiener process and in generalhas a nonstandardasymptotic distribution. As a special case, the resultsapply to a VAR with some roots equal to one but with fewer unit roots than variables,a case that has recentlycome to the fore as the class of cointegratedVAR models. Engle and Granger(1987) have pointed out that such models can be handled with a two-step procedure,in which the cointegratingvector is estimatedfirst and used to form a reduced, stationary model. The asymptoticdistribution theory for the reduced model is as if the cointegratingvector were known exactly. One implicationof our results is that such two-step proceduresare unnecessary,at least asymptotically:if the VAR is estimated on the originaldata, the asymptoticdistribution for the coefficients normalizedby T1l2 is a singularnormal and is identicalto that for a model in which the cointegratingvector is known exactlya priori. This result is important because the two-stepprocedures have so far beenjustified only by assumingthat the number of cointegratingvectors is known. This paper shows that, at a minimum,as long as one is not interestedin drawinginferences about intercepts or about linear combinationsof coefficientsthat have degeneratelimiting distri- butions when normalizedby T1l2, it is possible to avoid such two-step proce- dures in large samples. However,when there are unit roots in the VAR, the coefficientson any interceptsor polynomialsin time includedin the regression and their associatedt statisticswill typicallyhave nonstandardlimiting distribu- tions. In Sections 5 and 6, these general results are applied to several examples. Section 5 considersa univariateAR(2) with a unit root with and withouta drift; the Dickey-Fuller(1979) tests for a unit root in these models follow directlyfrom the more generalresults. Section 6 examinestwo common tests of linearrestric- tions performedin VAR's:a test for the numberof lags that enter the true VAR TIME SERIESMODELS 115 and a "causality"or predictabilitytest that laggedvalues of one variabledo not enter the equation for a second variable.These examples are developed for a trivariatesystem of integratedvariables with drift.In the test for lag length,the F test has a chi-squaredasymptotic distribution with the usual degreesof freedom. In the causalitytest, the statistichas a x2 asymptoticdistribution if the processis cointegrated;otherwise, its asymptoticdistribution is nonstandardand must be computednumerically. Some conclusionsare summarizedin Section7. 2. THE MODEL We consider linear time seriesmodels that can be writtenin first order form, (2.1) Yt= AYt- + G l212t (t = l9. T)q where Yt is a k-dimensionaltime series variable and A is a k x k matrix of coefficients.The N x 1 vector of disturbances{ t)} is assumedto be a sequence of martingaledifferences with E t = 0 and E[qtq ' ---, = IN for t = 1, .. ., T. The N x N matrix Q1/2 is thoughtof as the squareroot of the covariancematrix of some "structural"errors Q1/2 qt- The k X N constantmatrix G is thought of as known a priori, and typically contains ones and zeros indicatingwhich errorsenter which equations. Note that becauseN mightbe less than k, some of the elementsof Y, (or more generally,some linearcombinations of Yt) might be nonrandom.It is assumed that A has k1 eigenvalueswith modulus less than one and that the remainingk - k1 eigenvaluesexactly equal one. As is shown below, this formulationis sufficientlygeneral to includea VAR of arbitraryfinite orderwith arbitraryorders of integration,constants and finite order polynomials in t. The assumptionsdo not, however,allow complex unit roots so, for example,seasonal nonstationarity is not treated. The regressors Yt will in general consist of random variableswith various orders of integration,of constants,and of polynomialsin time. These compo- nents in general are of differentorders in t. Often there will be linear combina- tions of Yt having a lowerorder in probabilitythan the individualelements of Yt itself. ExtendingEngle and Granger's(1987) terminology,we referto the vectors that form these linear combinations as generalized cointegrating vectors. As long as the system has some generalizedcointegrating vectors, the calculationsbelow demonstratethat T-PYYY' will convergeto a singular(possibly random) limit, where p is a suitablychosen constant;that is, some elementsof Yt will exhibit perfect multicolinearity,at least asymptotically.Thus we workwith a transforma- tion of Yt,say Zt, that uses the generalizedcointegrating vectors of Y, to isolate those componentshaving different orders in probability.Specifically, let (2.2) Z,= DYt. (Note that in the datingconvention of (2.1) the actualregressors are Yt> or, after transformingby D, Zt-1.) The nonsingulark x k matrix D is chosen in such a way that Zt has a simple representationin terms of the fundamentalstochastic