This paper formulates a conformity test for cointegration in the context of a VAR specification for a multivariate I(1) process. The test statistic is a function of the characteristic roots of the sample covariance matrix of a linear transformation of the cointegral vector; the latter is obtained from unrestricted estimator of the underlying parameters of the VAR. It is further shown that this test procedure is also applicable to the case where the I(1) process is a MIMA(k), i.e. a multivariate integrated moving average process, the moving average being of order k < ∞. The test statistic, under the null of cointegration, has a normal limiting distribution

Dhrymes, Phoebus J.

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A Conformity Test for Cointegration

A Conformity Test for CointegrationbyPhoebus Dhrymes, Columbia UniversityOctober 19951994-95 Discussion Paper Series No. 750VA Conformity Test for CointegrationPHOEBUS J. DHRYMESColumbia UniversityOctober 9, 1995AbstractThis paper formulates a conformity test for cointegration inthe context of a VAR specification for a multivariate /(I) process.The test statistic is a function of the characteristic roots of thesample covariance matrix of a linear transformation of the cointe-gral vector; the latter is obtained from the unrestricted estimatorof the underlying parameters of the VAR. It is further shown thatthis test procedure is also applicable to the case where the 7(1)process is a MIMA(k), i.e. a multivariate integrated movingaverage process, the moving average being of order k < oo .The test statistic, under the null of cointegration, has a normallimiting distribution.Key Words: Cointegration, Cointegration test, characteristicroots, VAR, MIMA(k).1 Introduction and SummaryLet {Xt. : t £ J\f] be a stochastic sequence defined on some probabilityspace ( f i , A, V). If the sequence is taken to be / ( I ) , in the sensethat {(/ — L)Xt. : t 6 AT] is strictly stationary, the question oftenarises as to whether the sequence in question is cointegrated. The lattermeans, in this context that there exists a q x r matrix B of rank r < qsuch that Xt.B is (strictly) stationary. A number of tests have beenproposed in the literature, some formal some informal. The somewhatinformal tests involve running a regression of one of the elements onthe others and using the Dickey-Fuller test, Dickey and Fuller (1979),(1981), to test the hypothesis of cointegration. A more formal test isgiven in Phillips and Ouliaris (1990) and a set of such tests are given inJohansen (1988), (1991), to mention but a few. All such tests employ anindirect approach in that they explore an implication of the cointegrationhypothesis beyond the property that constitutes its definition.In this paper we explore a conformity test for cointegration, and givethe limiting distribution of the test statistic. We place the discussionin the VAR context of cointegration popularized by Johansen as notedabove, but the results are equally applicable to contexts that are less con-strained. Finally, the test statistic is shown to be asymptotically normaland thus tests for cointegration may be carried out in standard fashion,in contrast with other procedures that require special tabulations.2 Notation and Problem FormulationConsider the standard VAR3=0where Xt, is a q -element row vector, the error process being a MW N{T,),i.e. a multivariate white noise process with mean zero and covariance ma-trix £ > 0; normality is not necessary, as in the case of Johansen (1988),(1991)."Dividing" n(L) by (/ — L), where L is the usual lag operator wefind, after some rearrangement,(/ - L)Xt. = -X<-i.n(l) + xt.U* + e,., (2)wherext. = (AXt.1.,AXt,2.,...,AXt.n+1.), i = l , 2 , . . .T ,n- = (n;',...,^'.,)', n-= £ n- W) = itn>-i=j + l .7=0If the process is cointegrated of rank r then 11(1) is of (reduced) rankr < q. Hence, by the rank factorization theorem, see Dhrymes (1984)p. 23, there exist matrices F, B both of dimension q x r and rank rsuch that 11(1) = BT . A conformity test for cointegration consists ofestimating 11(1) without this restriction, say 11(1), and then formingthe sample covariance matrixn ( i ) p : i p _ 1 n ( i ) , p_1 = (x t_1 . ) , * = i , 2 , . . . r . (3)If the process is cointegrated of rank r then the rank of the limit ofM must be r ; if this hypothesis is rejected for any r < q then thehypothesis of coitengration is rejected.2To implement this procedure obtain the least squares estimatorN = Iq- X(X'X)~lX\ U = (6,.). (4)Define,+A1 2iAl2 = ~f22 T { r j ( ¥ ) () )( ( ( ( (T \ T 2 ) \ T ) \ T2 j \ T j \ T2and note thatd-> 0, andA/,-M^A/,-M, M = M,,, n ( 1 ) / p y- l I I ( 1 ) ^ M22. (6)Hence, the ordered characteristic roots of M converge to the (ordered)characteristic roots of M — Mzz , and it is evident that, as expected,the number of zero roots of 11(1), and thus of M , corresponds to thenumber of unit roots of |n(z)| = 0, and its rank corresponds to thecointegration rank.To examine the limiting distribution of such roots, we first note thatAi2, A22 both converge to zero at the rate of Ta , for a E [0,1). Thus,the limiting distribution of the roots depends entirely on the first term,and we obtainVT(M- MZZ) ~ VT [(^na/pl^na) - M«)] , (7)owing to the fact that\/TAu -i 0, VTA22 -i 0. (8)We note that- VTMZZ = 4= £ (Vi^-i . - Mzx) , (9)t=iwhere zt~\. = X i_1.II(l), and[z\zt. - M 2 z : t eis a zero mean strictly stationary process. If the MWN{Yi) that definesthe VAR(n) model is assumed in this context to be normal, the entitiesof Eq. (9) obeyE || z't.zt. - Mzz \\2+a< oc, a > 0. (10)If normality is not assumed the moment condition above needs to beassumed explicitly. It follows, from Corollary la in Chapter 5, Dhrymes(1995) that1 Tt-i.zt-i. - Mzz)t=iobeys a CLT. Consequently, its limiting distribution is given by,tf*), where V* = f\f(\)® f(\)]d\, (11)Joand / denotes the spectral matrix of the cointegral vector zt.. Thederivation of this particular covariance matrix may be found in Hannan(1970), p. 228.We have therefore provedTheorem 1. In the context of the discussion above, consider the unre-stricted estimator 11(1) and the matrixM = -]The followinmg statements are true:i. M —> M = Mzz, and thus in probability as well, by Proposition40, p. 263 in Dhrymes (1989);ii. vec [VT(M - M)\ - i N(0, #*), where ^* is as defined in Eq. (11)and / is the spectral matrix of the cointegral vector.We are now in a position to formulate a (conformity) test based on theresult above. Evidently, we have cointegration if and only if rank (M) <q. But this means that some of the characteristic roots of M must benull. More precisely, if there is cointegration of rank r then q — r of thecharacterstic roots of M must be zero. Therefore, to devise a test weneed to obtain the limiting distribution of such roots, a result that is notavailable in the literature. To this end, we haveTheorem 2. In the context of Theorem 1, the (ordered) characteristicroots of M, contained in the diagonal matrix A obeyy/T(k - A) ~ d*[y/TQ\M - M)Q],where the notation d*[A] indicates the diagonal elements of the ma-trix A, and Q is the orthogonal matrix of the decomposition M —QAQ'.Proof: Since M and M are at least positive semidefinite by Proposition52 in Dhrymes (1984), pp. 61-62, they have the (orthogonal) decompo-sitionM = QAQ\ M = QAQ'. (12)Moreover, by the results of Theorem 1, Proposition 28, Corollary 5, pp.242-244 in Dhrymes (1989), Q, A converge, respectively, to Q and A;in addition, y/T(Q — Q), and VT(A — A) have well denned limitingdistributions. Next, considerVT[Q'(M - M)Q] = VT(M* - A), M* = Q'QkQ'Q, (13)and putVf(Q'Q - Iq) = C, VT(A - A) = D, VT(M* -A) = G. (14)Note that by Theorem 1, C , D , G , are all a.c. finite random variables,in the sense that they have well denned limiting distributions. It followsimmediately thatVT{Mm - A) = G ~ D + AC - CA. (15)Since9 a = d a , gij = (\t - \ j ) c i h o r c%3 = # j / ( A t - - A j ) , i ^ j , ( 1 6 )teristic roots are different so that the elements c,-j are well defined. Wethis concludes the proof of the theorem, for the case where all charac-now examine the case whereLl 0A~ ' 0 O J 'where Aj is a diagonal matrix containing the (r ) positive roots underthe null of cointegration. Partitioning the other matrices conformablywe determine, in this caseConsequently, we have againgu = da, i = 1,2,..., q, gi3 = (At- - \3)ct3, i, j = 1,2,..., r, i ^ j ;gij = At-ct-j, i = 1, 2,. . ., r, j = r + 1, r + 2,. . . , q;= \3ctJ, i = r + l , r + 2 , . . . , ? , z = 1,2,. .. , r;= 0, z , j = r + l , r + 2 , . . . ,g , zVi - (18)q.e.d.Corollary 1. The characteristic roots of M , and hence their limitingdistributions, are exactly those of M* .Proof: Obvious since Q is a fixed orthogonal matrix.Corollary 2. The distribution of the (associated) characteristic vectorsis given by the distribution of QC .Corollary 3. A test on the rank of cointegration may be carried out asfollows: let A(2), i = 1,2,... ,q be the characteristic roots of- M\ = 0,arranged in decreasing order, and let it be desired to test the hypoth-esisHo : the rank of cointegration is ras against the alternativeHi : the rank of cointegration is r -f s < q .Consider the entityI A A Ar * = (A(r+i), A(r+2), • • • i A(r+S));the characteristic roots contained therein are, under Ho,where ^* is the submatrix of ^* corresponding to the roots in r* .Thus,r'*tf*-V ~ xl- (19)Corollary 4. For s = 1 , we have the simple rank test Ho : rank(M) =r, as against the alternative Hi : rank(M) = r-f 1. For s = q — r wehave the test for the existence of cointegration, i.e. Ho : rank(M) = r,as against the alternative Hi : rank(M) = q.Remark 1. The major computational burden entailed by this procedureis the estimation of the spectral matrix of the (estimated) cointegralvector zt. = X<_REFERENCESDhrymes, P. J. (1984), Mathematics for Econometrics, second edition,New York: Springer Verlag.Dhrymes, P. J. (1989), Topics in Advanced Econometrics: vol. I, Proba-bility Foundations, New York: Springer Verlag.Dhrymes, P. J. (1995), Topics in Advanced Econometrics: vol. Ill, Topicsin Time Series, unpublished.Dickey, D. A. and W. A. Fuller (1979), "Distribution of the Estimators forAutoregressive Time Series with a Unit Root", Journal of the AmericanStatistical Association, vol. 74, pp. 42-431.Dickey, D. A. and W. A. Fuller (1981), "Likelihood Ratio Statistics forAutoregressive Time Series with a Unit Root", Econometrica, vol.49, pp.1057-1072.Hannan, E. J. (1970), Multiple Time Series, New York: Wiley.Johansen, S. (1988), "Statistical Analysis of Cointegration Vectors", Jour-nal of Economic Dynamics and Control, vol. 12, pp. 231-254.Johansen, S. (1991), "Estimation and Hypothesis Testing of Cointegra-tion Vectors in Gaussian Vector Autoregressive Models", Econometrica,vol. 59, pp. 1551-1580.Phillips, P.C.B. and S. Ouliaris (1990), "Asymptotic Properties of Resid-ual Based Tests for Cointegration", Econometrica, vol. 58, pp. 165-93.1994-1995 Discussion Paper SeriesDepartment of EconomicsColumbia University1022 International Affairs Bldg.420 West 118th StreetNew York, N.Y., 10027The following papers are published in the 1994-95 Columbia University Discussion Paper serieswhich runs from early November to October 31 (Academic Year). Domestic orders for discussionpapers are available for purchase at $8.00 (US) each and $140.00 (US) for the series. Foreign orderscost $10.00 (US) for individual paper and $185.00 for the series. To order discussion papers, pleasesend your check or money order payable to Department of Economics, Columbia University to theabove address. 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