STATISTICAL ANALYSIS OF COINTEGRATION VECTORS (by Soren JOHANSEN)

Journal of Economic Dynamics and Control 12 (1988) 231-254. North-Holland STATISTICAL ANALYSIS OF COINTEGRATION VECTORS Soren JOHANSEN* University of Copenhagen, DK-2100 Copenhagen, Denmark Received September 1987, final version received January 1988 We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimensions. Further we test linear hypotheses about the cointegration vectors. The asymptotic distribution of these test statistics are found and the first is described by a natural multivariate version of the usual test for unit root in an autoregressive process, and the other is a x2 test. 1. Introduction The idea of using cointegration vectors in the study of nonstationary time series comes from the work of Granger (1981), Granger and Weiss (1983), Granger and Engle (1985), and Engle and Granger (1987). The connection with error correcting models has been investigated by a number of authors; see Davidson (1986), Stock (1987), and Johansen (1988) among others. Granger and Engle (1987) suggest estimating the cointegration relations using regression, and these estimators have been investigated by Stock (1987), Phillips (1985), Phillips and Durlauf (1986), Phillips and Park (1986a, b, 1987), Phillips and Ouliaris (1986,1987), Stock and Watson (1987), and Sims, Stock and Watson (1986). The purpose of this paper is to derive maximum likelihood estimators of the cointegration vectors for an autoregressive process with independent Gaussian errors, and to derive a likelihood ratio test for the hypothesis that there is a given number of these. A similar approach has been taken by Ahn and Reinsel (1987). This program will not only give good estimates and test statistics in the Gaussian case, but will also yield estimators and tests, the properties of which can be investigated under various other assumptions about the underlying data generating process. The reason for expecting the estimators to behave better *The simulations were carefully performed by Marc Andersen with the support of the Danish Social Science Research Council. The author is very grateful to the referee whose critique of the first version greatly helped improve the presentation. 0165-1889/88/$3,5001988, Elsevier Science Publishers B.V. (North-Holland) J.E.D.C.--B 232 S. Johansen, Statistical analysis of cointegration vectors than the regression estimates is that they take into account the error structure of the underlying process, which the regression estimates do not. The processes we shall consider are defined from a sequence {E,} of i.i.d. p-dimensional Gaussian random variables with mean zero and variance matrix A. We shall define the process X, by x,=n,x,_,+ ... +t~X,_,+E,, t=1,2 >.‘., (1) for given values of X_ k + i, . . , X0. We shall work in the conditional distribu- tion given the starting values, since we shall allow the process X, to be nonstationary. We define the matrix polynomial A(z)=I-qz- ... -n,z”, and we shall be concerned with the situation where the determinant IA(z)/ has roots at z = 1. The general structure of such processes and the relation to error correction models was studied in the above references. We shall in this paper mainly consider a very simple case where X, is integrated of order 1, such that AX, is stationary, and where the impact matrix A(z)l,=,=II=I-II,- ... -nk has rank r . - * > ip of SkOS&,?!$,, with respect to Skk, i.e., the solutions to the equation Ixs,, - S,,S&‘S,,,] = O, (9) and E the matrix of the corresponding eigenvectors, then S,,ED = S,,&&,,E, where E is normalised such that E’S,, E = I. Now choose p = Et where E is p X r, then we shall minimise IE’E - E’DEI/IE’EI. This can be accomplished by choosing E to be the first r unit vectors or by choosing B to be the first r eigenvectors of S,#!&‘SOk with reSpeCt to Skk, i.e., the first r columns of E. These are called the canonical variates and the eigenvalues are the squared canonical correlations of R, with respect to R,. For the details of these calculations the reader is referred to Anderson (1984, ch. 12). This type of analysis is also called reduced rank regression [see Ahn and Reinsel (1987) and Velu, Reinsel and Wichem (1986)]. Note that all possible choices of the optimal p can be found from B by j3 = /$I for p an r X r matrix of full rank. The eigenvectors are normalised by the condition B’SkkB = Z such that the estimates of the other parameters are given by & = - S,,,& ,&S,&& -’ = -S,,,& which clearly depends on the choice of the optimising p, whereas (10) ‘= -s,,fi(,@k&-lfj,= -SOk,j/ftr (11) 236 S. Johansen, Statistical analysis of cointegration vectors (12) L,Z’= IS,l fJ (1 -A,>, 03) r=l do not depend on the choice of optimising p. With these results it is easy to find the estimates of II and A without the constraint (3). These follow from (6) and (7) for r =p and /I = I and give in particular the maximised likelihood function without the constraint (3): L-2’T= ,S,, JJ (1 -A,>. max 04) If we now want a test that there are at most r cointegrating vectors, then the likelihood ratio test statistic is the rftio of (13) and (14) which can be expressed as (4) where ir+l > . . . > A,, are the p - r smallest eigenvalues. This completes the proof of Theorem 1. Notice how this analysis allows one to calculate all p eigenvalues and eigenvectors at once, and then make inference about the number of important cointegration relations, by testing how many of the h’s are zero. Next we shall investigate the test of a linear hypothesis about j3. In the case we have Y = 1, i.e., only one cointegration vector, it seems natural to test that certain variables do not enter into the cointegration vector, or that certain linear constraints are satisfied, for instance that the variables Xi, and X,, only enter through their difference Xi, - X,,. If r 2 2, then a hypothesis of interest could be that the variables Xi, and X,, enter through their difference only in all the cointegration vectors, since if two different linear combinations would occur then any coefficients to Xi, and X,, would be possible. Thus it seems that some natural hypotheses on p can be formulated as Hi: P=%, (15) where H( p x s) is a known matrix of full rank s and q(s x r) is a matrix of unknown parameters. We assume that r I s up. If s =p, then no restrictions are placed upon the choice of cointegration vectors, and if s = r, then the cointegration space is fully specified. Theorem 2. The maximum likelihood estimator of the cointegration space, under the assumption that it is restricted to sp( H), is given as the space spanned S. Johansen, Statistical analysis of cointegration vectors 237 by the canonical variates corresponding to the r largest squared canonical correlations between the residuals of H’X,_, and AX, corrected for the lagged diflerences of X,. The likelihood ratio test now becomes -2ln(Q)=Tiln((l-X:)/(1-i;)), i=l (16) where Xl;, . . . , A: are the r largest squared canonical correlations. Proof. It is apparent from the derivation of p^ that if p = Hq is fixed, then regression of R,, on -‘p’H’R,, is still a simple linear regression and the analysis is as before with R,, replaced by H’R,,. Thus the matrix ‘p can be estimated as the eigenvectors corresponding to the r largest eigenvalues of H’S,,S&,‘S,,, H with respect to H’S,,H, i.e., the solution to IXH’S,,H - H’S,,&&HI = 0. Let the s eigenvalues be denoted by hy, i = 1,. . . , s. Then the likelihood ratio test of Hi in H, can be found from two expressions like (13) and is given by (16), which completes the proof of Theorem 2. In the next section we shall find the asymptotic distribution of the test statistics (4) and (16) and show that the cointegration space, the impact matrix II and the variance matrix A are estimated consistently. 3. Asymptotic properties of the estimators and the test statistics In order to derive properties of the estimators we need to impose more precise conditions on the parameters of the model, such that they correspond to the situation we have in mind, namely of a process that is integrated of order 1, but still has r cointegration vectors fi. First of all we want all roots of \A( z)l = 0 to satisfy lz I > 1 or possibly z = 1. This implies that the nonstationarity of the process can be removed by differencing. Next we shall assume that X, is integrated of order 1, i.e., that AX, is stationary and that the hypothesis (3) is satisfied by some (Y and p of full rank r. Correspondingly we can express AX, in terms of the e’s by its moving average representation, AX,= f- C-p-/~ J=o for some exponentially decreasing coefficients C,. Under suitable conditions on these coefficients it is known that this equation determines an error correction 238 S. Johansen, Statistical analysis of cointegration vectors model of the form (5), where r,X,_, = -IIX,_, represents the error correc- tion term containing the stationary components of X,_k, i.e., B’Xt_k. More- over the null space for C’ = c,“=&.’ given by {[I C’[ = 0} is exactly the range space of r,l, i.e., the space spanned by the columns in B and vice versa. We thus have the following representations: II=@’ and C=yr8’, (17) where 7 is ( p - r) X (p - r), y and 6 are p X ( p - r), and all three are of full rank, and y’B = B’(Y = 0. We shall later choose 6 in a convenient way [see the references to Granger (1981) Granger and Engle (1985), Engle and Granger (1987) or Johansen (1988) for the details of these results]. We shall now formulate the main results of this section in the form of two theorems which deal with the test statistics derived in the previous section. First we have a result about the test statistic (4) and the estimators derived in (11) and (12). Theorem 3. Under the hypothesis that there are r cointegrating vectors the estimate of the cointegration space as well as IT and A are consistent, and the likelihood ratio test statistic of this hypothesis is asymptotically distributed as tr( /o’dBB’[ L’BB’du] -‘l’B dB’), (18) where B is a( p - r)-dimensional Brownian motion with covariance matrix I. In order to understand the structure of this limit distribution one should notice that if B is a Brownian motion with I as the covariance matrix, then the stochastic integral /ofB d B’ is a matrix-valued martingale, with quadratic variation process /‘var(BdB’)=/‘BB’duQI, 0 0 where the integral _/,,‘BB’du is an ordinary integral of the continuous matrix- valued process BB’. With this notation the limit distribution in Theorem 3 can be considered a multivariate version of the square of a martingale /ofB d B’ divided by its variance process /,,‘BB’du. Notice that for r =p - 1, i.e., for testing p - 1 cointegration relations one obtains the limit distribution with a one-dimensional Brownian motion, i.e., (~1BdB)~~1B2du=((B(1)2-1),Z)Y11B2dU, 0 S. Johansen, Statistical analysis of coiniegration vectors 239 Table 1 The quantiles in the distribution of the test statistic, where B is an m-dimensional Brownian motion with covariance matrix I. m 2.5% 5% 10% 50% 90% 95% 97.5% 1 0.0 0.0 0.0 0.6 2.9 4.2 5.3 2 1.6 1.9 2.5 5.4 10.3 12.0 13.9 3 7.0 7.8 8.8 14.0 21.2 23.8 26.1 4 16.0 17.4 19.2 26.3 35.6 38.6 41.2 5 28.3 30.4 32.8 42.1 53.6 57.2 60.3 which is the square of the usual ‘unit root’ distribution [see Dickey and Fuller (1979)]. A similar reduction is found by Phillips and Ouliaris (1987) in their work on tests for cointegration based on residuals. The distribution of the test statistic (18) is found by simulation and given in table 1. A surprisingly accurate description of the results in table 1 is obtained by approximating the distributions by cx2(f) for suitable values of c and f. By equating the mean of the distributions based on 10,000 observations to those of cx2 with f = 2m2 degrees of freedom, we obtain values of c, and it turns out that we can use the empirical relation c = 0.85 - 0.58/f. Notice that the hypothesis of r cointegrating relations reduces the number of parameters in the II matrix from p2 to rp + r( p - r), thus one could expect ( p - r)2 degrees of freedom if the usual asymptotics would hold. In the case of nonstationary processes it is known that this does not hold but a very good approximation is given by the above choice of 2( p - r)2 degrees of freedom. Next we shall consider the test of the restriction (15) where linear con- straints are imposed on j3. Theorem 4. The likelihood ratio test of the hypothesis Hi: P=& of restricting the r-dimensional cointegration space to an s-dimensional subspace of RP is asymptotically distributed as x2 with r( p - s) degrees of freedom. We shall now give the proof of these theorems, through a series of inter- mediate results. We shall first give some expressions for variances and their 240 S. Johansen, Statistical analysis ojcointegrution uectors limits, then show how the algorithm for deriving the maximum likelihood estimator can be followed by a probabilistic analysis ending up with the asymptotic properties of the estimator and the test statistics. We can represent X, as X, = c:=, A Xi, where X0 is a constant which we shall take to be zero to simplify the notatton. We shall describe the stationary process AX, by its covariance function q(i) = var(AX,, AX,+,), and we define the matrices p,, =$(i -j) = E(AX,_,AX,‘,), i, j=O ,...,k-1, i=O ,..., k- 1, and P kk= - /=-CC Finally define Note the following relations: J=o J=o r-k var(x,-k)= C (t-k- ljl)G(j), J=-t+k *--I COV( x,-k, Ax,-i) = c ‘b(j), j=k-1 S. Johansen, Sratistical analysis of cointegration vectors 241 which show that var(X,/T1/2) -+ f 4(j) =+, I=--m and cov( xTpk, AX,_,) --) f q(j) = P/cry j=k-r whereas the relation T-k T-k var(p’xT-k) = tTek) c b’+(j)/?- c i_db’+(j)P /= -T+k j= -T+k shows that since P’C = 0 implies that P’J, = 0, such that the first term vanishes in the limit. Note that the nonstationary part of X, makes the variance matrix tend to infinity, except for the directions given by the vectors in fi, since P’X, is stationary. The calculations involved in the maximum