Specifying the cointegrating rank

In practice, the cointegrating rank r is also usually unknown. It is commonly determined by a sequential testing procedure based on likelihood ratio (LR) type tests. Because for a given cointegrating rank Gaussian ML estimates for the reduced form VECM are easy to compute, as mentioned in Section 3.1, LR test statistics are also easily available. The following sequence of hypotheses may be considered:

H0(r0) : rank(n) = r0 versus H1(r0) : rank(n) > r0, r0 = 0,…, K – 1.


The testing sequence terminates if the null hypothesis cannot be rejected for the first time. If the first null hypothesis, H0(0), cannot be rejected, a VAR process in first differences is considered. At the other end, if all the null hypotheses can be rejected, the process is assumed to be I(0) in levels.

In principle, it is also possible to investigate the cointegrating rank by testing hypotheses H0 : rank(n) < r0 versus H1 : rank(n) > r0, r0 = K – 1,…, 1, 0, that is, by testing in reverse order from the largest to the smallest rank. The cointegrating rank is then chosen as the last r0 for which H0 is not rejected. From a theoretical viewpoint such a procedure has a drawback, however. Strictly speaking the criti­cal values for the tests to be discussed in the following apply for the situation that rank(n) is equal to r0 and not smaller than r0. Moreover, starting with the smallest rank as in (32.10) means to test the most restricted model first. Thus, a less restricted model is considered only if the data are strongly in favor of remov­ing the restrictions.

Although, under Gaussian assumptions, LR tests can be used here, it turns out that the limiting distribution of the LR statistic under H0(r0) is non-standard. It depends on the difference K – r0 and on the deterministic terms included in the DGP. In particular, the deterministic trend terms in the DGP have an impact on the null distribution of the LR tests. Therefore, LR type tests have been derived under different assumptions regarding the deterministic trend parameters. Fortu­nately, the limiting null distributions do not depend on the short-term dynamics if the latter are properly specified and, hence, critical values for LR type tests have been tabulated for different values of K – r0 under alternative assumptions for deterministic trend terms.

In this context it turns out that the model (32.4), where the deterministic and stochastic parts are separated, is a convenient point of departure. Therefore we consider the model

10 arbitrary

11 = 0

Ay, = Vo + Пу + Ш Г Ay,, +

Johansen (1995)

Johansen and Juselius (1990)



Подпись: I 0, I 1 arbitrary Подпись: Ay, Подпись: V + [П : V1] image848

Ay, = n(y,-i – До) + E^r.-Ay,.- + u,

Подпись:Ay, – {1 = n(y,-1 – {0 – {1(f – 1)) + Ер’=-11Г ( Ay,-j – {1) + Щ


A x, = П x,_1 + Г1А x,_1 + … + rp_1A x,_p+1 + u,. (32.12)

It is easy to see that the process yt has a VECM representation


Ay, = v0 + vlt + ny,_1 + E rAy,- + u,

Подпись: y,-1 Ї - 1

Подпись: = v + [П : v1]



Подпись:= v + n+y+_1 + E rjAy,-j + ut,


where v0 and v1 are as defined below (32.5), v = v0 + v1, П+ = [П : v1] and y+_1 = [ y P_1 : f _ 1]’. Depending on the assumptions for |0 and |1, different LR type tests can be obtained in this framework by appropriately restricting the parameters of the deterministic part and using RR regression techniques. An overview is given in Table 32.1 which is adopted from Table 1 of Hubrich, Lutkepohl and Saikkonen (2001) where more details on the tests may be found.

For instance, if p1 = 0 and p0 is unrestricted, a nonzero mean term is accommo­dated whereas a deterministic linear trend term is excluded by assumption. Three variants of LR type tests have been considered in the literature for this situation plus a number of asymptotically equivalent modifications. As can be seen from Table 32.1, the three statistics can be obtained easily from VECMs. The first test is obtained by dropping the v1t term in (32.13) and estimating the intercept term in the VECM in unrestricted form and hence, the estimated model may generate linear trends. The second test enforces the restriction that there is no linear deter­ministic trend in computing the test statistic by absorbing the intercept into the cointegration relations. Finally, in the third test the mean term p0 is estimated in a first step and is subtracted from yt. Then the estimation procedure with rank restriction for П is applied to (32.12) with xt replaced by xt = yt – "0. A suitable estimator "0 is proposed by Saikkonen and Luukkonen (1997) who also give the asymptotic distribution of the resulting test statistic under the null hypothesis. It is shown in Saikkonen and Lutkepohl (1999) that the latter test can have consid­erably more local power than the other two LR tests. Thus, based on local power it is the first choice if p 1 = 0.

If p 0 is arbitrary, p 1 Ф 0 and P’p 1 = 0, at least one of the variables has a deter­ministic linear trend because p 1 Ф 0, whereas the cointegration relations do not have a linear trend due to the constraint P’p1 = 0. The resulting tests are perhaps the most frequently used ones for determining the cointegrating rank in applied work. It may be worth emphasizing, however, that for the (K x r) matrix P to satisfy P’p1 = 0, p1 Ф 0 implies that r < K. Hence, if a trend is known to be present then it should also be allowed for under the alternative and con­sequently even under the alternative the rank must be smaller than K. In other words, in the present setting only tests of null hypotheses rank(n) = r0 < K – 1 make sense. This result is a consequence of the fact that a linear trend is assumed in at least one of the variables (p1 Ф 0) whereas a stable model with an intercept cannot generate a linear trend. Two different LR type tests are available for this case.

In the third case, both p0 and p1 are unrestricted so that the variables and the cointegrating relations may have a deterministic linear trend. Three different LR type tests and some asymptotically equivalent relatives have been proposed for this situation. Again, these test statistics can be obtained conveniently via the VECMs using the techniques mentioned in Section 3. The first model is set up in such a way so as to impose the linearity of the trend term. The second model includes the trend term in unrestricted form. As mentioned earlier, in principle such a model can generate quadratic trends. Finally, the last test in Table 32.1 is again based on prior trend adjustment and estimation of the resulting VECM for the trend adjusted variables. The trend parameters are again estimated in a first step by a generalized LS procedure. Critical values for all these tests may be found in the references given in Table 32.1.

Instead of the pair of hypotheses in (32.10) one may alternatively test H0(r0) : rank(n) = r0 versus H*(r0) : rank(n) = r0 + 1. LR tests for this pair of hypotheses were also pioneered by Johansen and are known as maximum eigenvalue tests. They can be applied for all the different cases listed in Table 32.1. They also have non-standard limiting distributions. Critical values can be found in the literature cited in the foregoing.

A comprehensive survey of the properties of LR type tests for the cointegrating rank as well as a substantial number of other tests that have been proposed in the literature is given by Hubrich et al. (2001). We refer the interested reader to that article for further details.

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