4.4. Estimating the number of cointegrating vectors
Rather than testing for the number of cointegrating vectors, we can also estimate it
consistently, as follows. Denote
1 Y1
q
∏λ t
Ik1
ifr ^0,
q-r
∏λ k
Ik1
Y 1(
n
√ I
ʌ
q ʌ
2 r∏ λ.
1 1 k, m
k=q~r+1
(26)
q
n 2 qIIλk,m ifrq.
where m is chosen from Table 1 for one of the three significance levels and the test result for
r, provided r < q, and m = q, say, if the test result is r = q . Then gm(r) converges in
probability to infinity if the true number of cointegrating vectors is unequal to r, and gm(r) =
Op(1) if the true number of cointegrating vectors is indeed r. Thus, taking rm =
argmin0≤r≤1{gm(r)} we have limn→∞P(rm = r) = 1. This approach may be useful as a double-
check on the test results for r.
5. The choice of the weight functions Fk
The best choice of the weight functions Fk is such that the power of the lambda-min
test is maximal, but again this is not feasible because the power depends on nuisance
parameters. However, Lemma 5 suggests that the second best choice is to choose the Fk’s as
to minimize the squared γk ’s, subject to the conditions (6) through (10). In doing so, it will
be convenient to replace first the conditions (6) and (7) by the weaker conditions
(27)
Fk(x ) dx о 0
and
15