becomes singular, we recommend the rule-of-thumb m =2q. The corresponding critical values
are presented in Table 3. As is easy to see, the same rule-of-thumb applies to the lambda-max
test. The critical values of the lambda-max test, for m =2q, and the weight functions Fk
chosen as in Lemma 3, are given in Table 4.
<Insert Tables 3-4 about here>
6.4. Estimation of the cointegrating vectors
The results in section 6.1 can also be used to derive consistent estimators of the
cointegrating vectors, as follows. Choose again m = 2q, and let H be the matrix of the r
eigenvectors corresponding to the r smallest eigenvalues of the generalized eigenvalue
problem
n2H AmH = Op(1). Moreover, using (37) we can write
^ - / ^ - ^ — 1 I
det[Am - λA^Lm + n 2Am j
0 0,
(36)
1 TT ∙ t 1 1 ∙ 1 1 t1 t TT tI Â
where H is standardized such that H IAm + n
-, 1 -1 ∖ 1 ^ _ __ . . _ _ , _ _ .
2/Am ) H = Ir. Then similarly to (33)
we can
write
√A √∖ /V
H = ‰γГ1 + Rr l
where rank(Γ1) = s ≥ 0, rank
Γ
2
V 2√
(37)
with Γ1 and Γ2 stochastically bounded matrices. It follows now similarly to Theorem 3 that
H W Γ rRHR,.,γi ÷ γir,-A,R i ÷ 1 ` R ar γ2 ÷ WRr,γi .
Therefore, it follows easily from part (15) of Lemma 4 that Γ1 = Op(1/n). Since by (37), Γ1 =
Rq-TH^, we now have Rq-rrH = Op(1/n). Thus:
Theorem 6. If there are r linear independent cointegrating vectors then the matrix H of
standardized eigenvectors corresponding to the r smallest eigenvalues of the generalized
eigenvalue problem (36) (with m chosen from Table 1 ) satisfies Rq-τTH = Op(1/n), where Rq-r
is the matrix of eigenvectors of C(1)C(1)T corresponding to the positive eigenvalues.
22
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