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
More intriguing information
1. The name is absent2. The name is absent
3. Detecting Multiple Breaks in Financial Market Volatility Dynamics
4. TLRP: academic challenges for moral purposes
5. The name is absent
6. The name is absent
7. Putting Globalization and Concentration in the Agri-food Sector into Context
8. Does Presenting Patients’ BMI Increase Documentation of Obesity?
9. The name is absent
10. Confusion and Reinforcement Learning in Experimental Public Goods Games