Nonparametric cointegration analysis

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 F_k
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 A_mH = Op(1). Moreover, using (37) we can write

H W Γ ^rRHR,.,^γi ÷ ^γ_i^r,-A,R ⁱ ÷ ¹ ` R ^{ar γ}2 ÷ WR_r,^γi .

Therefore, it follows easily from part (15) of Lemma 4 that Γ₁ = Op(1/n). Since by (37), Γ₁ =
R_q-TH^, we now have R_q-^r_rH = 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 R_q-^τTH = O_p(1/n), where R_q-r
is the matrix of eigenvectors of C(1)C(1)^T corresponding to the positive eigenvalues.

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