Nonparametric cointegration analysis

are just the reciprocals of n²λ_j,_m. Thus again referring to Andersen, Brons and Jensen (1983) it
follows that n²(λ_q-r+1 ,_m, .. , λ_q,_m ) converges in distribution to the ordered eigenvalues of the
matrix V_r,m² . Finally, observe that, with X^*_k defined by (18) and

Y" R {RT^τD(1)D(1)^trJ'÷R,^τD(1)Y_t   (- N_r[0,I_r]),                                    ⁽²¹⁾

we can write the matrix V_r,m as

V_r,_n, -    D(1)D(1) R’ Vr'mⁱ' D(1)D(1) ^τR_r)⁴

where

Y     YY

Thus we have:

Theorem 2. Under the conditions of Theorem 1, n²(λ_q-r+1,_m, .. , λ_q,_m) converges in distribution
to (λ^*_1,m²,...,λ^*_r,m²), where λ^*_1,m ≥ ... ≥ λ^*_r,m are the ordered solutions of the generalized eigenvalue
problem

det V,m - λ Ri_r^τD(1)D(1)^τRr) "

where the matrix V_r^*_{, m} is defined in (23) with the X^*_i ’s and Y^*_j^*’s independent q-r-variate and
r-variate, respectively, standard normally distributed random vectors.

4. Testing the number of cointegrating vectors

12

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