are just the reciprocals of n2λj,m. Thus again referring to Andersen, Brons and Jensen (1983) it
follows that n2(λq-r+1 ,m, .. , λq,m ) converges in distribution to the ordered eigenvalues of the
matrix Vr,m2 . Finally, observe that, with X*k defined by (18) and
Y" R {RTτD(1)D(1)trJ'÷R,τD(1)Yt (- Nr[0,Ir]), (21)
we can write the matrix Vr,m as
(22)
Vr,n, - D(1)D(1) R’ Vr'mi' D(1)D(1) τRr)4
where
V,
r,m
∑ γ 2 yΓyΓ
k- 1
(23)
Y YY
Λ,
Thus we have:
Theorem 2. Under the conditions of Theorem 1, n2(λq-r+1,m, .. , λq,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 RirτD(1)D(1)τRr) "
where the matrix Vr*, 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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