Nonparametric cointegration analysis



det HτA H λ λHτ(A

m             m m


+ n -21A ^1)^1 H
m J


(31)


times n2, converge jointly in distribution to the ordered solutions of the generalized

eigenvalue problem det[ W,m λ λVs;q r,m] = 0, where   Ws,m ς ∑ Z1Y 2YYT,

(

V

s s,q-r,m


∑γ 2y.“y;

vk1

(32)


1(


(

m

Ib 1


Y
m

k= 1


m

Ik=1


** T


[cf. (23)], and the Yk** and X*k involved are independent s-variate and q-r-variate, respectively,
standard normal random vectors.

Note that the matrix Vs*, q-r,m in (32) differs from the matrix Vs*,m defined by (23) with r
replaced by s in that in the latter case the vectors X*k are (q-s) × 1 rather than (q-r) × 1.

If the null hypothesis (29) is false, then the matrix H can be written as

H R Rq rΓ1 + RrΓ2, where rank(Γ1) ^ s 1 1, rank


(rʌ
γ1

2

V 2


s.


(33)


Then again it follows straightforwardly from (15), (17) and (30) that

D            m

H A. H →TτlRqrC(1)∑ XlXkτC(1)τRrrΓ,

k= 1

and

19



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