Nonparametric cointegration analysis



3. Convergence in distribution of a class of random matrices and their
generalized eigenvalues

Our tests will be based on the following pair of random matrices:

m

 = Va χιTk,
m               n, k n, k,

k= 1


m

Bn, ≈ Σ bn, fink

k= 1


depending on a natural number mq, where

m;(F„(•))/fin          b = fin M(F1(•))

an, k          .-------------------------------------------------------- ,       n, k            .---------------------

y ʃʃFk(x )Fk (y )min(x, y ) dxdy            y ʃFk(x )2 dx
with

nn

M(Fk) = fi∑F((t/n)z,,  Mfiz(F.) - ɪ Fk( t/n )ʌzt

t= 1                                                     t= 1

where {Fk } is a class of differentiable real functions on the unit interval [0,1]. As will be
shown below, the functions
Fk can be chosen such that

( ʌ

D .               .,,ɔ m _ .               .,,ɔ

C (1) C (1) T £ XkXkτ C (1) C (1) T) ,

(5)


C(1) C(1) 1/2 ∑ Y1Y1T C(1) C(1) T) 1/2,

yk= 1

where the Xk ’s and Yk ’s are independent q-variate standard normal random vectors, and D
indicates convergence in distribution. In order to apply the result of Andersen, Brons and
Jensen (1983), saying:

if for a pair of square random matrices Pn , Qn ,(Pn,Qn ) converges in distribution to
(P,Q), where Q is a.s. nonsingular, then the ordered solutions of the generalized
eigenvalue problem
det(Pn -λQn ) = 0 converge in distribution to the ordered solutions of



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