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:

depending on a natural number m ≥ q, where

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

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

y ʃʃF_k(x )F_k (y )min(x, y ) dxdy            y ʃF_k(x )2 dx
with

nn

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

t= 1                                                     t= 1

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

( ʌ

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

→ C (1) C (1) T £ X_kXk^τ C (1) C (1) T) ,

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

yk= 1

where the X_k ’s and Y_k ’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 P_n , Q_n ,(P_n,Q_n ) converges in distribution to
(P,Q), where Q is a.s. nonsingular, then the ordered solutions of the generalized
eigenvalue problem det(P_n -λQ_n ) = 0 converge in distribution to the ordered solutions of

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