The absence of the drift parameter vector μ in the right-hand side of (11) is due to conditions
(6) and (7). Since the matrix Σk in (12) is diagonal, due to condition (6), and the two
components on the right-hand side of (11) are linear functionals of a Wiener process and thus
normally distributed, they are independent. They are also independent over k, due to the
conditions (8), (9) and (10). Thus we have:
Lemma 1. Under Assumption 1 and conditions (6) through (10),
( ʌ
Mnz(Fk)/Tn
M∆ (Fk)F
k k √
.. Γrr ^^ ~ ' ' ' ’ ~
→D
C(1)Xk\ J JFk(x)Fk(У)min(x,У)dxdy
k C (1) Ykfj( x )2dx
jointly for k = 1 ,...,m, with m a fixed natural number, where the Xk ,s and Yk S are
independent q-variate standard normally distributed random vectors depending on Fk in the
following way:
JFk(x ) W(x ) dx
'-k ɪ , ---------------
J JJFk(x )Fk (y )min(x, y ) dxdy
Yk
Fk(1) W(1) - Jfk(x ) W(x ) dx
J JFk(x )2 dx
(13)
This result holds regardless the possible existence of cointegration. Thus Lemma 1 proves (5),
with C(1)Xk replaced by [C(1)C(1)T]1/2Xk , and similarly for Yk .
Next, assume that there are r linear independent cointegrating vectors. As is well-
known, we can write
where Λq-r is the diagonal matrix of the q-r positive eigenvalues, Rq-r is the corresponding
matrix of orthonormal eigenvectors, and Rr is the matrix of orthonormal eigenvectors
C(1)C(1)T R RΛR T = (Rq_r, Rr)
(.
Λ
q
к O
' Rr
ʌRr )