Nonparametric cointegration analysis

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 Σ_k 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),

.. Γrr ^^     ~ ' ' ' ’ ~

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 X_k ^,s and Y_k S are

independent q-variate standard normally distributed random vectors depending on F_k in the
following way:

This result holds regardless the possible existence of cointegration. Thus Lemma 1 proves (5),
with C(1)X_k replaced by [C(1)C(1)^T]^1/2X_k , and similarly for Y_k .

Next, assume that there are r linear independent cointegrating vectors. As is well-
known, we can write
where Λ_q-r is the diagonal matrix of the q-r positive eigenvalues, R_q-r is the corresponding
matrix of orthonormal eigenvectors, and R_r is the matrix of orthonormal eigenvectors

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