Nonparametric cointegration analysis

6. Testing linear restrictions

6.1. Design of the generalized eigenvalue problem, and asymptotic distribution theory

Following Johansen (1988,1991), we now focus on the problem of how to test whether
a cointegrating vector ξ satisfies a linear relation of the form

H0: ξ = Hφ , where rank(H) s s ≤ r, φ ∈ Bs.

Thus, the matrix H is of full column rank s. At first sight we may think of mimicking
Johansen’s test for these linear restrictions, on the basis of the matrices Am and IBm + n2Âm^-1.
However, that leads to a case-dependent asymptotic null distribution. Therefore we propose
the following alternative approach, on the basis of the matrix Aim only.

First, note that the null hypothesis (29) implies

H R R_rΓ,                                                                         (30)

where Γ is a r × s matrix of rank s. Then it follows straightforwardly from (15), (17) and (30)
that

D           m

n²HAmH n n²ΓTR,TA,R_rΓ →ΓTRrTD(1)£γ2YY_tTD(1)TR_rΓ
k= 1

and

HT  + n 2A ^¹ H = Γ^tR∕r(r⅛4  +n 2¹)r) RTR Γ→ ΓTV Γ.

∖ m           m∣              ^r ∖ ∖ m           m∣∣        r           r, m

Since similarly to (21) we can write

Y" - (γ^tR∕D(1)D(1)TR_rΓ■’ΓTR,TD(1)Y_t   (~ N,^[0,Is]),
we have that

Theorem 4. If there are r cointegrating vectors then under the null hypothesis (29) the
ordered solutions of the eigenvalues problem

18

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