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
(29)
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 RrΓ, (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,RrΓ →ΓTRrTD(1)£γ2YYtTD(1)TRrΓ
k= 1
and
HTÂ + n 2A ^1 H = ΓtR∕r(r⅛4 +n 2Â1)r) RTR Γ→ ΓTV Γ.
∖ m m∣ r ∖ ∖ m m∣∣ r r, m
Since similarly to (21) we can write
Y" - (γtR∕D(1)D(1)TRrΓ■’ΓTR,TD(1)Yt (~ 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
More intriguing information
1. The name is absent2. Developing vocational practice in the jewelry sector through the incubation of a new ‘project-object’
3. The name is absent
4. Direct observations of the kinetics of migrating T-cells suggest active retention by endothelial cells with continual bidirectional migration
5. Visual Artists Between Cultural Demand and Economic Subsistence. Empirical Findings From Berlin.
6. From music student to professional: the process of transition
7. Change in firm population and spatial variations: The case of Turkey
8. Reform of the EU Sugar Regime: Impacts on Sugar Production in Ireland
9. Iconic memory or icon?
10. Towards a Strategy for Improving Agricultural Inputs Markets in Africa