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. Olive Tree Farming in Jaen: Situation With the New Cap and Comparison With the Province Income Per Capita.2. The name is absent
3. Midwest prospects and the new economy
4. A COMPARATIVE STUDY OF ALTERNATIVE ECONOMETRIC PACKAGES: AN APPLICATION TO ITALIAN DEPOSIT INTEREST RATES
5. Großhandel: Steigende Umsätze und schwungvolle Investitionsdynamik
6. Empirical Calibration of a Least-Cost Conservation Reserve Program
7. The name is absent
8. The name is absent
9. Methods for the thematic synthesis of qualitative research in systematic reviews
10. Human Development and Regional Disparities in Iran:A Policy Model