Nonparametric cointegration analysis



∞∞

Σ α2k+ Σβ 2kk

4π2 -j1________j-1_____,

У α2k + A βi
ʌ'
2 2          2 2

j 1 j j 1 j

which we are going to minimize subject to the conditions in Lemma 6, as follows: First,
choose for some large natural number
N and all j,kN, αj,k =βj,k = 0. Denote θk =
(
α1,k,β1,k,α2,k,β2,k,...,αN,k,βN,k)T, J = diag(1,1,2,2,...,N,N). Then
hence the unconstrained minimum of
γk2 corresponds to the minimum solution of the
generalized eigenvalue problem det(
I - λJ-2) = 0. Taking k = 1, this minimum eigenvalue is 1
(twice), with corresponding normalized eigenvectors (1,0,0,...,0)
T and (0,1,0,...,0)T. Thus, the
unconstrained optimal solution satisfies
αj,1 = βj,1 = 0 forj > 1, and then the conditions in
Lemma 6 imply that also
β1,1 = 0, whereas without loss of generality we may take α1,1 =1.
Next, for
k = 2 the conditions in Lemma 6, except condition (31), imply that the optimal
solution corresponds to the minimum eigenvalue for which the corresponding eigenvector is
orthogonal to (1,0,0,...,)
T and (0,1,0,...,0)T. Clearly, this minimum eigenvalue is k2 = 4, and the
corresponding normalized eigenvectors are (0,0,1,0,...,0)
T and (0,0,0,1,...,0)T. Thus, αj,2 = βj,2 =
0 for
j2, and condition (31) implies that also β2,2 = 0, whereas again we may choose α2,2 =
1. Continuing this argument shows that the optimal solution for
Fk is the one in Lemma 3,
provided that the stronger conditions (6) and (7) also hold. The latter has already been
established in Lemma 3. Since
N was chosen arbitrary, it follows by induction that:

γ2


4π2


θTθk


θ J 2θk


Theorem 3. The choice Fk(x) = cos(2kπx) for the weight functions is optimal in the sense that
then for any fixed positive integer m the lower bound
(25) of the power of the lambda-min
test is maximal.

17



More intriguing information

1. The name is absent
2. The name is absent
3. Palkkaneuvottelut ja työmarkkinat Pohjoismaissa ja Euroopassa
4. Environmental Regulation, Market Power and Price Discrimination in the Agricultural Chemical Industry
5. LOCAL CONTROL AND IMPROVEMENT OF COMMUNITY SERVICE
6. Convergence in TFP among Italian Regions - Panel Unit Roots with Heterogeneity and Cross Sectional Dependence
7. EXPANDING HIGHER EDUCATION IN THE U.K: FROM ‘SYSTEM SLOWDOWN’ TO ‘SYSTEM ACCELERATION’
8. The name is absent
9. Death as a Fateful Moment? The Reflexive Individual and Scottish Funeral Practices
10. The Prohibition of the Proposed Springer-ProSiebenSat.1-Merger: How much Economics in German Merger Control?
11. Special and Differential Treatment in the WTO Agricultural Negotiations
12. 101 Proposals to reform the Stability and Growth Pact. Why so many? A Survey
13. Pupils’ attitudes towards art teaching in primary school: an evaluation tool
14. The name is absent
15. Cross-Country Evidence on the Link between the Level of Infrastructure and Capital Inflows
16. Nach der Einführung von Arbeitslosengeld II: deutlich mehr Verlierer als Gewinner unter den Hilfeempfängern
17. Crime as a Social Cost of Poverty and Inequality: A Review Focusing on Developing Countries
18. Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme
19. Non Linear Contracting and Endogenous Buyer Power between Manufacturers and Retailers: Empirical Evidence on Food Retailing in France
20. The name is absent