(compare also (5)), where Hii are the diagonal elements of H = T 2ε0D'εε0Dε. Finally, they
applied the triangle and Cauchy-Schwarz inequalities to get
d d 1
E(ZT,d) ≤ £ £ [e(Hii)E(γ-4)E(H4)E(γ-4)] 4 ,
i=1 j=1
so that it remains to bound E(Hi4i) and E(γj-4) (uniformly in T).
The major difficulty with the proof of Larsson et al. (2001) is that the authors seem
to ignore the randomness of the matrix G. They argue, for example, that ε = εG' has
the same distribution as ε since G is orthogonal; but G depends on ε (note that even for a
deterministic G the assumption ε ~ N(0, IT0Ω) would generally imply a different distribution
of ε: ε ~ N(0,IT 0 GΩG0)). More importantly, to bound E(γ-4) they state that Γ =
GATG0 = T-2ε0D0Dε follows some d-variate Wishart distribution with T — 1 degrees of
freedom. However, we do not see how the diagonal matrix Γ can be Wishart distributed.
Probably, the authors believe that AT = T -2ε0D0Dε is Wishart distributed and use the
orthogonality of G. As before, complications arise from the randomness of G. Moreover,
AT would be Wishart distributed if, for instance, the rows of Dε ~ N(0,DDl 0 Ω) are
independent or T -2D0D is a projection matrix, but both statements do obviously not hold.
As intended by Larsson et al. (2001), we establish the existence of the first two moments
of the asymptotic trace statistic by showing that the sequence {ZT2 ,d} is uniformly integrable.
However, our corrected proof of their Lemma 1 uses basically inequality (8) and thus (9),
where we have to choose a fixed value of m in an appropriate way. On the one hand the
moments of the inverted Wishart variable U-1 (with m degrees of freedom) must exist, and on
the other hand the eigenvalue λm must be of order T2 , which requires a careful investigation
of the eigenvalues of the matrix F = D0D .
References
Breitung, J. (2005). A parametric approach to the estimation of cointegration vectors in
panel data. Econometric Reviews, 24, 1-20.
Coope, I. D. (1994). On matrix trace inequalities and related topics for products of Hermitian
matrices. Journal of Mathematical Analysis and Applications, 188, 999-1001.
Groen, J. J. J. & Kleibergen, F. (2003). Likelihood-based cointegration analysis in panels of
vector error correction models. Journal of Business and Economic Statistics, 21, 295-318.
Johansen, S. (1988). Statistical analysis of cointegrating vectors. Econometric Reviews, 12,
231-254.
Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.
Oxford University Press: Oxford.
Larsson, R., Lyhagen, J., & Lothgren, M. (2001). Likelihood-based cointegration tests in
heterogeneous panels. Econometrics Journal, 4, 109-142.
Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. New York: Wiley.
Neumann, M. (2000). Inverses of Perron complements of inverse M-matrices. Linear Algebra
and its Applications, 313, 163-171.