Econometric Theory
- Econometric Theory / Volume 24 / Issue 05 / October 2008, pp 1443-1455
- Copyright © Cambridge University Press 2008
- DOI: http://dx.doi.org/10.1017/S0266466608080675 (About DOI), Published online: 09 July 2008
Research Article
NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC
James Davidsona1, Jan R. Magnusa2 c1 and Jan Wiegerincka3
a1 University of Exeter
a2 Tilburg University
a3 University of Amsterdam
Abstract
We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.
Correspondence
c1 Address correspondence to Jan R. Magnus, Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands; e-mail: magnus@uvt.nl
Footnotes
We are grateful to Dieter Balkenborg, Nigar Hashimzade, Michael Jansson, Chris Müris, Morten Nielsen, Paolo Paruolo, and Katsuto Tanaka for helpful advice and discussions and to two referees for constructive comments. Figure 1 is taken from Davidson (2008).
