Econometric Theory

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).

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