a1 University of Exeter
a2 Tilburg University
a3 University of Amsterdam
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.
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).