NOTES AND PROBLEMS A GENERAL BOUND FOR THE LIMITING DISTRIBUTION OF BREITUNG'S STATISTIC

We consider the Breitung (2002, <italic>Journal of Econometrics</italic> 108, 343–363) statistic ξ_null, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξ_null as <italic>n</italic> → ∞, 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.