Asymptotic normality and Cramér-type moderate deviations of Yule’s nonsense correlation statistic for Ornstein–Uhlenbeck processes

Abstract

In this paper, under discrete observations, we study the asymptotic consistency, asymptotic normality and Cramér-type moderate deviations of Yule’s nonsense correlation statistic for two Ornstein–Uhlenbeck processes. As applications, the global and local powers of the hypothesis testing for the independence between two Ornstein–Uhlenbeck processes are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results. Moreover, empirical applications illustrate the usefulness of the above mentioned statistic and the asymptotic theory. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals and asymptotic analysis techniques.