Testing for Common Arrivals of Jumps for Discretely Observed Multidimensional Processes

• Published in: The Annals of statistics 2009-08, Vol.37 (4), p.1792-1838
• Main Authors: Jacod, Jean, Todorov, Viktor
• Format: Article
• Language: English
• Subjects: 60H10; 60J60; 62F12; 62M05; Asymptotic methods; Bleeding time; Common jumps; Critical values; discrete sampling; Exact sciences and technology; Exchange rates; Finite volume method; General topics; Geometric lines; high-frequency data; Markov processes; Mathematics; Null hypothesis; Parametric inference; Perceptron convergence procedure; Poisson equation; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Simulation; Statistical theories; Statistical variance; Statistics; Studies; tests; Truncation
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/08-AOS624

We consider a bivariate process $X_{t} = (X_{t}^{1}, X_{t}^{2})$ , which is observed on a finite time interval [0, T] at discrete times 0, $\Delta_{n}$ , $2\Delta_{n}$ ,... Assuming that its two components X¹ and X² have jumps on [0, T], we derive tests to decide whether they have at least one jump occurring at the same time ("common jumps") or not ("disjoint jumps"). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh $\Delta_{n}$ goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.