Testing for common arrivals of jumps for discretely observed multidimensional processes

TL;DR

This paper develops statistical tests to determine whether two components of a bivariate process observed discretely share common jump times or have disjoint jumps, with proven asymptotic properties and practical applications.

Contribution

It introduces two novel tests for detecting common versus disjoint jumps in discretely observed multidimensional processes, with proven asymptotic levels and demonstrated finite-sample performance.

Findings

Tests have a prescribed asymptotic level as sampling frequency increases.

Simulations show good finite-sample performance of the tests.

Application to exchange rate data illustrates practical utility.

Abstract

We consider a bivariate process $X_t=(X^1_t,X^2_t)$, which is observed on a finite time interval $[0,T]$ at discrete times $0,\Delta_n,2\Delta_n,....$ Assuming that its two components $X^1$ and $X^2$ 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.