Testing for simultaneous jumps in case of asynchronous observations

Abstract

This paper proposes a novel test for simultaneous jumps in a bivariate Itô semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the jumps of the two processes which is estimated using bivariate power variations of Hayashi-Yoshida type without an additional synchronization step. An associated central limit theorem is shown whose asymptotic distribution is assessed using a bootstrap procedure. Simulations show that the test works remarkably well in comparison with the much simpler case of regular observations.

The null hypothesis of (common) jumps in case of irregular and asynchronous observations

“…Both from a theoretical and a practical point of view this is unsatisfactory, and one would like to understand what happens when the underlying observations come in randomly at irregular and asynchronous times. This is precisely what we aim at in this paper, and our study complements the previous work Martin and Vetter (2017) in which a test for the presence of common jumps (so under the opposite null hypothesis of no common jumps) was constructed.…”

“…s , ρ s and L n = ln(n) , M n = 10 √ n in the simulation of the Z n,m (s). For an explanation for the choice of these parameters see again Section 5 of Martin and Vetter (2017). We only use paths in the simulation where µ i ([0, 1], R) = 0 whenever α i = 0, i = 1, 2, 3.…”

“…As we have already applied most of the techniques used in the upcoming proofs in Martin and Vetter (2017), we keep the discussion of similar parts brief and add references to Martin and Vetter (2017) for more detailed arguments. Parts in the proofs which are new or specific to the tests introduced in this paper will be discussed in full detail.…”

Laws of large numbers for Hayashi–Yoshida-type functionals

“…The convergence (4.17) has already been shown in Proposition A.2 of Martin and Vetter (2018a). Without presenting detailed computations we state two more results to demonstrate that the limit in Theorem 4.6 after simplification sometimes has a much simpler representation compared to the general form in (4.16).…”

“…Looking at the proof of (A.6) in Martin and Vetter (2018a) we conclude that (6.11) vanishes in probability as first n → ∞ and then ρ → 0 if lim ρ→0 lim q→∞ lim sup…”