Open Access
June 2017 Testing for time-varying jump activity for pure jump semimartingales
Viktor Todorov
Ann. Statist. 45(3): 1284-1311 (June 2017). DOI: 10.1214/16-AOS1485


In this paper, we propose a test for deciding whether the jump activity index of a discretely observed Itô semimartingale of pure-jump type (i.e., one without a diffusion) varies over a fixed interval of time. The asymptotic setting is based on observations within a fixed time interval with mesh of the observation grid shrinking to zero. The test is derived for semimartingales whose “spot” jump compensator around zero is like that of a stable process, but importantly the stability index can vary over the time interval. The test is based on forming a sequence of local estimators of the jump activity over blocks of shrinking time span and contrasting their variability around a global activity estimator based on the whole data set. The local and global jump activity estimates are constructed from the real part of the empirical characteristic function of the increments of the process scaled by local power variations. We derive the asymptotic distribution of the test statistic under the null hypothesis of constant jump activity and show that the test has asymptotic power of one against fixed alternatives of processes with time-varying jump activity.


Download Citation

Viktor Todorov. "Testing for time-varying jump activity for pure jump semimartingales." Ann. Statist. 45 (3) 1284 - 1311, June 2017.


Received: 1 July 2015; Revised: 1 May 2016; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 1371.62062
MathSciNet: MR3662455
Digital Object Identifier: 10.1214/16-AOS1485

Primary: 62F12 , 62M05
Secondary: 60H10 , 60J75

Keywords: central limit theorem , high-frequency data , Itô semimartingale , jump activity index , jumps , Nonparametric test , power variation , stochastic volatility

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • June 2017
Back to Top