Trading-flow assisted estimation of the jump activity index
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https://doi.org/10.1007/s11425-018-9442-1
Trading-flow assisted estimation of the jump activity index Xinbing Kong1 , Guangying Liu1,∗ & Shangyu Xie2
2School
1School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China; of Banking and Finance, University of International Business and Economics, Beijing 100029, China
Email: [email protected], [email protected], [email protected] Received April 27, 2018; accepted August 20, 2018
Abstract
Existing estimators for the jump activity index only made use of the price dynamics of assets.
In this study, we incorporate trading information and propose a trading-flow-adjusted (TA) estimator for the jump activity index for pure-jump Itˆ o semimartingales observed at high frequencies. We derive the central limit theorem of the estimator and perform simulation studies that justify the theory. The new estimator is shown to be more efficient in terms of the convergence rate as compared with the existing estimators, which use only the price information under some realistic conditions. Empirical analysis shows estimates with lower standard errors than those that do not incorporate the trading information. Keywords MSC(2010)
jump activity index, pure-jump Itˆ o semimartingale, trading volume, power variation 60J75, 62M05
Citation: Kong X B, Liu G Y, Xie S Y. Trading-flow assisted estimation of the jump activity index. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9442-1
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Introduction
The jump activity index (JAI) is an important characteristic of jump processes. For a semimartingale X, it is defined as { } ∑ β = inf r; |∆s X|r < ∞ , 06s6T
where ∆s X = Xs − Xs− is the jump size of X at time s and T is a fixed time horizon. β is an indicator of the frequency of arrival of small jumps, and lies in the interval [0, 2] for semimartingales. The larger the β is, the more active the jumps are. In particular, the jump process for finite activity, finite variation, and infinite variation has β = 0, 0 < β < 1, and 1 6 β < 2, respectively. For β = 2, we make a convention that the process is an Itˆo process driven by Brownian motion. There are several estimators for β in the literature. A¨ıt-Sahalia and Jacod [1] proposed an estimator for the JAI of jump components in the simultaneous presence of a continuous martingale using the counts of large jumps. To gain more efficiency, Jing et al. [11] proposed a refinement of the latter by using the counts of both large and small jumps. Using the theory of power variation, Todorov and Tauchen [21,22] * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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2
Kong X B et al.
Sci China Math
and Todorov [19] studied the properties of a variation-based estimator for the JAI. Estimation optimality was established in [4]. Jing et al. [9] considered the estimation of the JAI using noisy high-frequency data. However, the aforementioned studies only used the observed price data while neglecting attributes th
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