August 2022 A CLT for second difference estimators with an application to volatility and intensity
Emil A. Stoltenberg, Per A. Mykland, Lan Zhang
Author Affiliations +
Ann. Statist. 50(4): 2072-2095 (August 2022). DOI: 10.1214/22-AOS2176


In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation (TSQC) estimators, is based on sums of increments of second differences of the observed processes, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are based on only one partition of the observational window. Moreover, a two-scales approach is employed to deal with asymptotic bias terms in a systematic manner, thus automatically giving consistent estimators without having to work out the form of the bias term on a case-to-case basis. We highlight the versatility of our central limit theorem by applying it to a novel leverage effect estimator that does not belong to the class of TSQC estimators. The principal empirical motivation for the present study is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its spot-volatility process. As an application of the TSQC estimators, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded and hence observed.

Funding Statement

The authors are grateful for the detailed comments of an anonymous referee and the Editor. The authors would also like to thank the United States National Science Foundation under grants DMS 17-13118 and DMS-2015530 (Zhang), and DMS 17-13129 and DMS-2015544 (Mykland).


Emil A. Stoltenberg would like to thank the Fulbright Foundation for financial support; The University of Chicago for their hospitality; Sylvie Bendier Decety for her kindness; Nils Lid Hjort for his encouragement; the statistics section at the Department of Mathematics, University of Oslo; and the PharmaTox Strategic Research Initiative at the Faculty of Mathematics and Natural Sciences, University of Oslo.


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Emil A. Stoltenberg. Per A. Mykland. Lan Zhang. "A CLT for second difference estimators with an application to volatility and intensity." Ann. Statist. 50 (4) 2072 - 2095, August 2022.


Received: 1 December 2020; Revised: 1 November 2021; Published: August 2022
First available in Project Euclid: 25 August 2022

MathSciNet: MR4474483
zbMATH: 1500.60014
Digital Object Identifier: 10.1214/22-AOS2176

Primary: 60F05 , 60G44 , 62M09 , 62M10
Secondary: 60G42 , 62G20 , 62P20

Keywords: Asynchronous times , central limit theorem , consistency , Convergence rates , counting processes , endogenous observation times , high frequency , intensity , irregular times , microstructure , observed asymptotic variance , overlapping intervals , rolling intervals , sufficiency , two-scales estimation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 4 • August 2022
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