Estimation of integrated volatility in the presence of dependent and endogenous microstructure noise

Xin Luo; Xin Luo

doi:10.3934/math.2026996

AIMS Mathematics

2026, Volume 11, Issue 8: 24733-24769. doi: 10.3934/math.2026996

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Research article

Estimation of integrated volatility in the presence of dependent and endogenous microstructure noise

Xin Luo ^{1,2
,
,}

1.
School of Economics and Management, Hubei University of Technology, Wuhan, China
2.
Hubei Circular Economy Development Research Center, Hubei University of Technology, Wuhan, China

Received: 12 April 2026 Revised: 18 July 2026 Accepted: 23 July 2026 Published: 12 August 2026
MSC : 62F12, 62M10, 62M20, 62P05

In this paper, a new econometric procedure is proposed to estimate the integrated volatility (Ⅳ) of the efficient price process in the presence of general dependent and endogenous market microstructure noise. A consistent estimator for Ⅳ, referred to as pre-averaging kernel realized volatility $ \bigl(\operatorname{PKRV}(Y)^{(n)} \bigr) $, is first developed, based on a combination of the pre-averaging method and the realized kernel technique. The stable convergence of the proposed estimator is proved with a rate of $ n^{-1/6} $. Next, to improve the performance of $ \operatorname{PKRV}(Y)^{(n)} $ in the presence of jumps, the threshold pre-averaging kernel realized volatility $ \bigl(\operatorname{TPKRV}(Z)^{(n)} \bigr) $ is further proposed as an estimator for Ⅳ under market microstructure noise and jumps. Furthermore, the asymptotic properties of $ \operatorname{PKRV}(Y)^{(n)} $ and $ \operatorname{TPKRV}(Z)^{(n)} $, such as the consistency and central limit theorems, are established. A scheme for determining the optimal window width, the only parameter in $ \operatorname{PKRV}(Y)^{(n)} $ and $ \operatorname{TPKRV}(Z)^{(n)} $, is also proposed. Extensive simulation studies demonstrate the excellent performance of the proposed estimators, while empirical studies further illustrate their validity and accuracy.
- high-frequency,
- integrated volatility,
- market microstructure noise,
- dependent and endogenous noise,
- pre-averaging method
Citation: Xin Luo. Estimation of integrated volatility in the presence of dependent and endogenous microstructure noise[J]. AIMS Mathematics, 2026, 11(8): 24733-24769. doi: 10.3934/math.2026996

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Abstract

In this paper, a new econometric procedure is proposed to estimate the integrated volatility (Ⅳ) of the efficient price process in the presence of general dependent and endogenous market microstructure noise. A consistent estimator for Ⅳ, referred to as pre-averaging kernel realized volatility $ \bigl(\operatorname{PKRV}(Y)^{(n)} \bigr) $, is first developed, based on a combination of the pre-averaging method and the realized kernel technique. The stable convergence of the proposed estimator is proved with a rate of $ n^{-1/6} $. Next, to improve the performance of $ \operatorname{PKRV}(Y)^{(n)} $ in the presence of jumps, the threshold pre-averaging kernel realized volatility $ \bigl(\operatorname{TPKRV}(Z)^{(n)} \bigr) $ is further proposed as an estimator for Ⅳ under market microstructure noise and jumps. Furthermore, the asymptotic properties of $ \operatorname{PKRV}(Y)^{(n)} $ and $ \operatorname{TPKRV}(Z)^{(n)} $, such as the consistency and central limit theorems, are established. A scheme for determining the optimal window width, the only parameter in $ \operatorname{PKRV}(Y)^{(n)} $ and $ \operatorname{TPKRV}(Z)^{(n)} $, is also proposed. Extensive simulation studies demonstrate the excellent performance of the proposed estimators, while empirical studies further illustrate their validity and accuracy.

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