## The Annals of Statistics

### Efficient estimation of integrated volatility functionals via multiscale jackknife

#### Abstract

We propose semiparametrically efficient estimators for general integrated volatility functionals of multivariate semimartingale processes. A plug-in method that uses nonparametric estimates of spot volatilities is known to induce high-order biases that need to be corrected to obey a central limit theorem. Such bias terms arise from boundary effects, the diffusive and jump movements of stochastic volatility and the sampling error from the nonparametric spot volatility estimation. We propose a novel jackknife method for bias correction. The jackknife estimator is simply formed as a linear combination of a few uncorrected estimators associated with different local window sizes used in the estimation of spot volatility. We show theoretically that our estimator is asymptotically mixed Gaussian, semiparametrically efficient, and more robust to the choice of local windows. To facilitate the practical use, we introduce a simulation-based estimator of the asymptotic variance, so that our inference is derivative-free, and hence is convenient to implement.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 156-176.

Dates
Revised: September 2017
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568585

Digital Object Identifier
doi:10.1214/18-AOS1684

Mathematical Reviews number (MathSciNet)
MR3909930

Zentralblatt MATH identifier
07036198

#### Citation

Li, Jia; Liu, Yunxiao; Xiu, Dacheng. Efficient estimation of integrated volatility functionals via multiscale jackknife. Ann. Statist. 47 (2019), no. 1, 156--176. doi:10.1214/18-AOS1684. https://projecteuclid.org/euclid.aos/1543568585

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#### Supplemental materials

• Supplement to “Efficient estimation of integrated volatility functionals via multiscale jackknife”. This appendix contains all mathematical proofs.