## The Annals of Statistics

### On the systematic and idiosyncratic volatility with large panel high-frequency data

Xin-Bing Kong

#### Abstract

In this paper, we separate the integrated (spot) volatility of an individual Itô process into integrated (spot) systematic and idiosyncratic volatilities, and estimate them by aggregation of local factor analysis (localization) with large-dimensional high-frequency data. We show that, when both the sampling frequency $n$ and the dimensionality $p$ go to infinity and $p\geq C\sqrt{n}$ for some constant $C$, our estimators of High dimensional Itô process; common driving process; specific driving process, integrated High dimensional Itô process, common driving process, specific driving process, systematic and idiosyncratic volatilities are $\sqrt{n}$ ($n^{1/4}$ for spot estimates) consistent, the best rate achieved in estimating the integrated (spot) volatility which is readily identified even with univariate high-frequency data. However, when $Cn^{1/4}\leq p<C\sqrt{n}$, aggregation of $n^{1/4}$-consistent local estimates of systematic and idiosyncratic volatilities results in $p$-consistent (not $\sqrt{n}$-consistent) estimates of integrated systematic and idiosyncratic volatilities. Even more interesting, when $p<Cn^{1/4}$, the integrated estimate has the same convergence rate as the spot estimate, both being $p$-consistent. This reveals a distinctive feature from aggregating local estimates in the low-dimensional high-frequency data setting. We also present estimators of the integrated (spot) idiosyncratic volatility matrices as well as their inverse matrices under some sparsity assumption. We finally present a factor-based estimator of the inverse of the spot volatility matrix. Numerical studies including the Monte Carlo experiments and real data analysis justify the performance of our estimators.

#### Article information

Source
Ann. Statist., Volume 46, Number 3 (2018), 1077-1108.

Dates
Revised: January 2017
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.aos/1525313076

Digital Object Identifier
doi:10.1214/17-AOS1578

Mathematical Reviews number (MathSciNet)
MR3797997

Zentralblatt MATH identifier
06897923

#### Citation

Kong, Xin-Bing. On the systematic and idiosyncratic volatility with large panel high-frequency data. Ann. Statist. 46 (2018), no. 3, 1077--1108. doi:10.1214/17-AOS1578. https://projecteuclid.org/euclid.aos/1525313076

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#### Supplemental materials

• Supplement to “On the integrated systematic and idiosyncratic volatility with large panel high-frequency data”. This supplement contains the technical proof of Lemmas 3–5, which is crucial in proving Theorem 1 and Theorem 2.