Abstract

It is an important task to model realized volatilities for high-frequency data in finance and

economics, while the heterogeneous autoregressive (HAR) model has dominated the applications in this

area. However, this model suffers from three drawbacks: (i.) its heterogeneous volatility components

are linear combinations of daily realized volatilities with fixed weights, which limits its flexibility for

different types of assets, (ii.)

it is still unknown what is the high-frequency probabilistic structure

for this model, as well as many other HAR-type models in the literature, and (iii.) there are no highdimensional theoretical results for vector HAR modeling although it is common to encounter many assets

in real applications. To overcome these drawbacks, this paper proposes a multilinear low-rank HAR

model by using tensor techniques, where a data-driven method is adopted to automatically select the

heterogeneous components. In addition, HAR-Itˆo models are introduced to interpret the corresponding

high-frequency dynamics. Moreover, non-asymptotic properties of the high-dimensional vector HAR

modeling are established, and a projected gradient descent algorithm with theoretical justifications is

suggested to search for estimates. Theoretical and computational properties of the proposed method

are verified by simulation experiments and real data analysis.

Key words and phrases: Diffusion process, Heterogeneous autoregressive model, High-dimensional anal- ysis, High-frequency data, Non-asymptotic property, Tensor technique

Information

Preprint No.SS-2025-0010
Manuscript IDSS-2025-0010
Complete AuthorsHuiling Yuan, Kexin Lu, Yifeng Guo, Lan Li, Guodong Li
Corresponding AuthorsLan Li
Emailsulilan@connect.hku.hk

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Acknowledgments

We are deeply grateful to the editor, an associate editor and two anonymous referees for their

valuable comments that led to the substantial improvement of the manuscript. Yuan’s research

was partially supported by National Natural Science Foundation of China (No. 72403089),

State Key Program of National Natural Science Foundation of China (No. 72531003), 2025

Shanghai Bai Yulan Talent Young Program. Li’s research was partially supported by GRF

grants 17313722 and 17309625 from the Hong Kong Research Grant Council.

Supplementary Materials

The online Supplementary Material includes tensor notation and Tucker decomposition; proofs

of Proposition 1 and Theorems 1 and 2, together with the auxiliary lemmas; a description

of the univariate HAR-Itˆo model; a table of selected ranks for the MLR-HAR and VHARI

models; and the definition and rolling estimates of the coefficient-based total spillover index

for the VHAR, VHARI, and MLR-HAR models.

Supplementary materials are available for download.