# HAR-Ito models and high-dimensional HAR modeling for high-frequency data

**Authors:** Huiling Yuan, Kexin Lu, Yifeng Guo, Guodong Li  
**arXiv:** [2303.02896](https://arxiv.org/abs/2303.02896)  
**Date:** 2023-03-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2303.02896v1/

## Key points

- The paper addresses the task of modeling realized volatilities for high-frequency data in finance and economics.
- The heterogeneous autoregressive (HAR) model has limitations due to fixed weights and unknown probabilistic structure.
- There is a lack of high-dimensional inference tools for HAR modeling despite multiple assets being common in real applications.
- The paper proposes a multilinear low-rank HAR model using tensor techniques to overcome these limitations.
- This approach incorporates a data-driven method to automatically select the heterogeneous components, enhancing flexibility for different asset types.
- HAR-Ito models are introduced to interpret the high-frequency dynamics of their proposed model and other HAR-type models.
- Non-asymptotic properties of high-dimensional HAR modeling are established, and a projected gradient descent algorithm is suggested for parameter estimation with theoretical justifications.
- Simulation studies verify theoretical and computational properties, while real data analysis illustrates the necessity of using a data-driven method for selecting heterogeneous components.
- The proposed multilinear low-rank HAR model offers increased flexibility and accuracy in capturing asset-specific characteristics while providing insights into high-frequency dynamics through HAR-Ito models.

## Summary

The paper titled "HAR-Ito models and high-dimensional HAR modeling for high-frequency data" addresses the task of modeling realized volatilities for high-frequency data in finance and economics. The heterogeneous autoregressive (HAR) model has been widely used in this field but suffers from several drawbacks. Firstly, its heterogeneous volatility components are linear combinations of daily realized volatilities with fixed weights, which limits its flexibility for different types of assets. Secondly, the high-frequency probabilistic structure of the HAR model as well as other HAR-type models is still unknown. Lastly, there is a lack of high-dimensional inference tools for HAR modeling despite the common occurrence of multiple assets in real applications. To overcome these limitations, the paper proposes a multilinear low-rank HAR model using tensor techniques. This approach incorporates a data-driven method to automatically select the heterogeneous components enhancing the flexibility of the model for different asset types. Additionally, the authors introduce HAR-Itô models to interpret the high-frequency dynamics of both their proposed model and other HAR type models. The paper establishes non-asymptotic properties of high dimensional HAR modeling and suggests a projected gradient descent algorithm with theoretical justifications to estimate parameters. Theoretical and computational properties of the proposed method are verified through simulation studies while real data analysis illustrates the necessity of using a data driven method for selecting heterogeneous components. Overall, this paper presents an innovative approach to modeling realized volatilities in high frequency finance and economics data by addressing limitations in existing models. The proposed multilinear low rank HAR model offers increased flexibility and accuracy in capturing asset specific characteristics while providing insights into high frequency dynamics through HAR Itô models.