Abstract

Advancements in technology have led to increasingly complex structures in high-frequency

data, necessitating the development of efficient models for accurately forecasting realized measures.

This paper introduces a novel approach known as the multilinear low-rank heterogeneous autoregressive (MLRHAR) model. Distinguishing itself from the conventional heterogeneous autoregressive

(HAR) model, our model utilizes a data-driven method to replace the fixed heterogenous volatility

components. To address the calendar effect, we utilize the fourth-order tensor technique, which simultaneously reduces dimensions in the response, predictor, and short-term and calendar temporal

directions. This not only reduces the parameter space but also enables the automatic selection of

heterogeneous components from both temporal directions. Moreover, we establish the non-asymptotic

properties of high-dimensional HAR models, and a projected gradient descent algorithm is proposed

with theoretical justifications for parameter estimation. Through simulation experiments, we evaluate

the efficiency of the proposed model. We apply our method to financial data on the constituent stocks

of the S&P 500 Index. The results obtained from both the simulation and real studies convincingly

demonstrate the significant forecasting advantages offered by our approach.

Information

Preprint No.SS-2024-0308
Manuscript IDSS-2024-0308
Complete AuthorsHuiling Yuan, Guodong Li, Kexin Lu, Alan T.K. Wan, Yong Zhou
Corresponding AuthorsGuodong Li
Emailsgdli@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, Shanghai Pujiang Program

(No. 23PJ1402400), and China Postdoctoral Science Foundation (No. 2023M741190). Li’s

research was partially supported by GRF grants 17313722 and 17309625 from the Hong

Kong Research Grant Council. Wan’s research was partially supported by National Natural

Science Foundation of China (No. 72273120). Zhou’s research was partially supported by

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

and Lu are co-first authors, and Li and Wan are co-corresponding authors.

Supplementary Materials

The online Supplementary Material contains the tensor notations and Tucker decomposition,

the proofs of the two theorems, Corollary 1, simulation results for the MLR-TT-HAR model,

and one Table for the selected ranks of the MLR-FT-HAR, MLR-TT-HAR and VHARI

models in Real data analysis.

Supplementary materials are available for download.