Heterogeneous Autoregressive Modeling with Flexible Cascade Structures

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.