It is an important task to model realized volatilities for high-frequency data in finance and economics and, as arguably the most popular model, 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 limit 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 is no high-dimensional inference tool for 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-It\^o models are introduced to interpret the corresponding high-frequency dynamics, as well as those of other HAR-type models. Moreover, non-asymptotic properties of the high-dimensional 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 studies, and the necessity of using the data-driven method for heterogeneous components is illustrated in real data analysis.

翻译：金融与经济领域中，对高频数据的已实现波动率进行建模是一项重要任务，而作为最流行的模型之一，异质自回归（HAR）模型在该领域的应用中占据主导地位。然而，该模型存在三个缺陷：（i）其异质波动成分是每日已实现波动率带固定权重的线性组合，限制了不同资产类型的灵活性；（ii）该模型以及文献中其他许多HAR类模型的高频概率结构仍未知；（iii）尽管实际应用中常涉及大量资产，但尚缺乏用于HAR建模的高维推断工具。为克服这些缺陷，本文提出一种利用张量技术的多线性低秩HAR模型，其中采用数据驱动方法自动选择异质成分。此外，引入HAR-Itô模型以解释相应的高频动态特性，以及其他HAR类模型的动态特性。同时，建立了高维HAR建模的非渐近性质，并提出一种具有理论保证的投影梯度下降算法以搜索估计量。通过仿真研究验证了所提方法的理论与计算性质，并在实际数据分析中阐明了使用数据驱动方法处理异质成分的必要性。