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

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.