  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.
