Factor GARCH-Itô models for high-frequency data with application to large volatility matrix prediction

Several novel large volatility matrix estimation methods have been developed based on the high-frequency financial data. They often employ the approximate factor model that leads to a low-rank plus sparse structure for the integrated volatility matrix and facilitates estimation of large volatility matrices. However, for predicting future volatility matrices, these nonparametric estimators do not have a dynamic structure to implement. In this paper, we introduce a novel Itô diffusion process based on the approximate factor models and call it a factor GARCH-Itô model. We then investigate its properties and propose a quasi-maximum likelihood estimation method for the parameter of the factor GARCH-Itô model. We also apply it to estimating conditional expected large volatility matrices and establish their asymptotic properties. Simulation studies are conducted to validate the finite sample performance of the proposed estimation methods. The proposed method is also illustrated by using data from the constituents of the S&P 500 index and an application to constructing the minimum variance portfolio with gross exposure constraints.