Unified discrete-time and continuous-time models and statistical inferences for merged low-frequency and high-frequency financial data

This paper introduces a unified model, which can accommodate both continuous-time Itô processes used to model high-frequency stock prices and GARCH processes employed to model low-frequency stock prices, by embedding a discrete-time GARCH volatility in its continuous-time instantaneous volatility. This model is called a unified GARCH-Itô model. We adopt realized volatility estimators based on high-frequency financial data and the quasi-likelihood function for the low-frequency GARCH structure to develop parameter estimation methods for the combined high-frequency and low-frequency data. We establish asymptotic theory for the proposed estimators and conduct a simulation study to check finite sample performances of the estimators. We apply the proposed estimation approach to Bank of America stock price data.