Statistical inference for GQARCH‐Itô‐jumps model based on the realized range volatility

Abstract

This article introduces a novel approach that unifies two types of models: one is the continuous‐time jump‐diffusion used to model high‐frequency market financial data, and the other is discrete‐time GQARCH for modeling low‐frequency financial data by embedding the discrete GQARCH structure with jumps in the instantaneous volatility process. This model is named GQARCH‐Itô‐Jumps model. Quasi‐likelihood functions for the low‐frequency GQARCH structure are developed for the parametric estimations. In the quasi‐likelihood functions, for high‐frequency financial data, the realized range‐based estimations are adopted as the ‘observations’, rather than the realized return‐based volatility estimators which entail the loss of intra‐day information of the price movements. Meanwhile, the asymptotic properties are mainly established for the proposed estimators in the case of finite activity jumps. Moreover, simulation studies and some financial data are implemented to check the finite sample performance of the proposed methodology.