Model verification for Lévy-driven CARMA(2,1) processes

Abstract

The Lévy-driven CARMA(2,1) process is a popular one with which to model stochastic volatility. However, there has been little development in statistical tools to verify this model assumption and assess the goodness-of-fit using high frequency realized volatility. When a Lévy-driven CARMA(2,1) is observed at high frequencies, the unobserved driving process can be approximated from the observed process. Since, under general conditions, the Lévy-driven CARMA(2,1) can be written as a sum of two dependent Lévy-driven CAR(1) process, the methods developed in Abdelrazeq, Ivanoff, and Kulik (2014, 2018) can be employed in order to use the approximated increments of the driving processes to test the assumption that the process is Lévy-driven. The performance of the tests is illustrated through simulation.

Keywords:

• Ornstein–Uhlenbeck process
• CARMA(2,1)
• Lévy-driven
• sampled process
• sample correlation