Double Asymptotics for Explosive Continuous Time Models

This paper develops a double asymptotic limit theory for the persistent parameter (k) in explosive continuous time models driven by Lévy processes with a large number of time span (N) and a small number of sampling interval (h). The simultaneous double asymptotic theory is derived using a technique in the same spirit as in Phillips and Magdalinos (2007) for the mildly explosive discrete time model. Both the intercept term and the initial condition appear in the limiting distribution. In the special case of explosive continuous time models driven by the Brownian motion, we develop the limit theory that allows for the joint limits where N ! 1 and h ! 0 simultaneously, the sequential limits where N ! 1 is followed by h ! 0, and the sequential limits where h ! 0 is followed by N ! 1. All three asymptotic distributions are the same.