Double Asymptotics for an Explosive Continuous Time Model

This paper develops a double asymptotic limit theory for the persistent parameter () in an explosive continuous time model with a large number of time span (N) and a small number of sampling interval (h). The limit theory 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. The initial condition, either …xed or random, appears in the limiting distribution. The simultaneous double asymptotic theory is derived by using results recently obtained in Phillips and Magdalinos (2007) for the mildly explosive discrete time model and so a invariance principle applies. However, our asymptotic distribution is di¤erent from what was reported in Perron (1991, Econometrica) where the sequential limits, h ! 0 followed by N ! 1, were considered. It is shown that the limit theory in Perron is not correct and the correct sequential asymptotic distribution is identical to the simultaneous double asymptotic distribution.