Asymptotic theory for explosive random coefficient autoregressive models and inconsistency of a unit root test against a stochastic unit root process

In this article, we develop the asymptotic theory of Hwang and Basawa (2005) for explosive random coefficient autoregressive (ERCA) models. Applying the theory, we prove that a locally best invariant (LBI) test in McCabe and Tremayne (1995), which is for the null of a unit root (UR) process against the alternative of a stochastic unit root (STUR) process, is inconsistent against a class of ERCA models. This class includes a class of STUR processes as special cases. We show, however, that the well-known Dickey–Fuller (DF) UR tests and a LBI test of Lee (1998) are consistent against a particular case of this class of ERCA models.