Asymptotic distributions related to mildly-explosive second order autoregressive models

In this paper, we consider the normalized least squares estimator of the parameter in a mildly-explosive first-order autoregressive model with dependent errors which are modeled as a mildly-explosive AR(1) process. We prove that the estimator has a Cauchy limit law which provides a bridge between moderate deviation asymptotics and the earlier results on the local to unity and explosive autoregressive models. In particular, the results can be applied to understand the near-integrated second order autoregressive processes. Simulation studies are also carried out to assess the performance of least squares estimation in finite samples.