Asymptotic Theory for a Stochastic Unit Root Model with Intercept and Under Mis-Specification of Intercept

Abstract

Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017) proposed a stochastic unit root model, where the source of the variation of the autoregressive coefficient is driven by a stationary process. In this paper, we study a new stochastic unit root model, in which the source of the variation of the autoregressive coefficient is driven by a (nearly) non-stationary process. The asymptotic theory for this model is established not only in the presence of an intercept but also under a mis-specification of intercept. Our study reveals some new findings which are different from those established in Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017). Our theoretical results demonstrate that the statistical properties of the estimators of the model parameters vary from case to case, and depend not only on whether the parameters are zero or not, and the validity of the model specification, but also on the degree of the persistence of the source of the variation of the autoregressive coefficient. The application to a hypothesis testing with null hypothesis of a deterministic unit root model is briefly discussed. Monte Carlo simulations are conducted to examine the finite-sample performance for the estimators of model parameters. Our theoretical findings are supported by the simulation results.