Asymptotic theory for a stochastic unit root model

Lieberman and Phillips (Journal of Time Series Analysis) proposed a stochastic unit root model in which the source of the variation of the autoregressive coefficient is driven by a stationary process. More recently, Lieberman and Phillips (Journal of Econometrics) generalized this model to the multivariate case and a hybrid case. Their studies revealed that these stochastic unit root models lead to a generalization of the Black-Scholes formula for derivative pricing. Inspired by their studies, in this paper, we propose 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. Our study reveals some new findings which are different from those established by Lieberman and Phillips. Results of Monte Carlo simulations are given to illustrate the finite-sample performance of estimators in the model. Moreover, a comparison between the stochastic unit root model proposed by Lieberman and Phillips and that proposed in this paper is conducted via an empirical study.