This paper treats estimation in a class of new nonlinear threshold autoregressive models with both a stationary and a unit root regime. Existing literature on nonstationary threshold models has basically focused on models where the nonstationarity can be removed by differencing and/or where the threshold variable is stationary. This is not the case for the process we consider, and nonstandard estimation problems are the result. This paper proposes a parameter estimation method for such nonlinear threshold autoregressive models using the theory of null recurrent Markov chains. Under certain assumptions, we show that the ordinary least squares (OLS) estimators of the parameters involved are asymptotically consistent. Furthermore, it can be shown that the OLS estimator of the coefficient parameter involved in the stationary regime can still be asymptotically normal while the OLS estimator of the coefficient parameter involved in the nonstationary regime has a nonstandard asymptotic distribution. In the limit, the rate of convergence in the stationary regime is asymptotically proportional to View the MathML source n to the power of -1/4, whereas it is n to the power of -1 in the nonstationary regime. The proposed theory and estimation method are illustrated by both simulated data and a real data example.
Gao, J., Tjostheim, D. B., & Yin, J. (2013). Estimation in threshold autoregressive models with a stationary and a unit root regime. Journal of Econometrics, 172(1), 1 - 13. https://doi.org/10.1016/j.jeconom.2011.12.006