Estimation in threshold autoregressive models with a stationary and a unit root regime

Jiti Gao,Dag Tjøstheim,Jiying Yin

doi:10.1016/j.jeconom.2011.12.006

Abstract

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 n−14, whereas it is n−1 in the nonstationary regime. The proposed theory and estimation method are illustrated by both simulated data and a real data example.

Highlights

Ordinary unit root models have just one regime, whereas ordinary threshold models have several regimes, but are stationary
We study a threshold model that has unit–root behavior on one regime and acts as a stationary process in another regime
We could possibly expect a rate for τ − τ of order T −1(n) which can be associated with n−. This is in agreement with the finite sample results for Case A of Example 4.3 below, which is an example where a threshold of this type was investigated by simulation

Summary

Introduction

Ordinary unit root models have just one regime, whereas ordinary threshold models have several regimes, but are stationary. Liu, Ling and Shao (2009) extend the discussion of Pham, Chan and Tong (1991) by establishing an asymptotic distribution of the OLS estimator of α2 for the case where Cτ = The nonstationarity of these models under the null hypothesis has been of a rather restricted form, typically regarding both yt − yt−1 and the threshold variable to be stationary. For simplicity, we only treat the first order case, but the theory can be extended to higher order and vector models, making it possible to introduce threshold cointegration models in this context. Mathematical proofs of some lemmas are given in Appendix B

OLS estimation method and asymptotic theory

Discussion about estimation of the τ parameter

Estimation in semiparametric threshold autoregressive models

Examples of implementation

Conclusions and discussion