Estimation in Threshold Autoregressive Models with a Stationary and a Unit Root Regime

Jiti Gao,Jiying Yin,Dag Tjøstheim

doi:10.2139/ssrn.1677744

Jiti Gao, Jiying Yin + Show 1 more

Open Access

https://doi.org/10.2139/ssrn.1677744

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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 have 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 much slower than that in the nonstationary regime. The proposed theory and estimation method are illustrated by both simulated and real data examples.

Highlights

Ordinary unit root models have just one regime, whereas ordinary threshold models have several regimes, but are stationary
Even though (1.1) is the simplest possible of the type of models we are discussing, it requires nonstandard techniques using the theory of null recurrent Markov chains
This paper has considered two classes of threshold autoregressive models with possible nonstationarity

Summary

Introduction

Ordinary unit root models have just one regime, whereas ordinary threshold models have several regimes, but are stationary. Pham, Chan and Tong (1991) consider a nonlinear unit–root problem and establish strong consistency results for the ordinary least squares (OLS) estimators of α1 and α2 for the case where (α1, α2) lie on the boundary, Hansen (1996) rigorously establishes an asymptotic theory for the likelihood ratio test for a threshold, Chan and Tsay (1998) discuss a related continuous–time TAR model, and Hansen (2000) proposes a new approach to estimating stationary TAR models. There have been other extensions to the nonstationary case, see in particular Caner and Hansen (2001), having a class of models that allow for both nonlinearity and nonstationarity, and where these properties can be (Caner and Hansen 2001) separately tested for 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. The mathematical proofs of our theory are given in Appendix B

OLS estimation method and asymptotic theory

Remarks about estimation of the τ parameter

Estimation in semiparametric threshold autoregressive models

Examples of implementation

Conclusions and discussion