On non-stationary threshold autoregressive models

Abstract

In this paper we study the limiting distributions of the least-squares estimators for the non-stationary first-order threshold autoregressive (TAR(1)) model. It is proved that the limiting behaviors of the TAR(1) process are very different from those of the classical unit root model and the explosive AR(1).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ306 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the least squares estimation of multiple-regime threshold autoregressive models

“…In model (5.1), although β 21 = 1, {y t } is not a unit-root process even not a partial unit-root process in Liu et al (2010). √ n( β 21,n − β 21,0 ) is still asymptotically normal.…”

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

“…By contrast, the OLS estimator of α 2 is asymptotically consistent with the super n-rate of convergence. In a related paper by Liu, Ling and Shao (2009), the authors have established similar results for α 2 , but have not established any asymptotic theory for α 1 .…”

“…This implies that the asymptotic theory for α 2 is the same as that for the unit-root case when µ 2 = ∞ −∞ yI[y ∈ C τ ]π s (dy) = 0, i.e., {y t } has some symmetrical structure in the stationary regime. In this symmetrical case, the asymptotic distribution in (2.15) corresponds to the main result in Theorem 2.1 of Liu, Ling and Shao (2009).…”

“…Among the more recent contributions, Chan (1990Chan ( , 1993 consider both estimation and testing problems for the case where {y t } of (1.1) is 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 α 1 and α 2 for the case where (α 1 , α 2 ) lies 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. More recently, Liu, Ling and Shao (2009) extend the discussion of Pham, Chan and Tong (1991) by establishing an asymptotic dis-tribution of the OLS estimator of α 2 for the case where C τ = (−∞, τ ] and either α 2 = 1 and α 1 < 1 or α 2 > 1 and α 1 ≤ 1 holds.…”

Statistical Inference for Structurally Changed Threshold Autoregressive Models

“…Seo and Linton (2007) proposed a smoothed LSE for the TAR/regression model and showed that the estimated threshold is asymptotically normal with a lower rate of convergence. Liu et al (2011) and Gao et al (2013) studied the LSE for the non-stationary first-order TAR model and Chan et al (2015) adopted the LASSO method to estimate TAR models with multiple thresholds. Gao et al (2017) proposed a non-nested test for TAR models vs.…”