Abstract: In this study, we examine the theory and methodology of statistical inferences of thresholds and change-points in threshold autoregressive models. We show that least squares estimators (LSEs) of thresholds and change-points are 𝓃-consistent, and that they converge weakly to the minimizer of a compound Poisson process and the location of minima of a two-sided random walk, respectively. When the magnitude of the change in the parameters of the state regimes or in the time horizon is small, we further show that these limiting distributions can be approximated by a class of known distributions. The LSEs of the slope parameters are -consistent and asymptotically normal. Furthermore, a likelihood-ratio based confidence set is given for the thresholds and change-points, respectively. A Simulation study is carried out to assess the performance of our procedure, and the proposed theory and methodology are illustrated using a tree-ring data set.

Key words and phrases: Brownian motion, change-point, compound poisson process, least squares estimation, threshold.