This project aims to extend hysteretic or buffered time series models.

The threshold time series models have emerged as a prominent class of nonlinear time series models over the past four decades. Recently, there has been growing interest in extending these models to incorporate hysteresis or a buffer zone to their regime-switching structure. While most of these hysteretic studies focus on two-regime models with a single threshold variable, there is a need for multiple regimes and multiple threshold variables in many economic and financial applications, especially for long time series or multivariate nonlinear time series. For instance, George Tiao and Ruey Tsay in their 1994 paper (J. Forecast.) recommended a fourregime model for U.S. quarterly real GNP growth rate, with the regimes based on the past growth rate and the first difference of two consecutive past growth rates. Ruey Tsay in his 2021 paper (Stat. Sin.) modeled the weekly growth rates of US. gasoline stocks and prices using a two-regime bivariate hysteretic model that requires simultaneous switching of both series. Without such a restriction when using multiple threshold variables, the threshold models often result in many regimes, and hence they would become too complicated and hard to interpret. This project aims to extend hysteretic or buffered time series models that (1) incorporate multiple regimes; (2) support multiple threshold variables with flexible regime-switching structure, and (3) are easy to interpret. To address these, we propose new hysteretic models with tree-structured multiple thresholds that are a generalization of the standard hysteretic models. This project will first examine basic hysteretic models with multiple regimes based on a single threshold variable, deriving conditions for stationarity and ergodicity, and developing statistical inference procedures for estimation and model selection. Subsequently, the project will consider hysteretic time series models with tree-structured multiple thresholds, including extensions to multivariate and conditional mean and variance models. Instead of a hard threshold (i.e., a sudden jump from one regime to another), the project will investigate a smooth transition extension of the tree-structured hysteretic models, providing a soft threshold in the sense that there is a smooth transition between adjacent regimes. It is anticipated that the general multiple-regime hysteretic time series models will offer increased model complexity in a systematic and flexible manner, and also provide valuable tools for studying nonlinear regime-switching mechanism in many real applications in economics and financial risk management.