Threshold Variable Determination and Threshold Variable Driven Switching Autoregressive Models

In this paper we propose a new class of nonlinear time series models, the threshold variable driven switching autoregressive models. It is a hierarchical model that combines two important nonlinear time series models, the threshold autoregressive (AR) models and the random switching AR models. The underlying time series process switches between two (or more) different linear models. The switching dynamics relies on an observable threshold variable (up to certain estimable parameters) as used in a threshold model, hence reveals the true nature of the switching mechanism. It also allows certain randomness in the switching procedure similar to that in a random switching model, hence provides some flexibility. Furthermore, we propose a model building procedure that concentrates on a fast determination of an appropriate threshold variable among a large set of candidates (and linear combinations of them). This procedure is applicable to the new models as well as the classical threshold models. A simulation study and two data examples are presented.