A flexible two-threshold ℤ-valued autoregressive process with signed binomial thinning operator

Single-threshold model is commonly used for modelling nonlinear time series. For industrial growth, enterprise operation or market demand, their development and changes may exhibit the switching between the four regimes of ‘expansion, recession, depression and recovery’, which renders the single-threshold model no longer applicable. To characterize these phenomena, we propose a flexible two-threshold Z-valued autoregressive process with signed binomial thinning operator, which accurately characterizes the nonlinear characteristics of Z-valued time series by means of the two dimensions of different stages of change and speed of change. Then the basic probabilistic properties of the model are given. We consider conditional least squares and conditional maximum likelihood estimators under known and unknown thresholds and discuss the asymptotic normality of these estimators. Moreover, the nonlinearity test of the model is well addressed using a Wald-type test. We extend the proposed model to two classes of pth-order models with covariates. The outstanding performance of the proposed model is verified through simulation and practical application.