On the Stationary Marginal Distributions of Subclasses of Multivariate Setar Processes of Order One

To introduce more flexibility in process-parameters through a regime-switching behavior, the classical autoregressive (AR) processes have been extended to self-exciting threshold autoregressive (SETAR) processes. However, the stationary marginal distributions of SETAR processes are usually difficult to obtain in explicit forms and, therefore, they lack appropriate characterizations. The stationary marginal distribution of a multivariate (d-dimensional) SETAR process of order one (MSETARd(1)) with multivariate normal innovations is shown to belong to the unified skew-normal (SUN) family and characterized under a fairly broad condition. This article also characterizes the stationary marginal distributions of a subclass of the MSETARd(1) processes with symmetric multivariate stable innovations. To characterize the stationary marginal distributions of these processes, the authors show that they belong to specific skew-distribution families, and for a given skew-distribution from the corresponding family, an MSETARd(1) process, with stationary marginal distribution identical to the given skew-distribution, can be associated. Furthermore, this article illustrates a diagnostic of an MSETAR2(1) model using the corresponding stationary marginal density.