Abstract
This paper studies stability problems and their consequences for the first-order S-periodic auto-regressive diagonal bilinear (PBL(1, 0, 1, 1)) model in which the parameters vary periodically over time. Necessary and sufficient conditions for strict periodic stationarity and periodic stationarity up to the rth-order (r >= 1) of the PBL equation are provided. As a consequence, the autocovariance structure of the model is shown to have the periodic ARMA autocovariance structure and the S kurtoses of the r th-order periodically stationary solution are revealed. In addition, almost sure invertibility and mean-square invertibility of the PBL(1, 0, 1, 1) model are also considered. The paper gives a unified framework of stability that encompasses most of probability properties studied for the PBL model.