On the existence of stationary threshold bilinear processes

Abstract

AbstractThis article investigates some statistical and probabilistic properties of general threshold bilinear processes. Sufficient conditions for the existence of a causal, strictly and weak stationary solution for the equation defining a self-exciting threshold superdiagonal bilinear $$\left( SETBL\right) $$ S E T B L process are derived. Then it is shown that under well-specified hypotheses the higher-order moments of the SETBL process are finite. As a result, the skewness and kurtosis indexes are explicitly computed. The exact autocorrelation function is derived with an arbitrarily fixed number of regimes. Also, the covariance functions of the process and its powers are evaluated and the second (respectively, higher)-order structure is shown to be similar to that of a linear process. This implies that the considered process admits an ARMA representation. Finally, necessary and sufficient conditions for the invertibility and geometric ergodicity of a SETBL model are established. Some examples illustrate the obtained theoretical results.