On Stationarity and Ergodicity of the Bilinear Model with Applications to GARCH Models

Abstract

We establish sufficient conditions for the bilinear time series model to be strictly stationary and ergodic in terms of its associated Lyapunov exponent. In two special cases, we verify that the conditions are also necessary. We then use these results to give necessary and sufficient conditions for stationarity of specific GARCH models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH amongst others. These results generalise the ones found in Bougerol and Picard (1992b), Duan (1997), and Nelson (1990). In many cases, the conditions are weaker than the ones found elsewhere in the literature.

Keywords: Bilinear models, GARCH, random coefficient models, strict stationarity.

JEL Classification: C22, C51