On stationarity and ergodicity of the bilinear model with applications to GARCH models

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 generalized autoregressive conditionally heteroskedastic (GARCH) models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH among others. These results generalize the ones found in the studies of, among others, Bougerol and Picard [Journal of Econometrics 52 (1992) 115], Duan [Journal of Econometrics 79 (1997) 97] and Nelson [Econometric Theory 6 (1990) 318]. In many cases, the conditions are weaker than the ones found elsewhere in the literature. Copyright 2009 The Author. Journal compilation 2009 Blackwell Publishing Ltd