The GARCH(1,1)-M model: results for the densities of the variance and the mean

This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307–327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.