RCA models with GARCH innovations

Rapid developments of time series models and methods addressing volatility in computational finance and econometrics have been recently reported in the financial literature. The non-linear volatility theory either extends and complements existing time series methodology by introducing more general structures or provides an alternative framework (see Abraham and Thavaneswaran [B. Abraham, A. Thavaneswaran, A nonlinear time series model and estimation of missing observations, Ann. Inst. Statist. Math. 43 (1991) 493–504] and Granger [C.W.J. Granger, Overview of non-linear time series specification in Economics, Berkeley NSF-Symposia, 1998]). In this work, we consider Gaussian first-order linear autoregressive models with time varying volatility. General properties for process mean, variance and kurtosis are derived; examples illustrate the wide range of properties that can appear under the autoregressive assumptions. The results can be used in identifying some volatility models. The kurtosis of the classical RCA model of Nicholls and Quinn [D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction, in: Lecture Notes in Statistics, vol. 11, Springer, New York, 1982] is shown to be a special case.