Structure and asymptotic theory for STAR(1)-GARCH(1,1) models

Nonlinear time series models, especially those with regime-switching and GARCH errors, have become increasingly popular in the economics and finance literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the processes or the associated asymptotic theory. In this paper we derive necessary and sufficient conditions for strict stationarity and ergodicity of three different specifications of the first-order STAR-GARCH model, and sufficient conditions for the existence of moments. This is important, among others, to establish the conditions under which the traditional LM linearity tests based on Taylor expansions are valid. Finally, we provide sufficient conditions for consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator.