Nonlinear time series models, especially those with regime-switching and conditionally heteroskedastic
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 first derive necessary conditions for strict stationarity and ergodicity of three different specifications of
the first-order smooth transition autoregressions with heteroskedastic errors. This is important, among other
reasons, to establish the conditions under which the traditional LMlinearity tests based on Taylor expansions
are valid. Second, we provide sufficient conditions for consistency and asymptotic normality of the Quasi-
Maximum Likelihood Estimator for a general nonlinear conditional mean model with first-order GARCH
errors
