Introduction Recent years have witnessed

Introduction Recent years have witnessed a vast development of nonlinear techniques for modelling the conditional mean and conditional variance of economic and financial time series. In the vast array of new technical developments for conditional mean models, the Smooth Transition AutoRegressive (STAR) specification, proposed by Chan and Tong (1986) and developed by Luukkonen et al. (1988) and Teräsvirta (1994), has found a number of successful applications (see Tweedie (1988) for a recent review). The term “smooth transition” in its present meaning first appeared in Bacon and Watts (1971). They presented their smooth transition specification as a model of two intersecting lines with an abrupt change from one linear regression to another at an unknown change-point. Goldfeld and Quandt (1972, pp. 263–264) generalized the so-called two-regime switching regression model using the same idea. In the time series literature, the STAR model is a natural generalization of the Self-Exciting Threshold Autoregressive (SETAR) models pioneered by Tong (1978) and Tong and Lim (1980) (see also Tong (1990)). In terms of the conditional variance, Engle\'s (1982) Autoregressive Conditional Heteroskedasticity (ARCH) model and Bollerslev\'s (1986) Generalized ARCH (GARCH) model are the most popular specifications for capturing symmetric time-varying volatility in financial and economic time series data. McAleer (2005) provide an overview of different univariate and multivariate conditional volatility models. Despite their popularity, the structural and statistical properties of these p2x7 antagonist models were not fully established until recently. Chan and Tong (1986) derived sufficient conditions for strict stationarity and geometric ergodicity of a two-regime STAR model, where the transition function is given by the cumulative Gaussian distribution. Although several papers have been published in the literature with general conditions for strict stationarity and ergodicity of nonlinear time series models, especially threshold-type models, few attempts have been made to comprehend the dynamics of more general smooth transition processes (see Chen and Tsay (1991) for an early reference on the ergodicity of threshold models). In general, only very restrictive sufficient conditions are provided. For general nonlinear homoskedastic autoregressions, see Bhattacharya and Lee (1995), An and Huang (1996), An and Chen (1997), and Lee (1998), among many others. Nonlinear models with ARCH errors (not GARCH) have been considered, for example, by Masry and Tjostheim (1995), Cline and Pu (1998, 1999, 2004), Lu (1998), Lu and Jang (2001), Chen and Chen (2001), Hwang and Woo (2001), Liebscher (2005), and Saikkonen (2007). Stability of nonlinear autoregressions with GARCH-type errors has been analyzed by Liu et al. (1997), Ling (1999), and Cline (2007). Of these articles, those of Liu et al. (1997) and Ling (1999) are restricted to threshold AR-GARCH models, whereas Cline (2007) analyzes a very general nonlinear autoregressive models with GARCH errors. Cline (2007) obtained sharp results for geometric ergodicity but a difficulty with the application of these results is that the assumptions employed are quite general and, hence are difficult to verify. A threshold AR-GARCH model is the only example that is explicitly treated in the paper. Furthermore, conditional heteroskedasticity is driven by the observed series instead of the autoregressive errors as in the usual GARCH specification. Ferrante et al. (2003) considered threshold bilinear Markov processes. Only recently, Meitz and Saikkonen (2008) study the stability of general nonlinear autoregressions or order p with first-order GARCH errors. However, they explicitly analyzed only a STAR model with two limiting regimes. Consistency and asymptotic normality of the nonlinear least squares estimator are given under the assumption that the errors are homoskedastic and independent. In a recent paper, Mira and Escribano (2000) derived new sufficient conditions for consistency and asymptotic normality of the nonlinear least squares estimator. However, estimation of the conditional variance was not considered in these papers.