In this paper, we consider subgeometric (specifically, polynomial) ergodicity of univariate nonlinear autoregressions with autoregressive conditional heteroskedasticity (ARCH). The notion of subgeometric ergodicity was introduced in the Markov chain literature in the 1980s, and it means that the transition probability measures converge to the stationary measure at a rate slower than geometric; this rate is also closely related to the convergence rate of $\beta $-mixing coefficients. While the existing literature on subgeometrically ergodic autoregressions assumes a homoskedastic error term, this paper provides an extension to the case of conditionally heteroskedastic ARCH-type errors, considerably widening the scope of potential applications. Specifically, we consider suitably defined higher-order nonlinear autoregressions with possibly nonlinear ARCH errors and show that they are, under appropriate conditions, subgeometrically ergodic at a polynomial rate. An empirical example using energy sector volatility index data illustrates the use of subgeometrically ergodic AR–ARCH models.

It is well known that stationary geometrically ergodic Markov chains are $\beta$ -mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$ -mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$ -mixing with subgeometrically decaying mixing coefficients. Although this result is simple, it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity cannot be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$ -mixing results for the self-exciting threshold autoregressive model.

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

This paper studies a class of Markov models that consist of two components. Typically, one of the components is observable and the other is unobservable or “hidden.” Conditions under which geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given. This implies existence of initial values such that the joint process is strictly stationary and β-mixing. In addition to this, conditions for the existence of moments are also obtained, and extensions to the case of nonstationary initial values are provided. All these results are applied to a general model that includes as special cases various first-order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated nonlinear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

We consider a family of GARCH(1,1) processes introduced in He and Teräsvirta (1999a, Journal of Econometrics 92, 173–192). This family contains various popular generalized autoregressive conditional heteroskedasticity (GARCH) models as special cases. A necessary and sufficient condition for the existence of a strictly stationary solution is given.This research was financially supported by the Jan Wallander's and Tom Hedelius' Foundation, Grant J03–41. The author thanks the editor, an anonymous referee, Pentti Saikkonen, and Timo Teräsvirta for useful comments.