Chapter 3 Theoretical Properties

3.1 Introduction

In this chapter, we study some theoretical properties of our suggested models. More specifically, we discuss the conditions of the models’ stability, so that their Markov chain representation is geometrically ergodic, and more precisely Q-geometrically ergodic, which suggests that an initial distribution which renders our models strictly stationary and β-mixing exists. Moreover, we consider conditions required in order to be able to establish consistency and asymptotic normality for the estimator vector.

In addition, we present the multivariate versions of our suggested models, which, despite the fact that they are not used in empirical applications in this thesis, could be interesting at least at theoretical level.

3.2 Geometric ergodicity

There have been several studies of the stability of nonlinear autoregressive models with heteroskedastic errors. However, most of them have concentrated mainly on Threshold Autoregressive models for the conditional mean with conditional heteroskedasticity (e.g. Ling (1999), Liu, Li and Li (1999)) or on nonlinear autoregressions with ARCH-type, and not GARCH-type, errors (e.g. Liebscher (2005), Masry and Tjøstheim (1995)).

In this section we adopt the analysis of Meitz and Saikkonen (2006), who, following the analysis of Liebscher (2005), considered the case of a general nonlinear autoregressive model of order s with first-order generalised autoregressive conditional heteroskedastic (GARCH(1,1)) errors and provided the conditions of its stability, in the sense of geometric ergodicity, and specifically Q-geometric ergodicity. Since such a model is possible to be represented as a Markov chain, their analysis is based on the theory for Markov chains (for a detailed analysis of Markov chains and their stability theory, see Meyn and Tweedie (2008)).

It can be highlighted that their results are restricted to smooth nonlinear functions for the conditional mean and conditional variance models, in the sense that their derivatives exist and are continuous. This condition, which is not convenient in the case of e.g. Threshold Autoregressive or Threshold GARCH models, is convenient in the case of our models, as they all have an exponential term and not discontinuities, and hence they are all smooth. This fact justifies our choice to follow their analysis.

It should be noted, though, that the choice of first, and not higher, order of the GARCH model is due to the arduousness when establishing irreducibility of the Markov chain, a necessary property for when proving geometric ergodicity (Meitz and Saikkonen, 2006). However, their results hold not only for nonlinear autoregressive models with GARCH errors, but also in the case of nonlinear autoregressive models combined with any smooth nonlinear GARCH-type model for the conditional variance.