Geometric ergodicity, regularity of the invariant distribution and inference for a threshold bilinear Markov process

In this paper we consider a first order threshold bilinear Markov process, which can be viewed as an AR model with ARCH-type errors and may be useful for modelling economic or financial time series. We study the main features of this process within a wider family of nonlinear models, where the threshold term is replaced by a smooth approximating function. Under suitable general assumptions, we provide sufficient conditions for the geometric ergodicity of the processes of this class and for the existence of their finite moments of a given order. Furthermore, we state regularity conditions for the invariant measures and we prove that the invariant measures of the smooth models weakly converge to that of the threshold one. The problem of estimating the parameters, including the threshold parameter, is studied and a simple semiparametric procedure based on the theory of optimal estimating functions is proposed.