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\noindent Statistica Sinica {\bf 13}(2003), 367-384
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\centerline{\large\bf GEOMETRIC ERGODICITY, REGULARITY OF THE}
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\centerline{\large\bf INVARIANT DISTRIBUTION AND INFERENCE FOR A}
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\centerline{\large\bf THRESHOLD BILINEAR MARKOV PROCESS}
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\centerline{Marco Ferrante, Giovanni Fonseca and Paolo Vidoni}
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\centerline{\it
Universit\`a degli Studi di Padova, Universit\`a degli Studi dell'Insubria}
\centerline{\it and Universit\`a degli Studi di Udine}
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\begin{quotation}
\noindent {\it Abstract:}
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.\par
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\noindent {\it Key words and phrases:}
Autoregressive conditional heteroskedasticity, estimating function, invariant
probability, stationary process, threshold model.\par
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