Estimation in nonlinear time series models. (English) Zbl 0598.62109

The author considers a number of nonlinear time series models generating data \(X_ t\). These include \[ X_ t=\{\psi +\pi \exp (-\gamma X^ 2_{t-1})\}X_{t-1}+e_ t, \] where the \(e_ t\) sequence is i.i.d. \(N(0,\sigma^ 2)\); \[ X_ t=\sum^{p}_{1}(a_ i+b_{ti})X_{t- i}+e_ t \] where the vector \(b_ t'=(b_{t1},...,b_{tp})\) is i.i.d. and totally independent of \(e_ t\). The estimation is to be by minimizing a function \(Q_ n(\beta_ i,X_ 1,...,X_ n)\) where \(\beta\) is the parameter vector, e.g. \(\psi\), \(\pi\), \(\gamma\), \(\sigma^ 2\) in the first example.
It is shown how the asymptotic properties of the estimate of \(\beta\) may be found via familiar Taylor series expansion techniques under suitable conditions. The criterion is chosen to be of the form \[ \sum [X_ t- E\{X_ t| F_{t-1}(m)\}]^ 2 \] where \(F_ t(m)\) is the \(\sigma\)- algebra generated by \(\{X_ s\), \(t-m+1\leq s\leq t\}\).
Reviewer: E.J.Hannan


62M09 Non-Markovian processes: estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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