## 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

### MSC:

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