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\(\sqrt{n}\)-consistent estimation in a random coefficient autoregressive model. (English) Zbl 0884.62099

Summary: This paper deals with \(\sqrt{n}\)-consistent estimation of the parameter \(\mu\) in the RCAR(1) model defined by the difference equation \[ X_j= (\mu+U_j) X_{j-1}+ \varepsilon_j \qquad (j\in \mathbb{Z}), \] where \(\{\varepsilon_j: j\in\mathbb{Z}\}\) and \(\{U_j: j\in \mathbb{Z}\}\) are two independent sets of i.i.d. random variables with zero means, positive finite variances and \(E[(\mu+ U_1)^2]<1\). A class of asymptotically normal estimators of \(\mu\) indexed by a family of bounded measurable functions is introduced. Then an estimator is constructed which is asymptotically equivalent to the best estimator in that class. This estimator, asymptotically equivalent to the quasi-maximum likelihood estimator derived by D. F. Nicholls and B. G. Quinn [Random coefficient autoregressive models: an introduction. Lecture Notes Stat. 11 (1982; Zbl 0497.62081)], is much simpler to calculate and is asymptotically normal without the additional moment conditions those authors impose.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators

Citations:

Zbl 0497.62081
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