### Abstract

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251-273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.

Original language | English |
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Pages (from-to) | 125-141 |

Number of pages | 17 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 53 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 Jan 1 |

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### Keywords

- Asymptotic optimality
- Autoregressive conditional heteroskedasticity models
- Classical moment estimator
- Conditional least squares estimator
- Estimating function
- Nonlinear time series models
- Random coefficient autoregressive models

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Annals of the Institute of Statistical Mathematics*,

*53*(1), 125-141. https://doi.org/10.1023/A:1017924722711