AISM 53, 125-141

© 2001 ISM

## Estimating functions for nonlinear time series models

### S. Ajay Chandra and Masanobu Taniguchi

Department of Mathematical Science, Graduate School of
Engineering Science, Osaka University, Machikaneyama-cho 1-3, Toyonaka, Osaka 560-8531, Japan, e-mail:chandra@sigmath.es.osaka-u.ac.jp; taniguti@sigmath.es.osaka-u.ac.jp

(Received April 10, 2000; revised August 17, 2000)

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.

Key words and phrases:
Nonlinear time series models, random coefficient autoregressive models,
autoregressive conditional heteroskedasticity models, conditional least
squares estimator, estimating function, classical moment estimator, asymptotic
optimality.

**Source**
( TeX , DVI )