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 Tj≤stheim (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.