Asymptotic Optimality of Estimating Function Estimator for CHARN Model

CHARN model is a famous and important model in the finance, which includes many financial time series models and can be assumed as the return processes of assets. One of the most fundamental estimators for financial time series models is the conditional least squares (CL) estimator. However, recently, it was shown that the optimal estimating function estimator (G estimator) is better than CL estimator for some time series models in the sense of efficiency. In this paper, we examine efficiencies of CL and G estimators for CHARN model and derive the condition that G estimator is asymptotically optimal. 1. Introduction The conditional least squares (CL) estimator is one of the most fundamental estimators for financial time series models. It has the two advantages which can be calculated with ease and does not need the knowledge about the innovation process (i.e., error term). Hence this convenient estimator has been widely used for many financial time series models. However, Amano and Taniguchi [1] proved it is not good in the sense of the efficiency for ARCH model, which is the most famous financial time series model. The estimating function estimator was introduced by Godambe ([2, 3]) and Hansen [4]. Recently, Chandra and Taniguchi [5] constructed the optimal estimating function estimator (G estimator) for the parameters of the random coefficient autoregressive (RCA) model, which was introduced to describe occasional sharp spikes exhibited in many fields and ARCH model based on Godambe's asymptotically optimal estimating function. In Chandra and Taniguchi [5], it was shown that G estimator is better than CL estimator by simulation. Furthermore, Amano [6] applied CL and G estimators to some important time series models (RCA, GARCH, and nonlinear AR models) and proved that G estimator is better than CL estimator in the sense of the efficiency theoretically. Amano [6] also derived the conditions that G estimator becomes asymptotically optimal, which are not strict and natural. However, in Amano [6], G estimator was not applied to a conditional heteroscedastic autoregressive nonlinear model (denoted by CHARN model). CHARN model was proposed by H？rdle and Tsybakov [7] and H？rdle et al. [8], which includes many financial time series models and is used widely in the finance. Kanai et al. [9] applied G estimator to CHARN model and proved its asymptotic normality. However, Kanai et al. [9] did not compare efficiencies of CL and G estimators and discuss the asymptotic optimality of G estimator theoretically. Since CHARN model is the important and rich model, which