Efficiency and influence function of estimators for ARCH models

TL;DR

This paper introduces a new closed-form estimator for linear ARCH models that is easier to implement, distribution-free, and robust, showing significant efficiency improvements over traditional ML and QML estimators in simulations.

Contribution

It proposes a novel estimating function estimator for ARCH models that simplifies implementation and enhances robustness and efficiency compared to existing methods.

Findings

EF estimator is easier to implement than ML and QML

EF estimator does not rely on distributional assumptions

Simulation shows substantial efficiency gains for certain distributions

Abstract

This paper proposes a closed-form optimal estimator based on the theory of estimating functions for a class of linear ARCH models. The estimating function (EF) estimator has the advantage over the widely used maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators that (i) it can be easily implemented, (ii) it does not depend on a distributional assumption for the innovation, and (iii) it does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance equation. In the case of normality, the asymptotic distribution of the ML and QML estimators naturally turn out to be identical and, hence, coincides with ours. Moreover, a robustness property of the EF estimator is derived by means of influence function. Simulation results show that the efficiency benefits of our estimator relative to the ML and QML estimators are substantial for some ARCH innovation distributions.