Continuous invertibility and stable QML estimation of the EGARCH(1,1) model

TL;DR

This paper introduces the concept of continuous invertibility for volatility models driven by SREs, proving the strong consistency and asymptotic normality of a stabilized QMLE for the EGARCH(1,1) model.

Contribution

It establishes the first strong consistency result for the QMLE of EGARCH(1,1) under explicit invertibility conditions and proposes a practical stabilization method called SQMLE.

Findings

Strong consistency of SQMLE for invertible EGARCH(1,1)

Asymptotic normality of SQMLE under minimal assumptions

Practical stabilization of QMLE via empirical invertibility constraints

Abstract

We introduce the notion of continuous invertibility on a compact set for volatility models driven by a Stochastic Recurrence Equation (SRE). We prove the strong consistency of the Quasi Maximum Likelihood Estimator (QMLE) when the optimization procedure is done on a continuously invertible domain. This approach gives for the first time the strong consistency of the QMLE used by Nelson in \cite{nelson:1991} for the EGARCH(1,1) model under explicit but non observable conditions. In practice, we propose to stabilize the QMLE by constraining the optimization procedure to an empirical continuously invertible domain. The new method, called Stable QMLE (SQMLE), is strongly consistent when the observations follow an invertible EGARCH(1,1) model. We also give the asymptotic normality of the SQMLE under additional minimal assumptions.