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

Olivier Wintenberger (1, 2)

1 LFA - Laboratoire de Finance Assurance
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Invertible models; volatility models; quasi maximum likelihood; strong consistency; asymptotic normality; exponential GARCH; stochastic recurrence equation.; stochastic recurrence equation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Olivier Wintenberger. Continuous invertibility and stable QML estimation of the EGARCH(1,1) model. Scandinavian Journal of Statistics, 2013, 40 (4), pp.846-867. ⟨10.1111/sjos.12038⟩. ⟨hal-00751706v2⟩