GARCH models without positivity constraints: Exponential or Log GARCH?

Christian Francq (1, 2) , Olivier Wintenberger (1, 3) , Jean-Michel Zakoïan (1, 2)

1 LFA - Laboratoire de Finance Assurance
2 EQUIPPE - Economie Quantitative, Intégration, Politiques Publiques et Econométrie
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH model, which are already known, with those of an asymmetric version of the log-GARCH. The quasi-maximum likelihood estimation of the log-GARCH parameters is shown to be strongly consistent and asymptotically normal. Similar estimation results are only available for particular EGARCH models, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.

Mots clés: EGARCH; log-GARCH; Quasi-Maximum Likelihood; Strict stationarity; Tail index

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

Christian Francq, Olivier Wintenberger, Jean-Michel Zakoïan. GARCH models without positivity constraints: Exponential or Log GARCH?. Journal of Econometrics, 2013, 177 (1), pp.34-46. ⟨10.1016/j.jeconom.2013.05.004⟩. ⟨hal-00750015v2⟩