GARCH models without positivity constraints: Exponential or Log GARCH?

TL;DR

This paper compares log-GARCH and EGARCH models, focusing on their probabilistic properties and estimation methods, highlighting the advantages of the asymmetric log-GARCH in volatility modeling without positivity constraints.

Contribution

It provides a comprehensive probabilistic comparison and establishes strong consistency and asymptotic normality for the quasi-maximum likelihood estimation of the log-GARCH model.

Findings

Log-GARCH and EGARCH models share similar probabilistic properties.

The quasi-maximum likelihood estimator for log-GARCH is strongly consistent and asymptotically normal.

Simulation and real data analysis demonstrate the models' comparative performance.

Abstract

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.