QMLE for Periodic Time-Varying Asymmetric $\log $ GARCH Models

Abstract

This paper establishes probabilistic and statistical properties of the extension of time-invariant coefficients asymmetric $\log $ GARCH processes to periodically time-varying coefficients ($P\log $ GARCH) one. In these models, the parameters of $\log -$volatility are allowed to switch periodically between different seasons. The main motivations of this new model are able to capture the asymmetry and hence leverage effect, in addition, the volatility coefficients are not a subject to positivity constraints. So, some probabilistic properties of asymmetric $P\log $ GARCH models have been obtained, especially, sufficient conditions ensuring the existence of stationary, causal, ergodic (in periodic sense) solution and moments properties are given. Furthermore, we establish the strong consistency and the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) under extremely strong assumptions. Finally, we carry out a simulation study of the performance of the QML and the $P\log $ GARCH is applied to model the crude oil prices of Algerian Saharan Blend.

Keywords

QML / Periodicity / $\log $ GARCH')">Asymmetric $\log $ GARCH / EGARCH / Stationarity / Asymptotic properties