On periodic \(\log GARCH\) model with empirical application

This article introduces a new class of volatility models known as Periodic log-Generalized Autoregressive Conditional Heteroscedastic (P-logGARCH) models, which incorporate periodic variations in the parameters. These parameter variations are particularly relevant when incorporating seasonality into economic decision-making theory. The P-logGARCH formulation demonstrates several desirable properties, including unconstrained positivity of parameters and effective management of extreme values, along with a volatility trend. Specifically, in this paper we investigate the probabilistic structure of this model and establish necessary and sufficient conditions for the existence of stationary solutions in a periodic sense. Additionally, we examine the strong consistency and asymptotic normality of the Generalized Quasi-Maximum Likelihood Estimator (GQMLE) under mild assumptions. To assess the performance of our model, we conduct a Monte Carlo study to examine the finite-sample properties of the GQMLE. Finally, we present empirical evidence by applying the P-log GARCH model to analyze the exchange rates of the Algerian Dinar against the U.S. dollar and the Euro, thereby demonstrating its practical utility.