Periodic models for volatility process constitute an alternative representation for the seasonal patterns observed in data exhibits a strong seasonal volatility driven by periodic coefficients of high and law variation. Moreover, these varying-parameters can arise also when seasonality is incorporated into the theory of economic decision-making So, in this paper, we propose an extension of time-invariant coefficients threshold GARCH (TGARCH) processes to periodically time-varying coefficients (PTGARCH) one. This parametrization allows us to describe the dynamic volatility through a TGARCH model within each regime (or season), and therefore a new stylized fact that characterize the volatility by seasonal patterns. Hence, theoretical probabilistic properties of this model are derived. The necessary and sufficient conditions which ensure the strict stationarity and ergodicity (in periodic sense) solution of PTGARCH are given. We extend the standard results of the popular quasi-maximum likelihood estimator (QMLE) for estimating the unknown parameters in model. More precisely, the strong consistency and the asymptotic normality of QMLE are studied for the cases when the innovation process is an i.i.d (Strong case) or is not (Semi-strong case). A Monte Carlo study is further conducted to examine the finite sample properties of the QMLE. The simulation results reveal that the QMLE is approximately unbiased and consistent for modest sample sizes when the stationarity conditions hold. Empirical work on the exchange rates of the Algerian Dinar against the single European currency (Euro) shows that our approach also outperforms and fits the data well.

波动率过程的周期性模型构成了数据中观察到的季节性模式的替代表示，这些数据表现出由高和法律变化的周期性系数驱动的强烈的季节性波动性。此外，当将季节性因素纳入经济决策理论时，这些变化参数也会出现。因此，在本文中，我们提出将时不变系数阈值GARCH（TGARCH）过程扩展为周期性时变系数（PTGARCH））一。这种参数化使我们能够通过TGARCH描述动态波动率每个方案（或季节）内的模型，因此有一个新的程式化事实，可以通过季节性模式来表征波动率。因此，推导了该模型的理论概率性质。给出了确保PTGARCH的严格平稳性和遍历性（周期性）解决方案的充要条件。我们扩展了流行的拟最大似然估计器（QMLE）的标准结果，用于估计模型中的未知参数。更精确地，强一致性和的渐近正态QMLE进行了研究的情况下，当创新过程是独立同分布的（强的情况下）或不是（半强的情况下）。进一步进行了蒙特卡洛研究，以检验QMLE的有限样本属性。仿真结果表明，当平稳性条件成立时，对于适度的样本量，QMLE近似无偏且一致。关于阿尔及利亚第纳尔对单一欧洲货币（Euro）的汇率的经验研究表明，我们的方法也胜过并适合数据。