QMLE for periodic absolute value GARCH models

Periodic generalized autoregressive conditionally heteroscedastic (PGARCH) models were introduced by Bollerslev and Ghysels [T. Bollerslev and E. Ghysels, Periodic autoregressive conditional heteroscedasticity, J. Bus. Econom. Statist. 14 1996, 2, 139–151]; these models have gained considerable interest and continued to attract the attention of researchers. This paper is devoted to extensions of the standard absolute value GARCH (AVGARCH) model to the periodically time-varying coefficients (PAVGARCH) one. In this class of models, the parameters are allowed to switch between different regimes. Moreover, these models allow to integrate asymmetric effects in the volatility, Firstly, we give necessary and sufficient conditions ensuring the existence of stationary solutions (in the periodic sense). Secondary, a quasi-maximum likelihood (QML) estimation approach for estimating the PAVGARCH model is developed. The strong consistency and the asymptotic normality of the estimator are studied given mild regularity conditions, requiring strict stationarity and the finiteness of moments of some order for the errors term. Next, we present a set of numerical experiments illustrating the practical relevance of our theoretical results. Finally, we apply our model to two foreign exchange rates: of Algerian Dinar to the European currency Euro (Euro/Dinar) and the American currency Dollar (Dollar/Dinar). This empirical work shows that our approach also outperforms and fits the data well.