On periodic bilinear threshold $GARCH$ models

Periodic Generalized Autoregressive Conditionally Heteroscedastic ($PGARCH$) models were introduced by Bollerslev et Ghysels. These models have gained considerable interest and continued to attract the attention of researchers. This paper is devoted to extensions of the standard bilinear threshold $GARCH$ ($BLTGARCH$) model to periodically time-varying coefficients ($PBLTGARCH$) one. In this class of models, the parameters are allowed to switch between different regimes. Moreover, these models are allowed to integrate asymmetric effects in the volatility. Firstly, we give necessary and sufficient conditions ensuring the existence of stationary solutions (in periodic sense). Secondly, a quasi maximum likelihood ($QML$) estimation approach for estimating $PBLTGARCH$ model is developed. More precisely, 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. The finite-sample properties of $QMLE$ are illustrated by a Monte Carlo study. Finally our proposed model is applied to model the exchange rates of the Algerian Dinar against the single European currency ($Euro$).