Asymptotic properties of QMLE for periodic asymmetric strong and semi-strong GARCH models.

In this paper, we propose a natural extension of time-invariant coefficients threshold GARCH (TGARCH) processes to periodically time-varying coefficients (PTGARCH) one. So some theoretical probabilistic properties of such models are discussed, in particular, we establish firstly necessary and sufficient conditions which ensure the strict stationarity and ergodicity (in periodic sense) solution of PTGARCH. Secondary, we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) for estimating the unknown parameters of the model. More precisely, the strong consistency and the asymptotic normality of QMLE are studied in cases when the innovation process is an i.i.d (Strong case) and/or is not (Semi-strong case). The finite-sample properties of QMLE are illustrated by a Monte Carlo study. Our proposed model is applied to model the exchange rates of the Algerian Dinar against the U.S-dollar and the single European currency (Euro).