This article establishes the strong consistency and asymptotic normality (CAN) of the quasi-maximum likelihood estimator (QMLE) for generalized autoregressive conditionally heteroscedastic (GARCH) and autoregressive moving-average (ARMA)-GARCH processes with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution of the periodic GARCH (PGARCH) equation. As a result, it is shown that the moment of some positive order of the PGARCH solution is finite, under which we prove the strong consistency and asymptotic normality of the QMLE for a PGARCH process without any condition on its moments and for a periodic ARMA-GARCH (PARMA-PGARCH) under mild conditions. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
In this paper, a class of bilinear time series models with time varying coefficients is considered. In this nonstationary and nonlinear framework, our aim is to study the structure of usual time series analysis tools, in particular the sample autocovariance function which has been developed for analyzing stationary linear time series. We use appropriately defined Markovian representations to derive a necessary and sufficient condition for the existence and uniqueness of a solution with bounded first and second order moments (BFSM). A more explicit sufficient condition for the existence of a BFSM solution is provided. An explicit expression of the autocovariance function is obtained. The existence of a weak time-varying ARMA representation of the bilinear model with time varying coefficients is shown. We also discuss the existence of higher order moments. Several subclasses of the model are shown to be quasi-stationary. Under this assumption of quasi-stationarity, the asymptotic distributions of the sample mean and sample covariances are obtained.
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