This paper establishes consistency and asymptotic normality of the generalized quasi-maximum likelihood estimate (GQM LE) for a general class of periodic conditionally heteroskedastic time series models (P CH). In this class of models, the volatility is expressed as a measurable function of the in…nite past of the observed process with periodically time-varying parameters, while the innovation of the model is an independent and periodically distributed sequence. In contrast with the aperiodic case, the proposed GQM LE is rather based on S instrumental density functions where S is the period of the model while the corresponding asymptotic variance is in a "sandwich" form. Application to the periodic GARCH and the periodic asymmetric power GARCH model is given. Moreover, we discuss how to apply the GQM LE to the prediction of power problem in a one-step framework and to P CH models with complex periodic patterns such as high frequency seasonality and non-integer seasonality.Keywords: Periodic conditionally heteroskedastic models, periodic asymmetric power GARCH, generalized QM L estimation, consistency and asymptotic normality, prediction of powers, high frequency periodicity, non-integer periodicity.

We propose a general class of non-linear mixed Poisson autoregressions whose form and parameters are periodic over time. Under a periodic contraction condition on the forms of the conditional mean, we show the existence of a unique nonanticipative solution to the model, which is strictly periodically stationary, periodically ergodic and periodically weakly dependent having in the pure Poisson case …nite higher-order moments. Applications to some well-known integer-valued time series models are considered.