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This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form.

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.

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