Periodic Autoregressive Conditional Heteroskedasticity

High frequency asset returns generally exhibit time dependent and seasonal clustering of volatility. This paper proposes a new class of models featuring periodicity in conditional heteroskedasticity explicitly designed to capture the repetitive seasonal time variation in the second order moments. The structures of this new class of Periodic ARCH, or P-ARCH, models share many properties with the periodic ARMA processes for the mean. The implicit relation between P-GARCH structures and time-invariant seasonal weak GARCH processes documents how neglected autoregressive conditional heteroskedastic periodicity may give rise to a loss in efficiency. The importance and magnitude of this informational loss are quantified for a variety of loss functions through the use of Monte Carlo simulation methods. An empirical example for the daily bilateral Deutschemark - British Pound spot exchange rate highlights the practical relevance of the new P-GARCH class of models. Extensions to other periodic ARCH structures, including P-IGARCH and P- EGARCH processes along with possible discrete time periodic representations of stochastic volatility models subject to time deformation, are also discussed, along with issues related to multivariate representations and the possibility of common persistence in the seasonal volatility across multiple time series