Periodic Autoregressive Conditional Heteroscedasticity

Abstract

Most high-frequency asset returns exhibit seasonal volatility patterns. This article proposes a new class of models featuring periodicity in conditional heteroscedasticity explicitly designed to capture the repetitive seasonal time variation in the second-order moments. This new class of periodic autoregressive conditional heteroscedasticity, or PARCH , models is directly related to the class of periodic autoregressive moving average (ARMA) models for the mean. The implicit relation between periodic generalized ARCH (P-GARCH) structures and time-invariant seasonal weak GARCH processes documents how neglected autoregressive conditional heteroscedastic periodicity may give rise to a loss in forecast 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. Two empirical examples with daily bilateral Deutschemark/British pound and intraday Deutschemark/U.S. dollar spot exchange rates highlight the practical relevance of the new P-GARCH class of models. Extensions to discrete-time periodic representations of stochastic volatility models subject to time deformation are briefly discussed.

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“…Since the introduction of ARCH models by Engle (1982) and their generalized version (GARCH) by Bollerslev (1986), univariate volatility modeling has been an important research topic. More recently, multivariate GARCH (MGARCH) models have gained relevance as interest in understanding volatility spillovers across different markets has increased (Bollerslev et al 1988, Engle andKroner 1995).…”

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