We develop a class of ARCH models for series sampled at unequal time intervals set by trade orquote arrivals. Our approach combines insights from the temporal aggregation for GARCH models discussed byDrost and Nijman (1993) and Drost and Werker (1996), and the autoregressive conditional duration model ofEngle and Russell (1996) proposed to model the spacing between consecutive financial transactions.The class of models introduced here will be called ACD-GARCH. It can be described as a random coefficientGARCH, or doubly stochastic GARCH, where the durations between transactions determine the parameterdynamics. The ACD-GARCH model becomes genuinely bivariate when past asset-return volatilities are allowedto affect transaction durations, and vice versa. Otherwise, the spacings between trades are consideredexogenous to the volatility dynamics. This assumption is required in a two-step estimation procedure. Thebivariate setup enables us to test for Granger causality between volatility and intratrade durations. Undergeneral conditions, we propose several Generalized Method of Moments (GMM) estimation procedures, somehaving a Quasi Maximum Likelihood Estimation (QMLE) interpretation. As illustration, we present anempirical study of the IBM 1993 tick-by-tick data. We find some evidence that volatility of IBM stock pricesGranger-causes intratrade durations. We also find that the persistence in GARCH drops dramatically onceintratrade durations are taken into account.
