Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

Abstract

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter d 2 [0; 1=2) to ensure that the corresponding counting process N(t) satisfies VarN(t) » Ct2d+1 (C > 0) as t ! 1, with the same memory parameter d 2 [0; 1=2) that was assumed for the durations. Thus, these conditions ensure that the memory parameter in durations propagates to the same memory parameter in the counts. We then show that any Autoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with and all finite moments yields long memory in counts, with the same d. Next, we present a result implying that the only way for a series of counts aggregated over a long time period to have nontrivial autocorrelation is for the counts to have long memory. In other words, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory. Finally, under assumptions on the pure-jump price process, we show that the memory parameter in durations propagates all the way to the realized volatility, under both calendar-time sampling and transaction-time sampling.