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 $d \in [0,1/2)$ to ensure that the corresponding counting process $N(t)$ satisfies $Var N(t) \sim C t^{2d+1}$ ($C>0$) as $t \rightarrow \infty$, with the same memory parameter $d \in [0,1/2)$ 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 $d>0$ and all finite moments yields long memory in counts, with the same $d$. Finally, we provide some results about the propagation of long memory to the empirically-relevant case of realized variance estimates affected by market microstructure noise contamination.