CONDITIONS FOR THE PROPAGATION OF MEMORY PARAMETER FROM DURATIONS TO COUNTS AND REALIZED VOLATILITY

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter$d \in [0,{\textstyle{1 \over 2}})$to ensure that the corresponding counting processN(t) satisfies VarN(t) ~Ct2d+1(C> 0) ast→ ∞, with the same memory parameter$d \in [0,{\textstyle{1 \over 2}})$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, whereas any long memory stochastic duration model withd> 0 and all finite moments yields long memory in counts, with the samed. 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.