CONDITIONS FOR THE PROPAGATION OF MEMORY PARAMETER FROM DURATIONS TO COUNTS AND REALIZED VOLATILITY
研究了持续期序列的记忆参数如何传播到计数过程和已实现波动率，给出了确保记忆参数不变的条件，并比较了ACD模型与长记忆随机持续期模型的记忆性质。
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 process N ( t ) satisfies Var N ( t ) ~ Ct 2 d +1 ( C > 0) as t → ∞, 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 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.