# Propagation of Memory Parameter from Durations to Counts

Rohit Deo (), Clifford Hurvich, Philippe Soulier and Yi Wang
Philippe Soulier: University of Paris X
Yi Wang: New York University

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 $\textmd{Var} \, N(t) \sim C t^{2d+1}$ ($C>0$) 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 in durations propagates to the same memory parameter in counts and therefore in realized volatility. 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 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 short-term counts to have long memory. In other words, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory.

Keywords: Long Memory Stochastic Duration; Autoregressive Conditional Duration; Rosenthal-type Inequality. (search for similar items in EconPapers)
JEL-codes: C1 C2 C3 C4 C5 C8 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm and nep-ets
Date: 2005-11-08
Note: Type of Document - pdf; pages: 27
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http://econwpa.repec.org/eps/em/papers/0511/0511010.pdf (application/pdf)

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