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.
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Paper provided by EconWPA in its series Econometrics with number
0511010.
Find related papers by JEL classification: C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics C5 - Mathematical and Quantitative Methods - - Econometric Modeling C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
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Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005.
"Estimating Long Memory in Volatility,"
Econometrica,
Econometric Society, vol. 73(4), pages 1283-1328, 07.
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