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Abstract:
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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) ~ C t**(2d+1) as t-> infinity, (C>0), so that the memory parameter in durations propagates to the same memory parameter in 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. Next, we motivate the need for long memory: aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory. Finally, for a simple pure-jump price process, we show that the memory parameter in durations propagates all the way to the realized volatility.
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