Effects of aggregation on estimators of long-range dependence

Modern technologies have made available huge amount of data from phenomena that exhibit long-range dependence (LRD). For example, network traffic data can be sampled at intervals as small as one nanosecond. The analysis of such a big amount of data can pose practical challenges. Temporal aggregation has been employed to cope with the limitations in the storage capacity and analysis tools.; One question that arises immediately is whether the aggregated process has the same long-range dependence characteristics of the underlying process. Under mild local assumptions on the spectral density of the LRD process, we show that the fractional order d of the processes is invariant under temporal aggregation.; A second related question is whether and how the estimators of long-range dependence are affected by aggregation. We focus our attention on the log-periodogram regression estimator of Geweke and Porter-Hudak (GPH) and on the local Whittle (LW) estimator. For such estimators, a trimming parameter m needs to chosen by the user so as to balance the trade-off between bias and variance. One optimal choice of m is the value that minimizes the asymptotic mean squared error (MSE) of the estimator. We derive the asymptotic MSE of the GPH estimator for the model in consideration. We show how the MSE-optimal choice of m varies under aggregation for both the GPH and LW estimators.; We consider also the case when the sum of the LRD process and a white noise process is observed. This model emerges from long-memory stochastic volatility models (LMSV). We derive an expression for the asymptotic MSE of LW estimator. A LW-type estimator (ELW) which accounts for the presence of noise is also considered. We derive a representation of the Hessian matrix of the ELW estimator as functionals of incomplete beta functions. We evaluate numerically the effect of aggregation on the LW and ELW estimators when the observed process is composed by a LRD process plus noise.; We perform an empirical analysis of the effects of aggregation on the UNC network data.