Variance-type estimation of long memory

The aggregation procedure when a sample of length N is divided into blocks of length m=o(N), m→∞ and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu et al. (1995, Fractals, 3, 785–798), and Teverovsky and Taqqu (1997, J. Time Ser. Anal., 18, 279–304) introduced an aggregated variance estimator of the long-memory parameter of a stationary sequence with long range dependence and studied its empirical performance. With respect to autocovariance structure and marginal distribution, the aggregated series is closer to Gaussian fractional noise than the initial series. However, the variance type estimator based on aggregated data is seriously biased. A refined estimator, which employs least-squares regression across varying levels of aggregation, has much smaller bias, permitting deriviation of limiting distributional properties of suitably centered estimates, as well as of a minimum-mean squared error choice of bandwidth m. The results vary considerably with the actual value of the memory parameter.