Statistica Sinica 28 (2018), 319-338

A THRESHOLDING-BASED PREWHITENED

LONG-RUN VARIANCE ESTIMATOR AND

ITS DEPENDENCE-ORACLE PROPERTY

Ting Zhang

Boston University

Abstract: Statistical inference of time series data routinely relies on the estimation of long-run variances, defined as the sum of autocovariances of all orders. The current paper considers a new class of long-run variance estimators that first soaks up the dependence by a decision-based prewhitening filter, then regularizes autocorrelations of the resulting residual process by thresholding, and finally recolors back to obtain an estimator of the original process. Under mild regularity conditions, we prove that the proposed estimator (i) consistently estimates the long-run variance; (ii) achieves the parametric convergence rate when the underlying process has a sparse dependence structure as in finite-order moving average models; and (iii) enjoys the dependence-oracle property in the sense that it automatically reduces to the sample variance if the data are actually independent. Monte Carlo simulations were conducted to examine its finite-sample performance and make comparisons with existing estimators.

Key words and phrases: Long-run variance, prewhitening, thresholding.