Variance estimation is important for statistical inference. It becomes nontrivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these nuisance structures. This paper develops a general framework for the estimation of the long-run variance for time series with nonconstant means. The building blocks are difference statistics. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first asymptotically optimal estimator is derived. Our proposed estimator is theoretically proven to be invariant to arbitrary mean structures, which may include trends and a possibly divergent number of discontinuities.
This research was partially supported by grants GRF-2130730 and GRF-2130788 provided by Research Grants Council of HKSAR.
The authors would like to thank the anonymous referees, an associate editor and the editor for their constructive comments that improved the scope and presentation of the paper.
"Optimal difference-based variance estimators in time series: A general framework." Ann. Statist. 50 (3) 1376 - 1400, June 2022. https://doi.org/10.1214/21-AOS2154