Optimal Difference-based Variance Estimators in Time Series: A General Framework

TL;DR

This paper introduces a comprehensive framework for estimating long-run variance in time series with complex mean structures, using difference-based estimators that are proven to be asymptotically optimal and invariant to various mean patterns.

Contribution

It develops a general, theoretically grounded class of difference-based variance estimators that handle non-constant means and establishes their optimality and invariance properties.

Findings

Proposes a new class of difference-based estimators.

Establishes necessary and sufficient conditions for consistency.

Derives the first asymptotically optimal estimator.

Abstract

Variance estimation is important for statistical inference. It becomes non-trivial 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 non-constant 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.