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

Chan, K. W. (2022) Optimal Difference-based Variance Estimators in Time Series: A General Framework. Annals of Statistics, 50, 1376–1400.

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 suboptimally 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.

R code for a simplified version of the proposed estimator (see the R-package for a better estimator):

dlrv = function(x){ n = length(x) l = ceiling(2*n^(1/5)) h = 2*l m = 3 d = c(0.1942, 0.2809, 0.3832, -0.8582) D = rep(0,n-m*h) for(j in 0:m) D = D + d[j+1] * x[(m*h+1-j*h):(n-j*h)] acvf = acf(D,type="cov",plot=FALSE ,lag.max=l)$acf acvf[1] + 2*sum(acvf[-1]*(1-((1:l)/l)^2))}