Exactly/Nearly Unbiased Estimation of Autocovariances of a Univariate Time Series With Unknown Mean

摘要

This article proposes an exactlyearly unbiased estimator of the autocovariance function of a univariate time series with unknown mean. The estimator is a linear function of the usual sample autocovariances computed using the observed demeaned data. The idea is to stack the usual sample autocovariances into a vector and show that the expectation of this vector is a linear combination of population autocovariances. A matrix that we label, A, collects the weights in these linear combinations. When the population autocovariances of high lags are zero (small), exactly (nearly) unbiased estimators of the remaining autocovariances can be obtained using the inverse of upper blocks of the A matrix. The A-matrix estimators are shown to be asymptotically equivalent to the usual sample autocovariance estimators. The A-matrix estimators can be used to construct estimators of the autocorrelation function that have less bias than the usual estimators. Simulations show that the A-matrix estimators can substantially reduce bias while not necessarily increasing mean square error. More powerful tests for the null hypothesis of white noise are obtained using the A-matrix estimators.