A Simple Nearly Unbiased Estimator of Cross-Covariances

In this article, we propose a simple estimator of cross-covariance matrices for a multi-variate time series with an unknown mean based on a linear combination of the circular sample cross-covariance estimator. Our estimator is exactly unbiased when the data generating process follows a vector moving average (VMA) model with an order less than one half of the sampling period, and is nearly unbiased if such VMA model can approximate the data generating process well. In addition, our estimator is shown to be asymptotically equivalent to the conventional sample cross-covariance estimator. Via simulation, we show that the proposed estimator can to a large extent eliminate the finite sample bias of cross-covariance estimates, while not necessarily increase the mean squared error.