Correcting the bias of the sample cross-covariance estimator

We derive the finite sample bias of the sample cross-covariance estimator based on a stationary vector-valued time series with an unknown mean. This result leads to a bias-corrected estimator of cross-covariances constructed from linear combinations of sample cross-covariances, which can in theory correct for the bias introduced by the first $h$ lags of cross-covariance with any $h$ not larger than the sample size minus two. Based on the bias-corrected cross-covariance estimator, we propose a bias-adjusted long run covariance (LRCOV) estimator. We derive the asymptotic relations between the bias-corrected estimators and their conventional Counterparts in both the small-$b$ and the fixed-$b$ limit. Simulation results show that: (i) our bias-corrected cross-covariance estimators are very effective in eliminating the finite sample bias of their conventional counterparts, with potential mean squared error reduction when the data generating process is highly persistent; and (ii) the bias-adjusted LRCOV estimators can have superior performance to their conventional counterparts with a smaller bias and mean squared error.