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.