  Lag windows are commonly used in time series, econometrics, steady-state
simulation, and Markov chain Monte Carlo to estimate time-average covariance
matrices. In the presence of positive correlation of the underlying process,
estimators of this matrix almost always exhibit significant negative bias,
leading to undesirable finite-sample properties. We propose a new family of lag
windows specifically designed to improve finite-sample performance by
offsetting this negative bias. Any existing lag window can be adapted into a
lugsail equivalent with no additional assumptions. We use these lag windows
within spectral variance estimators and demonstrate its advantages in a linear
regression model with autocorrelated and heteroskedastic residuals. We further
employ the lugsail lag windows in weighted batch means estimators due to their
computational efficiency on large simulation output. We obtain bias and
variance results for these multivariate estimators and significantly weaken the
mixing condition on the process. Superior finite-sample properties are
illustrated in a vector autoregressive process and a Bayesian logistic
regression model.
