  We establish new mean-squared error (MSE) bounds for long-run variance (LRV)
estimation, valid for both stationary and nonstationary sequences that are
sharper than previously established. The key element to construct such bounds
is to use restrictions onthe degree of nonstationarity. Unlike previous bounds,
they show how nonstationarity influences the bias-variance trade-off. Unlike
previously established bounds, either under stationarity or nonstationarity,
the new bounds depends on the form of nonstationarity. The bounds are
established for double kernel long-run variance estimators. The corresponding
bounds for classical long-run variance estimators follow as a special case. We
use them to construct new data-dependent methods for the selection of
bandwidths for (double) kernel heteroskedasticity autocorrelation consistent
(DK-HAC) estimators. These account more flexibly for nonstationarity and lead
to tests with The new MSE bounds and associated bandwidths help to to improve
good finite-sample performance, especially good power when existing LRV
estimators lead to tests having little or no or no power. The second
contribution is to introduce a nonparametric nonlinear VAR prewhitened LRV
estimator. This accounts explicitly for nonstationarity unlike previous
prewhitened procedures which are known to be unstable. Its consistency, rate of
convergence and MSE bounds are established. The prewhitened DK-HAC estimators
lead to tests with good finite-sample size while maintaining good monotonic
power.
