  We establish theoretical results about the low frequency contamination (i.e.,
long memory effects) induced by general nonstationarity for estimates such as
the sample autocovariance and the periodogram, and deduce consequences for
heteroskedasticity and autocorrelation robust (HAR) inference. We present
explicit expressions for the asymptotic bias of these estimates. We distinguish
cases where this contamination only occurs as a small-sample problem and cases
where the contamination continues to hold asymptotically. We show theoretically
that nonparametric smoothing over time is robust to low frequency
contamination. Our results provide new insights on the debate between
consistent versus inconsistent long-run variance (LRV) estimation. Existing LRV
estimators tend to be in inflated when the data are nonstationary. This results
in HAR tests that can be undersized and exhibit dramatic power losses. Our
theory indicates that long bandwidths or fixed-b HAR tests suffer more from low
frequency contamination relative to HAR tests based on HAC estimators, whereas
recently introduced double kernel HAC estimators do not super from this
problem. Finally, we present second-order Edgeworth expansions under
nonstationarity about the distribution of HAC and DK-HAC estimators and about
the corresponding t-test in the linear regression model.
