  We show that the nonstandard limiting distribution of HAR test statistics
under fixed-b asymptotics is not pivotal (even after studentization) when the
data are nonstationarity. It takes the form of a complicated function of
Gaussian processes and depends on the integrated local long-run variance and on
on the second moments of the relevant series (e.g., of the regressors and
errors for the case of the linear regression model). Hence, existing fixed-b
inference methods based on stationarity are not theoretically valid in general.
The nuisance parameters entering the fixed-b limiting distribution can be
consistently estimated under small-b asymptotics but only with nonparametric
rate of convergence. Hence, We show that the error in rejection probability
(ERP) is an order of magnitude larger than that under stationarity and is also
larger than that of HAR tests based on HAC estimators under conventional
asymptotics. These theoretical results reconcile with recent finite-sample
evidence in Casini (2021) and Casini, Deng and Perron (2021) who showing that
fixed-b HAR tests can perform poorly when the data are nonstationary. They can
be conservative under the null hypothesis and have non-monotonic power under
the alternative hypothesis irrespective of how large the sample size is.
