The fixed-b limiting distribution and the ERP of HAR tests under nonstationarity

We show that the limiting distribution of HAR test statistics under fixed-b asymptotics is not pivotal when the data are nonstationary (i.e., time-varying autocovariance structure). It takes the form of a complicated function of Gaussian processes and depends on the second moments of the relevant series (e.g., of the regressors and errors for the case of the linear regression model). Hence, 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. We show that the error in rejection probability (ERP) is an order of magnitude larger than that under stationarity and can be also larger than that of HAR tests based on HAC estimators under conventional asymptotics. These theoretical results reconcile with recent finite-sample evidence showing that existing 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. Based on the new nonstationary fixed-b distribution, we propose a feasible method that controls the null rejection rates well regardless of whether the data are stationary or not and of the strength of the serial dependence as verified for some representative data-generating processes in a simple location model.