We develop a theory of evolutionary spectra for heteroskedasticity and autocorrelation robust (HAR) inference when the data may not satisfy second-order stationarity. Nonstationarity is a common feature of economic time series which may arise either from parameter variation or model misspecification. In such a context, the theories that support HAR inference are either not applicable or do not provide accurate approximations. HAR tests standardized by existing long-run variance estimators then may display size distortions and little or no power. This issue can be more severe for methods that use long bandwidths (i.e., fixed-b HAR tests). We introduce a class of nonstationary processes that have a time-varying spectral representation which evolves continuously except at a finite number of time points. We present an extension of the classical heteroskedasticity and autocorrelation consistent (HAC) estimators that applies two smoothing procedures. One is over the lagged autocovariances, akin to classical HAC estimators, and the other is over time. The latter element is important to flexibly account for nonstationarity. We name them double kernel HAC (DK-HAC) estimators. We show the consistency of the estimators and obtain an optimal DK-HAC estimator under the mean squared error (MSE) criterion. Overall, HAR tests standardized by the proposed DK-HAC estimators are competitive with fixed-b HAR tests, when the latter work well, with regards to size control even when there is strong dependence. Notably, in those empirically relevant situations in which previous HAR tests are undersized and have little or no power, the DK-HAC estimator leads to tests that have good size and power.

翻译：本文针对数据可能不满足二阶平稳性的情况，发展了异方差与自相关稳健（HAR）推断的演化谱理论。非平稳性是经济时间序列的常见特征，可能由参数变化或模型误设引起。在此背景下，支持HAR推断的现有理论要么不适用，要么无法提供精确近似。使用现有长程方差估计量标准化的HAR检验因此可能出现尺度失真，并表现出微弱甚至无效的检验功效。对于采用较长带宽的方法（即固定带宽HAR检验），该问题可能更为严重。我们引入了一类具有时变谱表示的非平稳过程，该谱表示在除有限个时间点外的整个时间轴上连续演化。我们提出了经典异方差与自相关一致（HAC）估计量的扩展形式，该形式采用双重平滑程序：一是对滞后自协方差进行平滑（类似于经典HAC估计量），二是沿时间维度进行平滑。后者对于灵活处理非平稳性至关重要。我们将其命名为双核HAC（DK-HAC）估计量。我们证明了该估计量的一致性，并在均方误差（MSE）准则下获得了最优DK-HAC估计量。总体而言，采用所提DK-HAC估计量标准化的HAR检验，在固定带宽HAR检验表现良好的情况下，即使在强依赖性条件下，其尺度控制能力仍具有竞争力。值得注意的是，在以往HAR检验存在尺度不足且功效微弱甚至无效的实证相关情境中，DK-HAC估计量能够使检验保持良好的尺度性质与检验功效。