Theory of Evolutionary Spectra for Heteroskedasticity and Autocorrelation Robust Inference in Possibly Misspecified and Nonstationary Models

The literature on heteroskedasticity and autocorrelation robust (HAR) inference is extensive but its usefulness relies on stationarity of the relevant process, say Vt, usually a function of the data and estimated model residuals. Yet, a large body of work shows widespread evidence of various forms of nonstationarity in the latter. Also, many testing problems are such that Vt is stationary under the null hypothesis but nonstationary under the alternative. In either case, the consequences are possible size distortions and, especially, a reduction in power which can be substantial (e.g., non-monotonic power), since all such estimates are based on weighted sums of the sample autocovariances of Vt, which are inflated. We propose HAR inference methods valid under a broad class of nonstationary processes, labelled Segmented Local Stationary, which possess a spectrum that varies both over frequencies and time. It is allowed to change either slowly and continuously and/or abruptly at some time points, thereby encompassing most nonstationary models used in applied work. We introduce a double kernel estimator (DK-HAC) that applies a smoothing over both lagged autocovariances and time. The optimal kernels and bandwidth sequences are derived under a mean-squared error criterion. The data-dependent bandwidths rely on the \plug-in" approach using approximating parametric models having time-varying parameters estimated with standard methods applied to local data. Our method yields tests with good size and power under both stationary and nonstationary, thereby encompassing previous methods. In particular, the power gains are achieved without notable size distortions, the exact size being as good as those delivered by the best fixed-b approach, when the latter works well.