Tight differencing in spectral density estimation with centrosymmetric kernels

Summary

Mean-robust estimation of spectral density and long-run variance is crucial for many statistical inference procedures. However, existing methods often degrade when serially dependent data exhibit volatile, time-varying trends, or sudden jumps, particularly in small samples. While differencing and kernel averaging are standard tools for achieving mean robustness and consistency, they are not inherently compatible. Combining them can compromise optimality. Specifically, tight differencing, an operation of taking small-lag differences to enhance local de-trending, introduces strong correlations that distort the high-order properties of kernel-averaged estimators. To resolve this incompatibility, we introduce a novel class of centrosymmetric kernels explicitly designed to integrate with tight differencing. We demonstrate that the optimal tight difference sequence for serially dependent data differs from classical sequences designed for independent data. Notably, these proposed optimal sequences are data-independent and can be applied directly without pre-fitting. Finally, the proposed estimators are demonstrated to be useful across various statistical inference tasks, including tests for stationarity and white noise.