Long-run covariance matrix estimation is the building block of time series inference problems. The corresponding difference-based estimator, which avoids detrending, has attracted considerable interest due to its robustness to both smooth and abrupt structural breaks and its competitive finite sample performance. However, existing methods mainly focus on estimators for the univariate process while their direct and multivariate extensions for most linear models are asymptotically biased. We propose a novel difference-based and debiased long-run covariance matrix estimator for functional linear models with time-varying regression coefficients, allowing time series non-stationarity, long-range dependence, state-heteroscedasticity and their mixtures. We apply the new estimator to i) the structural stability test, overcoming the notorious non-monotonic power phenomena caused by piecewise smooth alternatives for regression coefficients, and (ii) the nonparametric residual-based tests for long memory, improving the performance via the residual-free formula of the proposed estimator. The effectiveness of the proposed method is justified theoretically and demonstrated by superior performance in simulation studies, while its usefulness is elaborated by means of real data analysis.

翻译：长期协方差矩阵估计是时间序列推断问题的基础。相应的基于差值的估计方法避免了去趋势化处理，因其对平滑突变和突发结构断点的稳健性以及具有竞争力的有限样本性能而备受关注。然而，现有方法主要关注单变量过程的估计量，而其对大多数线性模型的直接多元扩展存在渐近有偏性。我们针对变系数函数线性模型提出了一种新型的基于差值且去偏的长期协方差矩阵估计量，该模型允许时变回归系数、时间序列非平稳性、长程相依性、状态异方差性及其混合情形。我们将新估计量应用于：(i) 结构稳定性检验，克服了由分段光滑回归系数备择假设引起的著名非单调幂现象；(ii) 基于非参数残差的长记忆检验，通过所提估计量的无残差公式改进性能。所提方法的有效性在理论上得到证明，并通过模拟研究中的优越性能得到验证，其应用价值通过实际数据分析予以阐明。