Long-run covariance matrix estimation is the building block of time series inference. 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 via real data analysis. Our method is implemented in the R package mlrv.

翻译：长程协方差矩阵估计是时间序列推断的基础。基于差分的估计量无需去趋势处理，因其对平滑和突变结构断点的稳健性及优异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，而其对大多数线性模型的直接多元扩展存在渐近偏差。本文针对时变回归系数的函数线性模型提出一种新型差分去偏长程协方差矩阵估计量，该模型允许时间序列非平稳性、长程依赖性、状态异方差性及其混合形式。我们将新估计量应用于：(i) 结构稳定性检验，克服了由回归系数分段平滑备择导致著名的非单调功效现象；(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提升性能。理论分析证明了所提方法的有效性，模拟研究展示了其优越性能，并通过实际数据分析阐明了其实用价值。该方法已在R包mlrv中实现。