We consider the problem of testing for long-range dependence for time-varying coefficient regression models. The covariates and errors are assumed to be locally stationary, which allows complex temporal dynamics and heteroscedasticity. We develop KPSS, R/S, V/S, and K/S-type statistics based on the nonparametric residuals, and propose bootstrap approaches equipped with a difference-based long-run covariance matrix estimator for practical implementation. Under the null hypothesis, the local alternatives as well as the fixed alternatives, we derive the limiting distributions of the test statistics, establish the uniform consistency of the difference-based long-run covariance estimator, and justify the bootstrap algorithms theoretically. In particular, the exact local asymptotic power of our testing procedure enjoys the order $O( \log^{-1} n)$, the same as that of the classical KPSS test for long memory in strictly stationary series without covariates. We demonstrate the effectiveness of our tests by extensive simulation studies. The proposed tests are applied to a COVID-19 dataset in favor of long-range dependence in the cumulative confirmed series of COVID-19 in several countries, and to the Hong Kong circulatory and respiratory dataset, identifying a new type of 'spurious long memory'.

翻译：我们考虑长距离依赖时间变化系数回归模型的测试问题。 共差和差错假定是当地固定的, 允许复杂的时间动态和异变性。 我们根据非参数剩余量开发 KPSS、 R/ S、 V/ S 和 K/ S / S 类的统计, 并提出配有基于差异的长期差异矩阵测算仪的靴式方法, 以便实际执行。 在无效假设下, 本地替代物以及固定替代物, 我们得出测试统计数据的有限分布, 建立基于差异的长期变异估测器的统一一致性, 并在理论上证明靴式测算法的合理性。 特别是, 我们测试程序的确切本地的当地失常能力享有 $O (\log ⁇ -1} n 的顺序, 与经典的 KPSS 测试在严格固定序列中进行长期记忆, 我们通过广泛的模拟研究来证明我们测试的有效性。 拟议的测试应用于数个COVI-19 数据集, 有利于香港的长、 呼吸道和呼吸道依赖性数据序列 。