A general class of non-stationary time series is considered in this paper. We estimate the time-varying coefficients by using local linear M-estimation. For these estimators, weak Bahadur representations are obtained and are used to construct simultaneous confidence bands. For practical implementation, we propose a bootstrap based method to circumvent the slow logarithmic convergence of the theoretical simultaneous bands. Our results substantially generalize and unify the treatments for several time-varying regression and auto-regression models. The performance for tvARCH and tvGARCH models is studied in simulations and a few real-life applications of our study are presented through the analysis of some popular financial datasets.

本文考虑了一类一般的非平稳时间序列。我们通过使用局部线性 M 估计来估计时变系数。对于这些估计器，获得了弱 Bahadur 表示并用于构建同时置信带。为了实际实现，我们提出了一种基于引导的方法来规避理论同步频带的缓慢对数收敛。我们的结果基本上概括和统一了几个时变回归和自回归模型的处理。在模拟中研究了 tvARCH 和 tvGARCH 模型的性能，并通过对一些流行的金融数据集的分析展示了我们研究的一些实际应用。