Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time series analysis. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by a white-noise-driven auto-regressive (AR) process of slowly diverging order. To our best knowledge, it is the first time such a structural approximation result is established for general classes of non-stationary time series. A high dimensional $\mathcal{L}^2$ test and an associated multiplier bootstrap procedure are proposed for the inference of the AR approximation coefficients. In particular, an adaptive stability test is proposed to check whether the AR approximation coefficients are time-varying, a frequently-encountered question for practitioners and researchers of time series. As an application, globally optimal short-term forecasting theory and methodology for a wide class of locally stationary time series are established via the method of sieves.
翻译:了解复杂时间系统的时间变化结构是现代时间序列分析的主要挑战之一。 在本文中,我们表明,每一个统一正态不变和足够短距离依赖的非静止和非线性时间序列都可以通过白噪音驱动的缓慢变化的自动递减(AR)过程在全球范围内加以比较。据我们所知,这是第一次为非静止时间序列的普通类确定这种结构近似结果。为推断AR近似系数,提出了高维值($mathcal{L ⁇ 2$)测试和相关的倍增靴套件程序。特别是,提议进行适应性稳定测试,以检查AR近似系数是否具有时间变化性,这是时间序列从业人员和研究人员经常遇到的问题。作为一种应用,通过Sieves方法为广泛的当地固定时间序列建立了全球最佳短期预测理论和方法。