The paper considers simultaneous nonparametric inference for a wide class of M-regression models with time-varying coefficients. The covariates and errors of the regression model are tackled as a general class of nonstationary time series and are allowed to be cross-dependent. We construct $\sqrt{n}$-consistent inference for the cumulative regression function, whose limiting properties are disclosed using Bahadur representation and Gaussian approximation theory. A simple and unified self-convolved bootstrap procedure is proposed. With only one tuning parameter, the bootstrap consistently simulates the desired limiting behavior of the M-estimators under complex temporal dynamics, even under the possible presence of breakpoints in time series. Our methodology leads to a unified framework to conduct general classes of Exact Function Tests, Lack-of-fit Tests, and Qualitative Tests for the time-varying coefficients under complex temporal dynamics. These tests enable one to, among many others, conduct variable selection procedures, check for constancy and linearity, as well as verify shape assumptions, including monotonicity and convexity. As applications, our method is utilized to study the time-varying properties of global climate data and Microsoft stock return, respectively.

翻译：本文考虑了具有时变系数的一类广泛M回归模型的同时非参数推断。回归模型的协变量和误差被处理为一般类别的非平稳时间序列，并允许存在交叉依赖关系。我们构建了累积回归函数的$\sqrt{n}$一致推断，其极限性质通过Bahadur表示和高斯逼近理论揭示。提出了一种简单且统一的自卷积自助法。该方法仅需一个调优参数，即可在复杂时变动态下（即使时间序列可能存在断点）一致地模拟M估计量的期望极限行为。我们的方法为复杂时变动态下时变系数的一般类精确函数检验、拟合优度检验和定性检验提供了统一框架。这些检验能够实现变量选择过程的执行、常数性和线性性的检验，以及形状假设（包括单调性和凸性）的验证。作为应用实例，我们分别利用该方法研究了全球气候数据和微软股票回报的时变特性。