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. A novel and easy-to-implement self-convolved bootstrap procedure is proposed. With only one tuning parameter, the bootstrap facilitates a $\sqrt{n}$-consistent inference of the cumulative regression function for 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. These tests enable one to, among many others, conduct variable selection, 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：本文考虑了一类具有时变系数的M回归模型的同步非参数推断问题。回归模型的协变量和误差被视为一类广义非平稳时间序列，并允许存在交叉依赖性。我们提出了一种新颖且易于实现的自卷积自助法。该自助法仅需一个调优参数，即可在复杂时间动态下，即使时间序列可能存在断点，为M估计量的累积回归函数提供$\sqrt{n}$一致推断。我们的方法形成了一个统一框架，用于对时变系数进行广义的精确函数检验、失拟检验和定性检验。这些检验使得研究者能够进行变量选择、检验常数性和线性性，以及验证形状假设（包括单调性和凸性）等多种分析。作为应用，我们的方法分别用于研究全球气候数据和微软股票收益率的时变特性。