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.

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