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 piece-wise locally stationary time series and are allowed to be cross-dependent. We introduce an integration technique to study the M-estimators, whose limiting properties are disclosed using Bahadur representation and Gaussian approximation theory. Facilitated by a self-convolved bootstrap proposed in this paper, we introduce a unified framework to conduct general classes of Exact Function Tests, Lack-of-fit Tests, and Qualitative Tests for the time-varying coefficient M-regression under complex temporal dynamics. As an application, our method is applied to studying the anthropogenic warming trend and time-varying structures of the ENSO effect using global climate data from 1882 to 2005.

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