Self-convolved Bootstrap for M-regression under Complex Temporal Dynamics

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.