Abstract

The paper considers simultaneous nonparametric inference for M-regression models with time-varying coefficients, where the covariates and errors are both nonstationary time series. Complicated as they are, the M-type estimators can be obtained by local linear estimation with kernels and simplified using Bahadur representation. Furthermore, the limiting properties of the estimators can be retrieved via Gaussian Approximation techniques. We also consider the integrated process of the regression coefficients. Similar limiting properties of the estimators are disclosed accordingly, which prove to have the optimal convergence rate.

Facilitated by multiplier bootstrap, the Exact Function Test and Lack-of-fit Test are proposed for the integrated process of regression coefficients with asymptotically correct accuracy. These tests enable people to perform the classical variable selection procedure and to check whether the coefficient belongs to a particular parametric family. We also propose a unified framework to conduct a general class of qualitative hypothesis testing. This can be extremely useful in conducting shape tests such as monotonicity and convexity. As an application, our method is applied to studying the warming trend and time-varying structures of the ENSO effect using global climate data from 1882 to 2005.