Abstract:  We consider parameter estimation, hypothesis testing and variable selection for partially time-varying coecient models. Our asymptotic theory has the useful feature that it can allow dependent, nonstationary error and covariate processes.  With a two-stage method, the parametric component can be estimated with a n1=2- convergence rate. A simulation-assisted hypothesis testing procedure is proposed for testing signicance and parameter constancy. We further propose an information criterion that can consistently select the true set of signicant predictors. Our method is applied to autoregressive models with time-varying coecients. Simulation results and a real data application are provided. This is a joint work with Wei Biao Wu from The University of Chicago.