We consider estimating nonparametric time-varying parameters in linear models using kernel regression. Our contributions are threefold. First, we consider a broad class of time-varying parameters including deterministic smooth functions, the rescaled random walk, structural breaks, the threshold model and their mixtures. We show that those time-varying parameters can be consistently estimated by kernel regression. Our analysis exploits the smoothness of the time-varying parameter quantified by a single parameter. The second contribution is to reveal that the bandwidth used in kernel regression determines a trade-off between the rate of convergence and the size of the class of time-varying parameters that can be estimated. We demonstrate that an improper choice of the bandwidth yields biased estimation, and argue that the bandwidth should be selected according to the smoothness of the time-varying parameter. Our third contribution is to propose a data-driven procedure for bandwidth selection that is adaptive to the smoothness of the time-varying parameter.