  We consider estimating nonparametric time-varying parameters in linear models
using kernel regression. Our contributions are twofold. 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
time-varying parameters rather than their specific form. The second
contribution is to reveal that the bandwidth used in kernel regression
determines the trade-off between the rate of convergence and the size of the
class of time-varying parameters that can be estimated. An implication from our
result is that the bandwidth should be proportional to $T^{-1/2}$ if the
time-varying parameter follows the rescaled random walk, where $T$ is the
sample size. We propose a specific choice of the bandwidth that accommodates a
wide range of time-varying parameter models. An empirical application shows
that the kernel-based estimator with this choice can capture the random-walk
dynamics in time-varying parameters.
