  We investigate the time-varying ARCH (tvARCH) process. It is shown that it
can be used to describe the slow decay of the sample autocorrelations of the
squared returns often observed in financial time series, which warrants the
further study of parameter estimation methods for the model. Since the
parameters are changing over time, a successful estimator needs to perform well
for small samples. We propose a kernel normalized-least-squares (kernel-NLS)
estimator which has a closed form, and thus outperforms the previously proposed
kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The
kernel-NLS estimator is simple, works under mild moment assumptions and avoids
some of the parameter space restrictions imposed by the kernel-QML estimator.
Theoretical evidence shows that the kernel-NLS estimator has the same rate of
convergence as the kernel-QML estimator. Due to the kernel-NLS estimator's ease
of computation, computationally intensive procedures can be used. A
prediction-based cross-validation method is proposed for selecting the
bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap
scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain
pointwise confidence intervals for the kernel-NLS estimator. It is shown that
distributions of the estimator using the bootstrap and the ``true'' tvARCH
estimator asymptotically coincide. We illustrate our estimation method on a
variety of currency exchange and stock index data for which we obtain both good
fits to the data and accurate forecasts.
