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Normalised least
squares estimation in time-varying ARCH models

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
normalised-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.

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