NORMALISED LEAST-SQUARES ESTIMATION IN TIME-VARYING ARCH MODELS
Suhasini Subba Rao
Texas A&M University
We investigate the time-varying ARCH process. It is shown that itcan 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 (NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum-likelihood (QML) estimator for small samples. The new 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 new estimator has the same rate of convergence as the kernel QML. Due to the kernel NLS estimator's ease in 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 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.