Statistical Inference for Time - Inhomogeneous Volatility Models

This paper offers a new approach for estimating and forecasting the volatility of financial time series. No assumption is made about the parametric form of the processes. On the contrary, we only suppose that the volatility can be approximated by a constant over some interval. In such a framework, the main problem consists of filtering this interval of time homogeneity; then the estimate of the volatility can be simply obtained by local averaging. We construct a locally adaptive volatility estimate (LAVE) which can perform this task and investigate it both from the theoretical point of view and through Monte Carlo simulations. Finally, the LAVE procedure is applied to a data set of nine exchange rates and a comparison with a standard GARCH model is also provided. Both models appear to be capable of explaining many of the features of the data; nevertheless, the new approach seems to be superior to the GARCH method as far as the out-of-sample results are concerned. 1. Introduction. The aim of this paper is to offer a new perspective for the estimation and forecasting of the volatility of financial asset returns such as stocks and exchange rate returns. A remarkable amount of statistical research is devoted to financial time series, in particular, to the volatility of asset returns, where the term volatility indicates a measure of dispersion, usually the variance or the standard deviation. The interest in this topic is motivated by the needs of the financial industry, which regards volatility as one of the main reference numbers for risk management and derivative pricing. Actually, asset returns time series display very peculiar stylized facts, which are connected with their second moments. Graphically, they look like