Clustering heteroskedastic time series by model-based procedures

The interest toward the classification of time series has recently received a lot of contributions (see Piccolo, 2007, for a review). Most of these studies are devoted to capture the structure of the mean of the process hypothesized as generator of the data, whereas little attention has been devoted to the variance. When dealing with heteroskedastic time series, in which the (conditional) variance follows a stochastic process (typically a GARCH process), the comparison of the dynamics of the variances is fundamental. This is particularly important dealing with financial time series, when the investor has a very large opportunity set (hundreds of stocks) and he would like to have groups of series with similar characteristics (similar unconditional variance, similar dynamics, etc.). Moreover, the volatility of a return is generally considered as a proxy of the risk of the same return; in other words the classification of returns of several assets is equivalent to classify the assets in clusters with similar risk. In this paper we propose a clustering procedure based on simple statistical tools. In particular, we consider the squared disturbances of the returns of a financial time series as a measure of the volatility of the series. Then, we use the GARCH representation of the conditional variance to derive the model underlying the squared disturbances. We classify the series with similar unconditional volatility, similar time-varying volatility and/or similar volatility structure, using classical Wald statistics.