Fat tails and non-linearity in volatility models: what is more important?

Since the seminal works of Engle [7] and Bollerslev [3] about heteroskedastic return series models, many extensions of their (G)ARCH models have been proposed in the literature. In particular, the functional dependence of conditional variances and the shape of the conditional distribution of returns have been varied in several ways (see [1] and [5] for an extensive overview). These two issues have been addressed by the neural network community using multi-layer perceptrons (MLPs) and mixture density networks (MDNs) (see, e.g., [6, 8, 10]). In this paper we extend the concept of MDNs in a recurrent way to allow for “GARCH effects”. These recurrent MDNs (RMDNs) offer a consistent framework to analyze the impact of non-linearity and of non-gaussian (leptokurtic) conditional distributions on the explanatory power of volatility models. We present numerical experiments on a very large return data set the size of which allows to perform detailed statistical tests to compare the obtained results. In summary, conditional non-gaussian distributions (fat tails in the conditional distributions) tend to be more important than non-linear specifications for conditional means and variances in the likelihood framework. With respect to other error measures however, the application of non-linear neural networks seems to be promising. We think that the choice of a particular model for predicting volatility