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

Since the seminal works of R.F. Engle (1982) and T. Bollerslev (1986) 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 (A.K. Bera and M.L. Higgins, 1993; T. Bollerslev et al., 1992). These two issues have been addressed by the neural network community using multi-layer perceptrons (MLPs) and mixture density networks (MDNs). 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 nonlinearity 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 one 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 nonlinear specifications for conditional means and variances in the likelihood framework. With respect to other error measures however, the application of nonlinear neural networks seems to be promising. We think that the choice of a particular model for predicting volatility is closely related to the question of how to measure the prediction performance of a model