abstract: We aim at modeling the interplay between the tail behavior and the dependence structure of financial data. log-returns of financial data often exhibit long range dependence effects (which affect the sample autocorrelation behavior of the absolute log-returns) and erratic behavior which results in heavy-tailed marginal distributions. In practice one often observes that financial log-return series can have infinite 3rd, 4th, 5th,... moments. We consider too major classes of econometric models which try to capture the empirical behavior. Those include GARCH (generalized autoregressive conditionally heteroscedastic) and stochastic volatility processes. The GARCH case turns out to be a very complex one. We embed these processes in finite-dimensional stochastic recurrence equations. Following classical work of Kesten, we show that the finite-dimensional distributions of these processes are multivariate regularly varying, thus they have infinite power moments of a certain degree. Heavy tails cause the classical limit theory for the sample autocorrelations to break down. Therefore one has to modify this asymptotic theory: unfamiliar infinite variance limit distributions occur and rates of convergence can be extremely slow. Point process techniques turn out to be important in this context.
The long range dependence effect of real-life data cannot be explained by GARCH models. A possible explanation for this effect is non-stationarity of the underlying time series. We will learn how statistical tools behave under non-stationarity and how they can fool us to see things which are not there.