Tail behavior and dependence structure the APARCH model

The APARCH model is a generalization of the GARCH model that attempts to capture asymmetric responses of returns and of volatility to positive and negative `news shocks' -the phenomenon known as the leverage effect. Despite its potential, the model's mathematical properties have not yet been fully investigated. While the capacity of the model to account for the leverage effect is clear from its defining structure, little is known how the effect is quantifed in terms of the model's parameters. The same applies to the quantifcation of heavy tails and time dependence. Here, in an attempt to fill this void, we study the model in further detail. We obtain sufficient conditions of its existence in different metrics as well as explicit forms of important characteristics: skewness, kurtosis, correlations and leverage. Utilizing these results, we analyze the roles of the parameters and discuss statistical inference. We also propose a natural extension by introducing an additional parameter and discuss how it affects the model. Through theoretical results and a Monte Carlo study we demonstrate that the model can produce heavy-tailed data. We illustrate these properties using S&P500 data as well as country indices for dominant European economies