We are evaluating a model for risk management based on extreme value theory using peaks over threshold and markov chain monte carlo methods.
In doing this, we are firstly fitting a GARCH (we have tried GARCH(1,1), E-GARCH, Asymmetric GARCH, GJR-GARCH, ...) model in order to filter the return series.
We are encountering a problem here however, wherein our filtered return distribution is far less leptokurtic than the original one, even when we use e.g. Student T or Generalized Hyperbolic distributions for the innovations. Informally speaking, we feel the GARCH model is over-reactive and negatively affects the subsequent POT step.
Are we missing something here? Is it, in practice, better to use a simpler (e.g. EWMA) volatility model? Any insight is highly appreciated.