Modeling tail data using Generalized Pareto distribution

Question

I just estimated a ARMA(1,1)+GARCH(1,1)+Threshold order(1) equation for time series of stock prices. Now I'm going to estimate the residuals' marginal distributions using the kernel density estimator in the interior of the distribution and the POT method in the tails using 10% of the data points for each tail. I calculated cdf for all tail points using estimated parameters $(\mu=-5.98,\sigma=36.342 ,\varepsilon=-7.04)$ but the function doesn't support all of data points in tails. $(\mu < x < \mu-\frac{\sigma}{\varepsilon})$ Am I supposed to fit GPD to all data not just tails? what are the other distributions appropriate for modeling tail data?

Answer 1

You might be interested in this ARTICLE (published in Quantitative Finance 2016) and citations therein. The authors consider different distributions to model tails in financial time series and in particular focus on EVT/GPDs.

GPDs are used to specifically model tails and hence are fitted after some threshold that separates the tail from the central region of the distribution. Threshold choice is a separate discussion, the literature provides several methods (e.g. see article above).