Fitting Generalized Pareto Distributions to Financial Tails

Summary

The document concerns fitting a residual distribution after an ARMA–GARCH model, using a kernel density estimate for the center and peaks-over-threshold methods for the tails. The question asks whether a generalized Pareto distribution should cover every observation or only observations beyond a chosen threshold, after encountering support limits when evaluating the fitted tail model.

The response clarifies that generalized Pareto models are intended for tail exceedances beyond a threshold, with the central distribution handled separately. It notes that selecting the threshold is a distinct modeling decision and points to statistical literature for methods and comparisons of financial tail distributions. The response does not diagnose the specific support-bound issue or recommend a single threshold, so implementation requires additional analysis of the data and fitted parameters.

Key ideas

• A generalized Pareto distribution is fitted to observations beyond a threshold in the tail.
• A separate model can describe the central part of the distribution.
• Threshold selection is a distinct problem and affects which observations enter the tail fit.
• The response does not resolve the specific support-limit issue or choose a threshold for the example.

Tags

# Modeling tail data using Generalized Pareto distribution


# Modeling tail data using Generalized Pareto distribution












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 by CFW (score 1)

https://quant.stackexchange.com/a/31442

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).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.