Choosing a Skewed Generalized Error Distribution for GARCH Models

Summary

The document raises a specification problem when using the skewed generalized error distribution in GARCH estimation. It compares the SGED formulation attributed to Theodossiou across several papers, noting that the stated probability density functions and parameter definitions differ. It also observes that two formulations produce nearly identical results in the particular case under consideration, and distinguishes them from the Fernández–Steel skewed distribution used by the rugarch package.

The post asks which density should be selected and where to find the Fernández–Steel definition. It supplies references to papers containing alternative formulations, but does not provide a resolution, derive the densities, or establish that the formulations are generally interchangeable. The practical lesson is to identify the exact parameterization required by an estimation method or software implementation before fitting a GARCH model. Similar names do not guarantee identical distributions, and agreement in one application is not evidence of equivalence across data or parameter settings.

Key ideas

• Several papers describe SGED formulations with different density and parameter definitions.
• The document distinguishes Theodossiou’s SGED specifications from the Fernández–Steel distribution used by rugarch.
• Alternative definitions may give similar results in a particular case without being generally equivalent.
• A GARCH implementation should match the intended distribution’s precise parameterization.
• The post references sources but leaves the choice of density unresolved.

Tags

# Skewed Generalized Error Distribution's (SGED) pdf


# Skewed Generalized Error Distribution's (SGED) pdf












I want to use the SGED distribution of Theodossiou for GARCH estimation, however, I am struggling to understand which is the correct pdf function of the distribution. Let me just say that the `rugarch` package cannot help me, as it estimates the GARCH with the SGED of Fernandez and Steel.

The distribution is introduced in the following paper in equation (10): https://www.researchgate.net/profile/Panayiotis_Theodossiou/publication/228262554_Skewed_Generalized_Error_Distribution_of_Financial_Assets_and_Option_Pricing/links/546a5d070cf2397f783017af.pdf

This is the latest paper by Theodossiou and also has the pdf of the SGED distribution described as in equation (49): http://www.mfsociety.org/modules/modDashboard/uploadFiles/journals/MJ~0~p1a4fjq38m1k2p45t6481fob7rp4.pdf

This is a bit different definition of the SGED distribution, but the results are almost identical for the case I am considering.

There is one more specification in this paper on page 16: https://www.researchgate.net/profile/J_Mcdonald4/publication/227347472_Robust_estimation_with_flexible_parametric_distributions_estimation_of_utility_stock_betas/links/00b495225f8fe3511c000000.pdf The A parameter is defined differently than in the first paper and the density looks differently.

Could you please advise me which pdf should I use as I am confused by such a number of definitions for the same thing and Theodossiou is the author or co-author of all the above-mentioned papers. If you know where I can find the definition of SGED by Fernandez and Steel I would also be grateful, as I am unable to find the pdf of the function. The help files for `rugarch` are not helpful on this matter.

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