# 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.

• Did you see this recent sgt R package : cran.r-project.org/web/packages/sgt/vignettes/sgt.pdf and cran.r-project.org/web/packages/sgt/sgt.pdf . You may find it usefull – Malick Dec 26 '15 at 18:57
• @Malick I haven't seen this package, thank you! Although the pdf is differently specified I compared the results for the same parameters with the one from Theodossiou (2000) and the results are identical. Now I will try to incorporate this pdf into my GARCH calculations, as the previous one did not give satisfactory results. Could you please tell me, whether I should scale the densite of this distribution (as given on page 8 of the vignette) when calculating the likelihood function? – Masher Dec 26 '15 at 22:06
• see my answer here quant.stackexchange.com/questions/22490/… – Malick Dec 27 '15 at 12:48