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.