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I have just learned GARCH model. One condition distribution of it is "sstd". One question of my coursework is to justify if the conditional distribution is skewed. I have seen another example sheet and it says the skew parameter must equal to one if the distribution is symmetric. I don't know why it is equal to 1 and I really don't what is a skewed-t distribution here.

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    $\begingroup$ This could be about a noncentral t distribution. For example when you have two normal distributions with the same variance the t statistic will have a central t distribution under the null hypothesis that the means are equal. The central t is symmetric. When the means differ the t statistic has a noncentral t distribution which is not symmetric. Skewness measures the degree of asymmetry. But when the distribution is symmetric the skewness is 0 (for this example). I don't know how this would come up when dealing with GARCH models. $\endgroup$ – Michael R. Chernick Apr 27 '17 at 19:25
  • $\begingroup$ @MichaelChernick, in GARCH models the distribution would be centered at zero but potentially skewed. Monier, see section 2.3.4 of the R package "rugarch" vignette and also check out help files of relevant functions (perhaps ugarchspec) in that package. $\endgroup$ – Richard Hardy Apr 27 '17 at 19:33
  • $\begingroup$ @RichardHardy If the distribution is t and centered at 0 how can it be skewed? Just after equation 62 on the page describing the t distribution it says that the skewness is 0. $\endgroup$ – Michael R. Chernick Apr 27 '17 at 20:13
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    $\begingroup$ @MichaelChernick, I should look at the details before commenting further, but in general I do not see why skewness should be anyhow related to noncentrality. $\endgroup$ – Richard Hardy Apr 28 '17 at 5:25
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    $\begingroup$ You can find answers among stats.stackexchange.com/search?q=skew+t and on wikipedia en.wikipedia.org/wiki/Skewed_generalized_t_distribution $\endgroup$ – kjetil b halvorsen Feb 9 '18 at 9:37
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I have found the definition of the skewed distribution. Here the "skewed parameter" is not skewness.

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    $\begingroup$ Please cite the source from which this was taken. $\endgroup$ – gung - Reinstate Monica Jan 8 '18 at 17:46
  • $\begingroup$ This is another way of introducing skewness into a distribution than the one I referenced in a comment after the Question. $\endgroup$ – kjetil b halvorsen May 27 '18 at 21:36
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It seems like skew-t is used as name for different distributions. The mostly used one (I believe) is in https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution, while the answer by @Monier is another one.

A Azallini introduced a general way of "skewing" a distribution, start with some density function $f$ (such as standard normal or $t_\nu$) symmetric about zero and transform it by $$2 f(x) F(\alpha x)$$ (where $F$ is the cdf corresponding to $f$), and $\alpha$ is a new transform parameter modeling the skewness. Observe that when $\alpha=0$ we get back $f$, since by symmetry $F(0)=1/2$. The skew normal case is https://en.wikipedia.org/wiki/Skew_normal_distribution. Many other symmetric distribution families can be skewed the same way, and location and scale parameters can be added.

For the skew normal case there is many posts on this site, see https://stats.stackexchange.com/search?q=+skew+distribution+normal+answers%3A1+

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