# What's the skewed-t distribution?

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.

• 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. – Michael R. Chernick Apr 27 '17 at 19:25
• @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. – Richard Hardy Apr 27 '17 at 19:33
• @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. – Michael R. Chernick Apr 27 '17 at 20:13
• @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. – Richard Hardy Apr 28 '17 at 5:25
• You can find answers among stats.stackexchange.com/search?q=skew+t and on wikipedia en.wikipedia.org/wiki/Skewed_generalized_t_distribution – kjetil b halvorsen Feb 9 '18 at 9:37 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.