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.
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+