# Johnson SU distribution parameters values from R/rugarch

I have fit a GARCH model where I set the distribution to the Johnson's SU. I don't fully understand the distribution parameters returned by the model.

To begin with, from Wikipedia, Johnson's SU is defined as:

$$\begin{equation} z = \gamma + \delta \sinh^{-1}\left( \frac{x - \xi}{\lambda} \right) \end{equation}$$

where $$x \sim N(0,1)$$.

The parameters of the distribution are $$\gamma, \xi, \delta, \lambda$$. My model outputs the following where the skew and shape parameters are the distribution parameters and the rest ($$\alpha, \beta, \omega$$ are GARCH parameters).

            Estimate   Std. Error       t value     Pr(>|t|)
mu     -1.747010e-17 0.0009983114 -1.749965e-14 1.000000e+00
alpha1  5.015195e-02 0.0090862109  5.519567e+00 3.398366e-08
beta1   8.999928e-01 0.0160119111  5.620771e+01 0.000000e+00
skew    3.265134e-02 0.0503597628  6.483618e-01 5.167510e-01
shape   1.008973e+00 0.0292731396  3.446753e+01 0.000000e+00
omega   8.894715e-05


First, the model suggests that the skew is 0 (p-value is 0.5). If skew here means 3rd moment then I'm even more confused because I know that the data has a negative skew.

Second, I am struggling to understand how the skew and shape parameters fit with the parameters of the distribution. As I understand it, under the hood, rugarch uses GAMLSS which defines distributions in terms of 4 paremeters: µ, σ, ν, τ.

So overall, I don't know what skew and shape mean here and how to interpret them.

• Here is a document (PDF) from the authors of GAMLSS in which they detail the arametrization of the distributions. Maybe you'll find your answers in there. Jul 31, 2021 at 14:30
• Wikipedia seems to have a typographical error: it is $z$ that is supposed to have a standard Normal distribution. The description of random variable generation makes this clear. That would make $\lambda$ a scale parameter and $\xi$ a location parameter for $x.$ The Wikipedia example, which varies $\gamma$ while fixing the other parameters, clearly shows a regular change in skewness, whence we might infer $\gamma$ is a "skew" parameter and $\delta$ a "shape" parameter.
– whuber
Jul 31, 2021 at 15:26