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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.

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  • $\begingroup$ 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. $\endgroup$ Jul 31 at 14:30
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Jul 31 at 15:26
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I think rugarch uses the gamlss JSU(mu, sigma,nu,tau) parameterisation, in which mu is the mean, sigma is the standard deviation, nu is a skewness parameter and tau is a kurtosis parameter.

The parameterisation is given on page 391 of ‘Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R’ R. A. Rigby, M. D. Stasinopoulos, G. Z. Heller and F. De Bastiani. Chapman and Hall/CRC, Boca Raton, 2019

Paperback (2021) www.routledge.com//9780367278847

I think perhaps (mu,omega,skew,shape) = (mu, sigma, nu, tau).

The JSU parameterisation is different from the gamlss JSUo parameterisation. JSUo is the original Johnson (1947) parameterisation and is defined by your Wikipedia formula.

The reason you don't get skewness, could be that JSU is modelling the conditional distribution of the response, not the marginal distribution.

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