[R-SIG-Finance] In rugarch, is Johnson's SU distribution properly scaled to mean=0, variance=1?

[alexios galanos]

Simple to quickly test:

library(rugarch)

f1 <- function(x) x * ddist(distribution = "jsu", x, mu = 0, sigma = 1, 
skew = -10, shape = 0.5)
f2 <- function(x) x^2 * ddist(distribution = "jsu", x, mu = 0, sigma = 
1, skew = -10, shape = 0.5)

# Mean
integrate(f1, -Inf, Inf, rel.tol = 1e-12)$value
 >-1.145875e-16

# Variance
integrate(f2, -Inf, Inf, rel.tol = 1e-12)$value
 >1

Page 24 of the vignette 
(https://cran.r-project.org/web/packages/rugarch/vignettes/Introduction_to_the_rugarch_package.pdf)
clearly states that the re-parameterization  of this distribution is 
from the Rigby and Stasinopoulos (2005) as implemented in their
gamlss package (and checked prior to implementing).

Alexios


On 2/16/22 4:22 AM, Richard Hardy wrote:
> Dear all,
>
> I have a question about the `ugarchspec` and `ugarchfit` functions from the
> `rugarch` package in R. I wonder if the likelihood function of the
> univariate GARCH model specifies the standardized residuals to have zero
> mean and unit variance when the (standardized) residuals follow Johnson's
> SU distribution -- as in
> uspec=ugarchspec(mean.model=list(armaOrder=c(0,0)),
> variance.model=list(model="sGARCH"), distribution.model="jsu")
>
> My question is partly motivated by Simonato "GARCH processes with skewed
> and leptokurtic innovations: Revisiting the Johnson Su case" (2012). The
> paper shows that care needs to be taken to parameterize Johnson's SU
> distribution properly when using it in GARCH models. A counterexample is
> given where an earlier paper has made some mistakes in that regard,
> invalidating the model to an extent.
>
> The `rugarch` manual and vignette are fairly brief when it comes to
> Johnson's SU distribution, so I am struggling to find the answer there.
> There is no reference to Simonato (2012) there. The relevant source codes
> are available e.g. here
> https://github.com/cran/rugarch/tree/master/R
> and more specifically here
> https://github.com/cran/rugarch/blob/master/R/rugarch-distributions.R,
> but they are a bit challenging to follow.
>
> My simulations show the resulting empirical means and variances of
> standardized residuals to fluctuate a fair bit (e.g. empirical variance
> being anywhere between 0.95 and 1.05). I am not sure if this is due to
> estimation imprecision or some other reason. I observe this not only in the
> Johnson's SU case but also in other cases (e.g. normal).
>
> Thank you in advance for your help!
>
> 	[[alternative HTML version deleted]]
>
> _______________________________________________
> R-SIG-Finance using r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions should go.