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!
In rugarch, is Johnson's SU distribution properly scaled to mean=0, variance=1?
2 messages · Richard Hardy, Alexios Ghalanos
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]]
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