[R-SIG-Finance] Rugarch package using external regressors

[Luigi Maria Briglia]

I’m using the rugarch package and I’m having troubles understanding how the external.regressors work.

For example I would expect that fitting a time series with gjr-garch(1,1) should give the same results as fitting the same time series with the plain vanilla garch(1,1) augmented with S_(t-1)*eps_(t-1)^2 as an external regressor.

However I'm not getting the same results.

Specifically this is the code I'm running:

rm(list = ls()) # empty memory 

library(rugarch) 
library(xts) 
data(sp500ret)
spx <- xts(sp500ret, as.Date(rownames(sp500ret)))
t = length(spx)
# assuming mu = 0; r_t = eps_t
s = rep(0,t)
for(i in 1:t){
  if(spx[i]<0){s[i]=1}
}
# eps.neg represents the leverage effect regressor
eps.neg <- xts(spx*s, as.Date(rownames(sp500ret)))
colnames(eps.neg)<-"eps.neg" 
# lag eps.neg
eps.neg.lag = lag(eps.neg,1)
inputs<-na.omit(cbind(spx, eps.neg.lag, join="left")) 
# gjrgarch(1,1)
gjr.spec <- ugarchspec(variance.model = list(model='gjrGARCH', garchOrder=c(1,1),
                      external.regressors = NULL, variance.targeting = T),
                      mean.model = list(armaOrder=c(0,0)),fixed.pars=list(mu = 0))
gjr.fit <- ugarchfit(spec=gjr.spec, data=inputs[,1],
                     solver.control=list(trace = 1))
# garch(1,1) augmented with inputs[,2]
aug.s.spec <- ugarchspec(variance.model = list(model='sGARCH', garchOrder=c(1,1),
                        external.regressors = inputs[,2]^2, variance.targeting = T),
                        mean.model = list(armaOrder=c(0,0)),fixed.pars=list(mu = 0))
aug.s.fit <- ugarchfit(spec=aug.s.spec, data=inputs[,1],
                       solver.control=list(trace = 1))
#results
gjr.fit
aug.s.fit 
However these are the results:

GJR-GARCH(1,1)

Robust Standard Errors:
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000000          NA         NA       NA
alpha1  0.007933    0.000184     43.139        0
beta1   0.909048    0.000008 117316.512        0
gamma1  0.139258    0.004006     34.764        0
omega   0.000002          NA         NA       NA
Augmented-Garch(1,1)

Robust Standard Errors:
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000000          NA         NA       NA
alpha1  0.085378    0.002123 4.0223e+01  0.00000
beta1   0.904696    0.000001 1.0116e+06  0.00000
vxreg1  0.000000    0.000060 1.6700e-04  0.99987
omega   0.000001          NA         NA       NA
clearly the two fits are not equivalent. Is there something I’m missing about the external.regressors?



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