## garch() uncertainty

As part of an on-going paper with Kerrie Mengersen and Pierre Pudlo, we are using a GARCH(1,1) model as a target. Thus, the model is of the form

$y_t=\sigma_t \epsilon_t \qquad \sigma^2_t = \alpha_0 + \alpha_1 y_{t-1}^2 + \beta_1 \sigma_{t-1}^2$

which is a somehow puzzling object: the latent (variance) part is deterministic and can be reconstructed exactly given the series and the parameters. However, estimation is not such an easy task and using the garch() function in the tseries package leads to puzzling results! Indeed, simulating data shows some high variability of the procedure against starting values:

genedata=function(para,nobs){

pata=epst=sigt=rnorm(nobs)
sigt[1]=sqrt(para[1])
pata[1]=epst[1]*sigt[1]
for (t in 2:nobs){
sigt[t]=sqrt(para[1]+para[2]*pata[t-1]^2+para[3]*sigt[t-1]^2)
pata[t]=epst[t]*sigt[t]
}
list(pata=pata,sigt=sigt,epst=epst)
}
> x = genedata(c(1, 0.3, 0.2),1000)$pata > garch(x,trace=FALSE) Call: garch(x = x, trace = FALSE) Coefficient(s): a0 a1 b1 4.362e+00 1.976e-01 6.805e-14 > garch(x,trace=FALSE,start=c(1,.3,.2)) Call: garch(x = x, trace = FALSE, start = c(1, 0.3, 0.2)) Coefficient(s): a0 a1 b1 0.8025 0.2592 0.3255 > simgarch=genedata(c(1, 0.2, 0.7),1000) Call: garch(x = simgarch$pat, trace = FALSE)

Coefficient(s):
a0         a1         b1
8.044e+00  1.826e-01  4.051e-14
> garch(simgarch$pat,trace=FALSE,star=c(1, 0.2, 0.7)) Call: garch(x = simgarch$pat, trace = FALSE, star = c(1, 0.2, 0.7))

Coefficient(s):
a0      a1      b1
1.1814  0.2079  0.6590


The above code clearly shows the huge impact of the starting value on the final estimate….

### 3 Responses to “garch() uncertainty”

1. I’ve never used the package tseries, but have you considered using rugarch or fGarch, they seem way more robust for this kind of models

• Thanks. I just tried to install both of them but they both failed for failed dependencies…

2. William Volterman Says:

It seems to be a convergence issue, I ran the code and looked at the likelihood. Pity that the function doesn’t return any information on convergence like most optimizers.

> genedata=function(para,nobs){
+
+ pata=epst=sigt=rnorm(nobs)
+ sigt[1]=sqrt(para[1])
+ pata[1]=epst[1]*sigt[1]
+ for (t in 2:nobs){
+ sigt[t]=sqrt(para[1]+para[2]*pata[t-1]^2+para[3]*sigt[t-1]^2)
+ pata[t]=epst[t]*sigt[t]
+ }
+ list(pata=pata,sigt=sigt,epst=epst)
+ }
>
> set.seed(1000)
> x y y$coef;y$n.likeli
a0 a1 b1
1.795077e+00 1.681971e-01 1.422666e-14
[1] 797.3634
>
> y y$coef;y$n.likeli
a0 a1 b1
1.10351778 0.33430225 0.08950214
[1] 780.3891
>
> y y$coef;y$n.likeli
a0 a1 b1
1.10349913 0.33430273 0.08950677
[1] 780.3891
>
> y y$coef;y$n.likeli
a0 a1 b1
2.492743e+00 1.026425e-01 1.242472e-15
[1] 838.3297