garch() uncertainty

May 16, 2012

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

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….

Filed under: R, Statistics, University life Tagged: GARCH, R, times series, tseries