I am fitting a GARCH(1,1) model to the data and want to look at the innovation distribution.
#generate the data
set.seed(1)
N = 5000
omega = 0.5
alpha = 0.08
beta = 0.91
X1 = rep(0,N)
X2 = rep(0,N)
sig1 = rep(0,N)
sig2 = rep(0,N)
for(i in 2:N){
sig1[i] = sqrt(omega + alpha * X1[i-1] + beta * sig1[i-1]^2)
X1[i] = sig1[i] * rnorm(1)
sig2[i] = sqrt(omega + alpha * X2[i-1] + beta * sig2[i-1]^2)
X2[i] = sig2[i] * rt(1, df = 8)
}
X1 = X1[-c(1:1000)]
I first generate the data and fit it to GARCH(1,1) model with t innovation.
spec = ugarchspec(mean.model=list(armaOrder=c(0,0),include.mean=F),
distribution.model="std") # GARCH(1,1) model
myfit = ugarchfit(spec, X1)
Suppose the fitted model is called myfit then I can get the error terms by z1 = myfit@fit$z which is calculated as
$(z_1,...z_n) = (\frac{X_1}{\hat{\sigma}_1},...,\frac{X_n}{\hat{\sigma}_n})$
However, if I only extract the parameters $\omega,\alpha$, and $\beta$ estimated by the rugarch package and calculate the error terms manually as
$\sigma_t^2 = \omega + \alpha X_{t-1}^2 + \beta \sigma_{t-1}^2$
$z_t = X_t / \sigma_t$
The code is:
omega1 = myfit@fit$coef[1]
alpha1 = myfit@fit$coef[2]
beta1 = myfit@fit$coef[3]
z2 = rep(0,(N-1000))
sighat1 = rep(0,(N-1000))
sighat1[1] = 1
z2[1] = X1[1]/sighat1[1]
for(i in 2:(N - 1000)){
sighat1[i] = sqrt(omega1 + alpha1 * X1[i-1]^2 + beta1 * sighat1[i-1]^2)
z2[i] = X1[i]/sighat1[i]
}
I got very different results between these two approaches by comparing the Q-Q plot of z1 and z2.
qqnorm(z1)
qqline(z1)
qqnorm(z2)
qqline(z2)
z1 seems to be normally distributed following the data generating process, while z2 has a heavy tail following the model specification. I was wondering why they are so different? I arbitrarily chose 1 for for first two terms of $\hat{\sigma}$, so does the difference come from the initial values?
And more generally, how does rugarch package fit the GARCH model and choose initial values? My understanding is that we need to find parameters using either QMLE or MLE and then find error terms iteratively using my second approach. But I am not sure how is the initial value chosen.
Thanks!
