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How are errors terms calculated in GARCH model by rugarch package?

2 messages · Xie Yijun, Alexios Ghalanos

Original
Xie Yijun
# 
Hi,

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 at fit$z
###################

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 at fit$coef[1]
     alpha1 = myfit at fit$coef[2]
     beta1 = myfit at fit$coef[3]

     z2 = rep(0,(N-1000))
     sighat1 = rep(0,(N-1000))
     sighat1[2] = 1
     sighat1[1] = 1
     for(i in 2:(N - 1000)){
       sighat1[i] = sqrt(omega1 + alpha1 * X1[i-1] + 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!
Patrick
Alexios Ghalanos
# 
1. The initialization of the recursion (and options for doing so) is
described in the vignette...please READ IT.

The default it to use the mean of the squared residuals.

Many questions have already been answered over the years on this mailing
list regarding
the estimation and related issues so you might also like to search the
archives.

2. For some reason you are multiplying alpha1 by the value of X1 rather
than X1^2.
Have you pre-squared X1 somewhere and I missed it?

Here is the code to get exactly the same values:
#########################
z2 = rep(0,(N-1000))
sighat1 = rep(0,(N-1000))
sighat1[1] = sqrt(mean(X1^2))
z2[1] = X1[1]/sighat1[1]
for(i in 2:length(X1)){
  sighat1[i] = sqrt(omega1 + alpha1 * X1[i-1]^2 + beta1 * sighat1[i-1]^2)
  z2[i] = X1[i]/sighat1[i]
}
all.equal(z1, z2)
#########################

3. You may want to revisit your use of "rt". This is not the
standardized distribution
and hence you will not have a st.deviation of 1.

In rugarch, rdist("std",mu,sigma,shape,skew) is the standardized student
distribution
i.e. rt(1, df=nu)/(sqrt(nu/(nu-2)))


-Alexios
On 08/06/2016 17:58, Xie Yijun wrote: