[R-SIG-Finance] Restriction test (H0: alpha1+beta1 = 1, H1:alpha1 + beta1 ≠ 1) on GARCH model in R not working

[alexios galanos]

You posted a lot of questions/follow up questions on stack exchange, but 
you have not taken the time to summarize what these are in this forum or 
provide data with which we can replicate the problem. The vignette of 
the package already addresses most of your questions as well as provide 
the suggested forum for posting such questions (here).

I will try to provide a brief summary of your questions:

1. Your restricted model's log likelihood is higher than the 
unrestricted model.
2. You are having trouble understanding how the log likelihood is 
calculated.
3. You do not understand why the beta coefficient in the restricted 
iGARCH model is printed
out and has NA in the standard errors.

As I do not have your dataset, the only comment I can make is that the 
small differences arise from the optimization not reaching an optimal. 
This is already discussed in the vignette and in many previous posts to 
this forum. There are many choices of solver, strategies to help out and 
tuning/tolerance parameters to use. For completeness however I will 
provide a similar example with the package's default dataset which shows 
that this in general works:

######################
library(rugarch)
data(sp500ret)
library(xts)
spx<-xts(as.numeric(sp500ret[,1]), as.Date(rownames(sp500ret)))
# restricted iGARCH model
spec1<-ugarchspec(mean.model=list(armaOrder=c(0,0), archm=TRUE, 
archpow=1), variance.model=list(model="iGARCH"))
# unrestricted sGARCH model
spec2<-ugarchspec(mean.model=list(armaOrder=c(0,0), archm=TRUE, 
archpow=1), variance.model=list(model="sGARCH"))
mod1<-ugarchfit(spec1, spx, solver="solnp")
mod2<-ugarchfit(spec2, spx, solver="solnp")

# restricted LL < unrestricted LL
likelihood(mod1)<likelihood(mod2)
 >TRUE

round(sum(dnorm(spx,mean = fitted(mod1), sd=sigma(mod1), log = 
TRUE)),5)==round(likelihood(mod1),5)
round(sum(dnorm(spx,mean = fitted(mod2), sd=sigma(mod2), log = 
TRUE)),5)==round(likelihood(mod2),5)

So in reply to the 3 questions:
1. Most likely not found the optimal in your dataset for the 
unrestricted model. Try changing the optimization tolerance.
2. Read Section 2.3 (Conditional Distributions) of the vignette and then 
try calculating:
#log(dnorm(residuals(mod1,TRUE))/sigma(mod1))#
as in equation 64 of the vignette to convince yourself that it is 
equivalent to what is shown above.
3. Beta is returned and printed out, but since it is not calculated, the 
standard error is NA. Perhaps it is redundant to print out what beta is 
(since it is imposed to be 1-\alpha), but it is and should not confuse you.

Regards,

Alexios

On 3/14/18 12:15 PM, Woo-Young Kang wrote:
> Dear R-Sig-Finance Committee,
>
>
> Hi, my name is Woo-Young Kang and I would like to ask a question regarding the "rugarch" package in R program if you don't mind.
>
>
> The title is "Restriction test (H0: alpha1+beta1 = 1, H1:alpha1 + beta1 $B!b(B 1) on GARCH model in R not working"
>
> which has been raised on the "Stack Exchange - Cross Validated" online questioning community which also deals with R programming issues.
>
>
> However, even the well-known site administrators were not able to solve this issue.
>
> In the end, they suggested me to ask you which I suppose to be my last resort.
>
>
> The question is quite long that I am sending the link that leads you to my R programming question as below:
>
> https://stats.stackexchange.com/questions/333256/restriction-test-h0-alpha1beta1-1-h1alpha1-beta1-%e2%89%a0-1-on-garch-model-in
>
>
> I would greatly appreciate if I can have your precious help on resolving this issue.
>
>
> Thank you so much in advance.
>
>
> Sincerely
>
> Woo-Young Kang
>
>
>
> 	[[alternative HTML version deleted]]
>
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