ARJI-GARCH-M model

[Michelle]

Dear Tom,

I would like to modify the Chan and Maheau (2002) ARJI-GARCH to an ARJI-GARCH-M model.As for the mean equation, I decompose the variance of the return into the diffusive part and the jump part and add these two parts into the mean equation sepatately, expressed as
rt=a1+ a2*rt-1+ a3*ht+ a4* (θ2+δ2) λt.
the code for this part is as follows:

dec real mu a1 a2
nonlin(parmset=meanparms) mu
frml uf = r-mu-a1*h-a2*lambda_t*(theta_t^2+deltasq_t)
compute a1=a2=0.0
nlsystem(parmset=meanparms,frml=uf) 3 1212
nonlin(parmset=meanparms) mu a1 a2 a3

The estimation result seems wrong. The standard error of all parameters is very small. I can not find what is wrong with this code.
Please, I implore you to help me in this regard,

Michelle

[TomDoan]

First, I'm not sure I understand how you generalize that to include an "M" effect. The jump GARCH model has a standard observable base variance and then the unobservable jump process where the number of jumps is unknown. For evaluating the likelihood, the latter requires summing across the number of jumps so there isn't a specific variance number which could affect the mean.

Second, it looks like you're taking the variance as given which wouldn't really be the correct procedure even if it worked.

[Michelle]

First, I'm not sure I understand how you generalize that to include an "M" effect. The jump GARCH model has a standard observable base variance and then the unobservable jump process where the number of jumps is unknown. For evaluating the likelihood, the latter requires summing across the number of jumps so there isn't a specific variance number which could affect the mean.

Second, it looks like you're taking the variance as given which wouldn't really be the correct procedure even if it worked.

Sorry to confuse you. Here I briefly describe my model. Most of the part is the same as Chan and Maheau (2002) except that I add the observable variance and the unobservable jump variance into the mean equation as the "M" part.

Rt=a1+a2* Rt-1+ a3* ht+ a4* (θ2+δ2)*λt+ε1,t+ε2,t
where ε1,t is the normal innovation and ε2,t is the jump innovation.
1. for the normal innovation part, ht=var(ε1,t| Rt-1) and ht has the expression as:
ht=ω+exp(b1+b2*Nt-1+I(εt-1)*(b3+b4* Nt-1)) *εt-12+β* ht-1, where Nt-1=E[nt|Rt]
2.As to the jump part, the jump size Yt,k is normal distributed with mean θ and variance δ2;
the number of jumps nt is a Poisson distribution with parameter λt, where
λt=γ0+γ1λt-1+γ2ζt

for the mean equation, i first try :

Code: Select all

dec series h_r
dec series j_r
set h_r = h
set j_r = lambda_t*(theta_t^2+deltasq_t)
nonlin(parmset=meanparms) mu a1 a2
frml uf = r-mu-a1*h_r-a2*j_r

but it doesn't work. I have found some code of Multivariate-GARCH-M model and use it as a reference to rewrite my mean equation part as prensented below:


Still, it does not seem the right because the standard error of all parameters is very small.

[TomDoan]

Sorry to confuse you. Here I briefly describe my model. Most of the part is the same as Chan and Maheau (2002) except that I add the observable variance and the unobservable jump variance into the mean equation as the "M" part.

I'm not sure I understand why the jump intensity would enter into the mean---if it's an "M" model, wouldn't the actual variance (which depends upon the number of jumps) be involved. Have you asked Chan and/or Maheu about your idea? It's possible that either they've thought about this, or at least have run across someone else who has thought about it?

Have you fit the model without the added "M" effect?

[Michelle]

Dear Tom,
Thanks for your reply. As for why the jump intensity enters into the mean. It is because that the jump part contains a jump arrival rate which is Poission distributed with parameter λt and a jump size Yt,k which is normal distributed with mean θ and variance δ^2. That is to say the variance of jump innovation var(ε1,t| Rt-1)=(θ^2+δ^2)*λt.
I have already fit the model without "M" effect and can get the estimation which seems correct. I can't figure out what is wrong with my code. I have tried to run the code without "M" parts first to get the variance first and then include the "M" part, but it still didn't work.

[Michelle]

As for a normal GARCH-M model, only ht enters the mean equation. But to a ARJI-GARCH model, the variance of the return is var(Rt| Rt-1)=ht+(θ^2+δ^2)*λt. The first part is the variance of observable base variance and the second part is the variance of the unobservable jump process. What I want to do is to add these two parts as "M" part into the mean equation to study the risk-return trade-off between them.

[TomDoan]

But the whole point of the jump model is that the distribution is a mixture of Normals---each value of k for the Poisson gives a different Normal with a different mean and different variance. Shifting the mean with the unconditional variance doesn't make sense. Why wouldn't you further shift the mean in the mixture distributions based upon the variance conditional on k?