bivariate GARCH-X

[fadimohamed]

Hi tomm,
i am using a bivariate GARCH-X model with two exogenous variables, the exogenous variables have unit roots in levels, can i use them in levels or i should take first differences?

thanks

Fadi

[TomDoan]

Hi tomm,
i am using a bivariate GARCH-X model with two exogenous variables, the exogenous variables have unit roots in levels, can i use them in levels or i should take first differences?

thanks

Fadi

Are the non-stationary variables in the mean equations, or in the variance equations?

[fadimohamed]

Dear Tomm,

The two non-stationary exogenous variables are in the variance equations

[TomDoan]

Why would you ask whether they should be in differences or in levels? The two have completely different effects and they can't both be right.

From a theoretical standpoint, there is a issue with a variance shift dummy in a non-exponential GARCH model being non-stationary since it should at some point force the variance to be negative. In a finite sample, that may not prove to a practical problem.

[hasanov]

Dear Tom,

I have a similar question:

As observed in the literature, many papers seem to disregard the strict positivity requirement in non-exponential GARCH (i.e. standard GARCH) models. These papers estimated, for example, the standard GARCH(1,1) model and incorporated regime dummies in variance equations. And, they consider the models are valid (i.e. the variances are strictly positive) although the coefficients on dummies negatively signed. Is there any study which supports this notion? Does the strict positivity not require the coefficients on dummies to be positive together with omega, alpha, and beta.

Thanks a lot,
Akram Hasanov

[TomDoan]

Dummies are a very different situation. The problem I was describing with regard to the non-stationary X variable is that random walks realizations will take long and large deviations in each signed direction at some point. No matter which sign is on the X variable in the variance, there will be (eventually) a run which forces the variance to be negative.

With dummies, however, the sign depends upon how the dummy is put in. For instance, if we have just one dummy D, then we can do either D or 1-D as the representation. If c+dD is the representation using D, then (c+d)-d(1-D) is the other. The two have different signs on the dummies but both are perfectly reasonable models. What's important is that c and c+d (which are the intercepts in the two regimes) are positive.

[fadimohamed]

Dear tomm,

i am asking this because i have a doubt if using exogenous variables with unit roots in the variance equations is empirically problematic,
so we need to take first difference to remove these unit roots?
what i understand from your reply is that there is no reason why not to use the exogenous variables in the variance equations in levels,
am i right?

when you say "The two have completely different effects and they can't both be right.", would you explain more what you mean?

Thank you
fadi

[TomDoan]

Dear tomm,

i am asking this because i have a doubt if using exogenous variables with unit roots in the variance equations is empirically problematic,
so we need to take first difference to remove these unit roots?
what i understand from your reply is that there is no reason why not to use the exogenous variables in the variance equations in levels,
am i right?

when you say "The two have completely different effects and they can't both be right.", would you explain more what you mean?

Thank you
fadi

If you use the level, you're saying that the variance shifts with the level of the variable, and if you put in the first difference, you're saying that the variance shifts with the change. Those aren't even close to being the same things.