Fitting ARCH and GARCH models using MLE

[yiimaisui]

Hi,

I am still unclear on how the pdf is derived (on fitting ARCH and GARCH models using MLE) on page 18 of chapter 13.

For an ARCH model, how is:

g[x_t|X_(t-1)]= 1/sigma_t * f(x_t / sigma_t)?

I got the impression the pdf is derived, using the fact that:

X_t = sigma_t * Z_t --> Z_t = X_t / sigma_t ; hence f(X_t / sigma_t) is the normal pdf evaluated at X_t / sigma_t . Since it is an ARCH(1) model, my understanding is that f(X_t / sigma_t) is g(x_t / x_(t-1)) itself (since x_(t-1) is used for the calculation of variance).

But where does the 1/sigma_t come from? Is this the pdf g(x_(t-1))?

Thanks in advance for the help.

 

[DevonMatthews]

Hi,

I am still unclear on how the pdf is derived (on fitting ARCH and GARCH models using MLE) on page 18 of chapter 13.

For an ARCH model, how is:

g[x_t|X_(t-1)]= 1/sigma_t * f(x_t / sigma_t)?

I got the impression the pdf is derived, using the fact that:

X_t = sigma_t * Z_t --> Z_t = X_t / sigma_t ; hence f(X_t / sigma_t) is the normal pdf evaluated at X_t / sigma_t . Since it is an ARCH(1) model, my understanding is that f(X_t / sigma_t) is g(x_t / x_(t-1)) itself (since x_(t-1) is used for the calculation of variance).

But where does the 1/sigma_t come from? Is this the pdf g(x_(t-1))?

Thanks in advance for the help.

It's not so clear is it? There is a question on this in the Q&A Bank which should clear it up. Have a look at part 3 Q&A.