Are the epsilons independent from each other?

[St41n]

[tex]\varepsilon _t = z_t \sigma _t [/tex]

[tex]
z_t \sim IID\,N\left( {0,1} \right)
[/tex]

[tex]
\sigma _t^2 = f\left( {\sigma _{t - 1} ,z_{t - 1} } \right)
[/tex]Are the epsilons independent from each other? Why?

 

[winterfors]

[tex]\varepsilon _t = z_t \sigma _t [/tex]

[tex]
z_t \sim IID\,N\left( {0,1} \right)
[/tex]

[tex]
\sigma _t^2 = f\left( {\sigma _{t - 1} ,z_{t - 1} } \right)
[/tex]


Are the epsilons independent from each other? Why?

I can't see that they could be independent in the general case of [itex] f (\sigma ,z)[/itex] being any real-valued function, since [itex] p(\varepsilon_{t+1} | \sigma_t ,z_t)[/itex] is dependent on [itex]\sigma_t[/itex] and [itex]z_t[/itex], just like [itex] p(\varepsilon_{t} | \sigma_t ,z_t) = \delta(\varepsilon_{t} - \sigma_t z_t)[/itex] ([itex] p[/itex] representing probability densities and [itex] \delta[/itex] the Dirac delta distribution). This implies that

[tex]
p(\varepsilon _t,\varepsilon _{t+1}) = \int\int{
p(\varepsilon_{t} | \sigma_t ,z_t)
p(\varepsilon_{t+1} | \sigma_t ,z_t)
p(\sigma_t)p(z_t)
d\sigma_t dz_t}
[/tex]

cannot, in general, be written as a product [itex]p(\varepsilon _t,\varepsilon _{t+1}) = p(\varepsilon_{t})p(\varepsilon_{t+1})[/itex] as is the case for independent variables.

-Emanuel

 

[St41n]

Ok, this makes sense
Thank you very much