In a GARCH model like the following $$ \begin{aligned} y_t &= \sigma_tz_t,\\ \sigma_t^2 &=\omega(1-\alpha-\beta)+\alpha y_{t-1}^2+\beta \sigma_{t-1}^2 \end{aligned} $$ where $z_t$ is assumed to be iidN(0,1), we say that conditional on past information $y_t$ has the Gaussian density $$f(y_t|y_{t-1},\sigma_{t-1}^2)=\frac{1}{\sqrt{2\pi\sigma^2_t}}\exp\left(\frac{1}{2\sigma^2_t}y_t^2\right)$$ Am I correct in making the following conclusion?

When conditioning on the past, we know what the value of $\sigma^2_t$ is.
Consequently, we can state that $y_t$ is conditionally normally distributed as $N(0,\omega(1-\alpha-\beta)+\alpha y_{t-1}^2+\beta \sigma_{t-1}^2)$.


bumped to the homepage by Community yesterday

This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.

  • $\begingroup$ to make this more clear I recommend stating what is being conditioned in the "conditionally normally distributed" statement. For example, conditioning on $y_{t-1}, \sigma^2_{t-1}$ makes your conclusion correct. $\endgroup$ – Lucas Roberts Mar 24 '18 at 14:13

In the density for $y_t$ you should substitute $\sigma_t$ with its expression above.
Once you have done that, it becomes more transparent that your conclusion holds.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.