# Understanding the conditioning in a GARCH process

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 $N(0,\omega(1-\alpha-\beta)+\alpha y_{t-1}^2+\beta \sigma_{t-1}^2)$.

• 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. – Lucas Roberts Mar 24 '18 at 14:13

In the density for $y_t$ you should substitute $\sigma_t$ with its expression above.