Showing that a GARCH(1, 1) model is an ARMA(1, 1) process for squared errors Consider a GARCH(1, 1) model: $$ \sigma_t^2 = \alpha_0 + \alpha_1 u_{t-1}^2 + \beta \sigma_{t-1}^2 $$
Where $ \sigma_t $ is the conditional variance at time $ t $, $ u_{t}^2 $ is the error term in time $t$. In "In Introductory Econometrics for Finance" by Brooks (pg. 418/674), section 8.8, it is shown that this can be represented as an ARMA(1, 1) model for the squared errors. That is, the above can be written as:
$$ u_t^2 = \alpha_0 + ( \alpha_1 + \beta) u_{t-1}^2 - \beta \epsilon_{t-1} + \epsilon_t $$
To do so, the author starts with the statement:
$$ \epsilon_t = u_t^2 - \sigma_t^2 $$
What does this statement mean and how is it valid?
 A: The derivation is as follows: You start with the conditional variance equation
\begin{equation}
\sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2
\end{equation}
Now add $w_t=\epsilon_t^2-\sigma_t^2$ on both sides. You obtain:
\begin{align}
&\sigma_t^2+w_t=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+w_t \\
\leftrightarrow &\sigma_t^2+\epsilon_t^2-\sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+w_t\\
\leftrightarrow &\epsilon_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+ w_t \\
\end{align}
Notice that $w_{t-1}=\epsilon_{t-1}^2-\sigma_{t-1}^2$ and therefore $\sigma_{t-1}^2=\epsilon_{t-1}^2-w_{t-1}$. You obtain:
\begin{align}
\epsilon_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1(\epsilon_{t-1}^2-w_{t-1})+ w_t \\
\leftrightarrow \epsilon_t^2=\alpha_0+(\alpha_1+\beta_1)\epsilon_{t-1}^2 -\beta_1w_{t-1} +w_t
\end{align}
This is an ARMA(1,1) for the squared shocks, if $w_t$ is a white noise process. Check that $E(w_t)=0$ and $Cov(w_t,w_{t-h})=0 , h\leq 1$. If $E(\epsilon_t^4)<\infty$ than $V(w_t)<\infty$ and then $w_t$ is a weak white noise. One thing that gets obvious when looking at this equation is that the $\epsilon_t$ are uncorrelated (you can check that by yourself) but they are not independent because the squared shocks  follow an ARMA(1,1) process. Furthermore it is now easy to derive the ACF of $\epsilon_t^2$. You can see that the ACF is always positive and converges at rate $\alpha_1+\beta_1$ to zero.
