Suppose I want to model some returns by
$$
\begin{aligned}
r_t &= \mu_t + a_t \\
a_t &=\sigma_t \epsilon_t \\
\sigma_t^2 &= \alpha_0 + \alpha_1 a_{t-1}^2 + \dots + \alpha_m a_{t-m}^2
\end{aligned}
$$
where $\mu_t$ denotes a stationary, low-order ARMA process and the error terms $a_t$ follows an ARCH process.
The literature says that the standardized residuals of the ARCH model have to be white noise for the model to be well specified.
Can someone please explain me
- how the residuals of the ARCH model are precisely defined within the above setup?; and
- why I have to check the standardized residuals instead of the normal ones?
Best Answer
The standardized residuals are $\hat\epsilon_t$, i.e. the fitted values of $\epsilon_t$.