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

  1. how the residuals of the ARCH model are precisely defined within the above setup?; and
  2. why I have to check the standardized residuals instead of the normal ones?

1 Answer 1

  1. The (regular) residuals are $\hat a_t$, i.e. the fitted values of $a_t$.
    The standardized residuals are $\hat\epsilon_t$, i.e. the fitted values of $\epsilon_t$.
  2. The model assumes that the standardized errors have a certain distribution (e.g. Normal, Student-$t$ or the like) with zero mean and unit variance. The likelihood function for the model is built using this assumption. If the assumption does not hold, the likelihood function is misspecified and the maximum likelihood estimator (MLE) might not have the desirable properties (although it still might work alright as quasi MLE in some cases). Therefore, you check the empirical counterpart of the standardized (rather than regular) errors which is the standardized residuals.
  • $\begingroup$ So if I want to check the specification of the ARMA model, I check if $\hat{\alpha_t}$ is white noise, and if I want to check the specification of the ARCH model, I check if $\hat{\epsilon_t}$ is white noise, right? $\endgroup$
    – Joe
    Aug 3, 2016 at 8:47
  • $\begingroup$ @Joe, Yes, you are right. But if you have an ARMA-ARCH model, then checking whether $\hat\alpha_t$ is white noise is redundant (because $\hat\alpha_t$ is supposed to follow an ARCH process) and you only check whether $\hat\epsilon_t$ is white noise. $\endgroup$ Aug 3, 2016 at 9:09
  • $\begingroup$ How is $\hat a_t$ a "residual"? I think it can be e.g. return or something like that. $\endgroup$
    – Aqqqq
    May 16, 2021 at 17:10
  • 2
    $\begingroup$ @Aqqqq, the definition is $a_t=r_t-\mu_t$ where $r_t$ is the return and $\mu_t$ is its conditional mean. Hence, $a_t$ is the error or the residual . $\hat{a}_t$ is the fitted value of $a_t$. If $\mu_t\equiv0$, then $a_t\equiv r_t$ and so it coincides with the return itself. But this is just a special case. $\endgroup$ May 16, 2021 at 17:14

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