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Suppose we have a GARCH (1,1) equation as follows:

$$\sigma_t^2 = \alpha_0 + \alpha\epsilon_{t-1}^2 + \beta\sigma_{t-1}^2$$

  1. How are the lagged variance terms calculated?
  2. Suppose I am also implementing an ARMA-GARCH process sequentially, so I am using the residuals from ARMA to optimize GARCH. (I believe) Then it is assumed that the $\epsilon_i$ in both ARMA and GARCH follow the same distribution. If this is the case, since $\epsilon_i$ and $\sigma_i$ are known, why can't this parameter identification be down through OLS?
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In general the GARCH(1,1) model is given by: \begin{align} r_t&=\mu_t+\epsilon_t \\ \epsilon_t&=\sigma_tz_t \quad z_t \overset{iid}{\sim} D(0,1) \\ \sigma_t^2&=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 \end{align} Where $\mu_t=E(r_t \vert {\cal F_{t-1}})$ is the conditional mean of $r_t$. The conditional variance of $r_t$ is given by $\sigma_t^2$, so the lagged variance is included through the second term in the variance equation $\beta_1\sigma_{t-1}^2$. When you specify an ARMA-GARCH model, you model $\mu_t$ via an ARMA model. The residuals for fitting the GARCH model are then given by $r_t-\hat{\mu}_t$, where $\hat{\mu}_t$ are the fitted values from the ARMA model. It is not possible to estimate the GARCH parameters, i.e. $\alpha_0$, $\alpha_1$ and $\beta_1$, via OLS due to the iterative character of the variance equation. Therefore the parameters are usually estimated via ML.

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  • $\begingroup$ Thanks a lot! I have two more questions if you don't mind. First, I am still unclear how the conditional variance associated with $\beta_1$ is computed in practice, could you elaborate on that? Secondly, after the ARMA GARCH model has been fitted, how do we predict values? As in, how do we choose a value for the $\epsilon_i$ if they are random? $\endgroup$
    – CBBAM
    Commented Apr 27, 2021 at 8:01
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    $\begingroup$ Sure, assume that you estimated a GARCH(1,1) model. Than you need an inital value for $\sigma_0^2$. There are different possiblities but one commonly used is: $\sigma_0^2=\frac{1}{T}\sum_{t=1}^Tr_t^2$ (because this is a consistent estimator). Now you can calculate $\sigma_1^2=\hat{\alpha}_0+\hat{\alpha}_1\epsilon_0^2+\hat{\beta}_1\sigma_0^2$ and so on. In most cases you have a very large $T$ and the effect of $\sigma_0^2$ is negligible. $\endgroup$
    – Count
    Commented Apr 27, 2021 at 10:29
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    $\begingroup$ Related to your second question: You observe the returns $r_t$ and you fitted the values for the conditional mean quation $\hat{\mu}_t$. You can use these values to calculate the $\hat{\epsilon}_t$ as $\hat{\epsilon}_t=r_t-\hat{\mu}_t$. When it comes to prediction you have to differ between predictions for $r_t$ or the conditional variance $\sigma_t^2$. In most cases you are interested in predicions for $\sigma_t^2$. The MSE optimal $l$-step ahead prediction is given by: $E(\sigma_{T+l}^2\vert {\cal F_{T}})$. $\endgroup$
    – Count
    Commented Apr 27, 2021 at 10:38
  • $\begingroup$ @Jonas_Dim, it is worth noting that the estimator of $\sigma_0^2$ you provide above is consistent under the assumption that $\mu_t\equiv 0$ but not necessarily otherwise. Could you also indicate briefly how GARCH can be estimated by OLS? E.g. provide a formula for the estimator like you would for a linear model where $\beta=(X^\top X)^{-1}X^\top y$? $\endgroup$
    – Richard Hardy
    Commented Apr 27, 2021 at 13:39
  • $\begingroup$ @RichardHardy, I corrected my answer, I mean it not possible to estimate a GARCH model via OLS, my thoughts were with ARCH models at this moment. Sorry for the confusion. $\endgroup$
    – Count
    Commented Apr 27, 2021 at 15:30

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